Algebra I. Linear Equations. Slide 1 / 267 Slide 2 / 267. Slide 3 / 267. Slide 3 (Answer) / 267. Slide 4 / 267. Slide 5 / 267

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1 Slide / 67 Slide / 67 lgebra I Graphing Linear Equations -- Slide / 67 Table of ontents Slide () / 67 Table of ontents Linear Equations lick on the topic to go to that section Linear Equations lick on the topic to go to that section Graphing Linear Equations Using Intercepts Horizontal and Vertical Lines Slope of a Line Point-Slope Form Slope-Intercept Form Proportional Relationships Solving Linear Equations Scatter Plots and the Line of est Fit PR Sample Questions Glossar and Standards Slide / 67 Graphing Linear Equations Using Intercepts Horizontal and Vertical Lines Vocabular Words are bolded Slope of a Line in the presentation. The tet Point-Slope Form bo the word is in is then Slope-Intercept Form linked to the page at the end of the presentation with the Proportional Relationships word defined on it. Solving Linear Equations Teacher Notes Scatter Plots and the Line of est Fit PR Sample Questions Glossar and Standards Slide / 67 Linear Equations Linear Equations Return to Table of ontents n equation must have at least one variable. Linear equations have either one or two variables and ma also have a constant. The variables in a linear equation are not raised to an power (beond one): the are not squared, cubed, etc. The standard form of a linear equation is + = Where: and are variables and are coefficients and is a constant,, and are integers

2 Slide 6 / 67 Linear Equations There are an infinite number of solutions to a linear equation. In general, each solution is an ordered pair of numbers representing the values for the variables that make the equation true. For each value of one variable, the value of the other variable is determined. Slide 7 / 67 Linear Equations The fact that the solutions of linear equations are an infinite set of ordered pairs helped lead to the idea that those could be treated as points on a graph, and that those points would then form a line. Which is wh these are called "Linear Equations." The idea of merging algebra and geometr led to analtic geometr in the mid 6's. Slide 8 / 67 Graphing Equations Slide 9 / 67 Graphing Equations naltic Geometr powerful combination of algebra and geometr. Independentl developed, and published in 67, b Rene escartes and Pierre de Fermat in France. The artesian Plane is named for escartes. How would ou describe to someone the location of these five points so the could draw them on another piece of paper without seeing our drawing? iscuss. Slide 9 () / 67 Graphing Equations Slide / 67 Graphing Equations How would ou describe to someone the location of these five points so the could draw them on another piece of The question on this slide addresses paper without seeing MP.. our drawing? iscuss. Math Practice dding this artesian coordinate plane makes that task simple since the location of each point can be given b just two numbers: an - and -coordinate, written as the ordered pair (, )

3 Slide / 67 Graphing Equations Slide / 67 Graphing Equations With the artesian Plane providing a graphical description of locations on the plane, solutions of equations (as ordered pairs) can be analzed using algebra. The artesian Plane is formed b the intersection of the -ais and -ais, which are perpendicular. It's also called a oordinate Plane or an XY Plane. The -ais is horizontal (side-to-side) and the -ais is a vertical (up and down). The aes intersect at the origin. Slide / 67 Graphing Equations Slide / 67 Graphing Equations n ordered pair represents a solution to a linear equation, and a point on the plane. The numbers represent the - and - coordinates: (,). The point (, 8) is shown. (, 8) linear equation has an infinite set of solutions. Graphing the pairs of and values which satisf a linear equation forms a line (hence the name "linear" equation). Slide / 67 Graphing Equations Slide 6 / 67 Graphing Equations One wa to graph the line that represents the solutions to a linear equation is to use a table to find a few sets of solutions. Since a line is uniquel defined b an two points, finding three or more points provides the line, and a check to make sure the points are correct. Let's graph the line: = + We'll make a table, pick some -values and then calculate the matching -values to create ordered pairs to graph. We can pick an values for, but will choose them so that the resulting points: re eas to plot. re far enough apart to allow us to draw an accurate line.

4 Slide 7 / 67 Slide 8 / 67 - = + While we onl need two points to determine the line, it's good to check with some etra points. Use the equation to fill in the -values in the table. Graphing Equations = These are just a few points on the line. There are an infinite number of ordered pairs that satisf the equation. Graphing Equations Let's draw the line that represents the infinite set of solutions to this equation. Slide 9 / 67 Slide / 67 = + The arrows on both ends of the line indicate that it continues forever in both directions. ecause it is a line, it includes an infinit of points representing all the real numbers. Graphing Equations = click - click 9... click 9 + click click Note: When the coefficient of our term is a fraction, it's helpful to select points that are multiples of the denominator. In this eample, -, and 6 had -values Note: click that to were reveal integers. Graphing Equations lick on the points that are integers & the line to graph Slide / 67 Given the equation, = -, what is when =? -7 Slide () / 67 Given the equation, = -, what is when =?

5 Slide / 67 Given an equation of = -, what is if =? - Slide () / 67 Given an equation of = -, what is if =? Slide / 67 Is (, ) on the line = -? es no Slide () / 67 Is (, ) on the line = -? es no not enough information not enough information Slide / 67 Which point is on the line - =? (-, ) (, ) Slide () / 67 Which point is on the line - =? (-, ) (, ) (-, -) (, ) (-, -) (, )

6 Slide / 67 Which point lies on the line whose equation is - = 9? Slide () / 67 Which point lies on the line whose equation is - = 9? (, ) (, ) (-, ) (-, ) (-, ) (6, ) (-, ) (6, ) Slide 6 / 67 6 Point (k, -) lies on the line whose equation is - = -. What is the value of k? -8 Slide 6 () / 67 6 Point (k, -) lies on the line whose equation is - = -. What is the value of k? Slide 7 / 67 Slide 8 / 67 - and -intercepts To graph a line, two points are required. One technique uses the - and - intercepts. Graphing Linear Equations Using Intercepts The -intercept is where a graph of an equation passes through the -ais. The coordinates of the -intercept are (a, ), where a is an real number. The -intercept of the linear equation shown is (, ) Return to Table of ontents

7 Slide 9 / 67 - and -intercepts Slide / 67 7 What is the -intercept of this line? To graph a line, two points are required. One technique uses the - and - intercepts. The -intercept is where a graph of an equation passes through the -ais. The coordinates of the -intercept are (, b), where b is an real number. The -intercept of the linear equation shown is (, -) Slide () / 67 7 What is the -intercept of this line? Slide / 67 8 What is the -intercept of this line? (, -) Slide () / 67 8 What is the -intercept of this line? Slide / 67 9 What is the -intercept of this line? (, )

8 Slide () / 67 9 What is the -intercept of this line? Slide / 67 What is the -intercept of this line? (, ) Slide () / 67 What is the -intercept of this line? Slide / 67 What is the -intercept of this line? (, ) Slide () / 67 What is the -intercept of this line? Slide / 67 What is the -intercept of this line? (, )

9 Slide () / 67 What is the -intercept of this line? (-, ) Slide 6 / 67 Graphing Linear Equations Using Intercepts The technique of using intercepts works well when an equation is written in Standard Form. Recall that a linear equation written in standard form is + =, where,, and are integers and. Slide 7 / 67 Graphing Linear Equations Using Intercepts Slide 8 / 67 Graphing Linear Equations Using Intercepts Eample: Find the - and -intercepts in the equation + =. Then graph the equation. -intercept: Let = : -intercept: Let = : + () = + = = = so -intercept is (, ) () + = + = = = so -intercept is (, ) Eample: Find the - and - intercepts in the equation + =. Then graph the equation. -intercept is (, ) lick -intercept is (, ) lick lick on the points & the line in the coordinate plane to reveal. Slide 9 / 67 Graphing Linear Equations Using Intercepts Slide / 67 Graphing Linear Equations Using Intercepts Eample: Find the - and -intercepts in the equation - =. Then graph the equation. Eample: Find the - and -intercepts in the equation - =. Then graph the equation. -intercept: Let = : -intercept: Let = : - () = = = so -intercept is (, ) () - = - = = - so -intercept is (, -) -intercept is (, ) lick -intercept is (, -) lick lick on the points & the line in the coordinate plane to reveal. slide down

10 Slide / 67 Graphing Linear Equations Using Intercepts Slide () / 67 Graphing Linear Equations Using Intercepts oes anone see a shortcut to finding the - and - intercepts? How could our shortcut make the problem easier? oes anone see a shortcut to finding the - and - intercepts? How could our shortcut make the problem easier? The question on this slide addresses MP.8. Math Practice Note: The answer is on the net slide. Slide / 67 Graphing Linear Equations Using Intercepts Given the equation - =, another wa to look at the intercept method is called the "cover-up method." Slide / 67 Graphing Linear Equations Using Intercepts If =, we can cover that up and solve the remaining equation. If =, we can cover - up (because zero times anthing is ) and solve the remaining equation. press - = - = that leaves us with = lick solve for the -intercept is (, ) lick press - leaves us with - = solve for lick the -intercept is (, -) lick Slide / 67 Slide () / 67 Graphing Linear Equations Using Intercepts Graphing Linear Equations Using Intercepts Tr This: Find the - and - intercepts of = - 9. Then graph the equation. (, ) Tr This: -intercept: Let = Find the - and - = - 9 intercepts of = - 9. Then graph the equation. 9 = = -intercept is (, ) -intercept: Let = = () - 9 = -9 -intercept is (, -9) (, ) lick on the points & the line in the coordinate plane to reveal. (, -9) lick on the points & the line in the coordinate plane to reveal. (, -9)

11 Slide / 67 Given the equation = - 7, what is the -intercept? Slide () / 67 Given the equation = - 7, what is the -intercept? (, ) Slide 6 / 67 Given the equation = - 7, what is the -intercept? Slide 6 () / 67 Given the equation = - 7, what is the -intercept? (, -7) Slide 7 / 67 Given the equation - = ( + ), what is the intercept? Slide 7 () / 67 Given the equation - = ( + ), what is the intercept? (-.7, ) (, )

12 Slide 8 / 67 6 Given the equation - = ( + ), what is -intercept? Slide 8 () / 67 6 Given the equation - = ( + ), what is -intercept? (, ) Slide 9 / 67 7 Given the equation + =, what is the -intercept? (, ) (, ) Slide 9 () / 67 7 Given the equation + =, what is the -intercept? (, ) (, ) (, ) (, ) (, ) (, ) Slide / 67 8 Given the equation + =, what is the -intercept? (, ) (, ) Slide () / 67 8 Given the equation + =, what is the -intercept? (, ) (, ) (, ) (, ) (, ) (, )

13 Slide / 67 Slide / 67 Horizontal & Vertical Lines Horizontal and vertical lines are different from slanted lines in the coordinate plane. Horizontal and Vertical Lines Return to Table of ontents vertical line goes "up and down". Select random points on each line shown to the left. What are the similarities and differences between the points on the vertical lines? iscuss! Slide () / 67 Horizontal & Vertical Lines Slide / 67 Horizontal & Vertical Lines Horizontal and vertical lines are different from slanted The question on this slide addresses lines in the coordinate MP.7 & MP.8. plane. dditional Questions that address MP's: vertical line goes o "up ou and see a pattern? an ou eplain down". the pattern? (MP.7 & MP.8) Math Practice Select random points on each line shown to the left. What are the similarities and differences between the points on the vertical lines? iscuss! Notice that each point on the line furthest to the left all have -coordinates of -7. Eamples of points on this line are (-7, ), (-7, ), (-7, -), etc. The same holds true for the points on all of the vertical lines that follow. What is the common -coordinate shared on the remaining lines? nd line from the left: - rd line from the left: th line from the left 8 slide to view Slide / 67 Horizontal & Vertical Lines Slide / 67 Horizontal & Vertical Lines vertical line has the equation = a, where a is the -intercept and the common -coordinate shared b all of the points on the line. Notice no is contained in the equation. = -7 = - = = 8 horizontal line goes "sidewas". Select random points on each line shown to the left. What are the similarities and differences between the points on the horizontal lines. iscuss!

14 Slide () / 67 Horizontal & Vertical Lines Slide 6 / 67 Horizontal & Vertical Lines horizontal line goes "sidewas". The question on this slide addresses Select random points MP.7 on & MP.8. each line shown to the left. What are the similarities dditional and Questions that address MP's: differences between o the ou see a pattern? an ou eplain points on the horizontal the pattern? lines. (MP.7 & MP.8) Math Practice iscuss! Notice that each point on the top line have -coordinates of. Eamples of points on this line are (, ), (-, ), (, ), etc. The same holds true for the points on all of the horizontal lines that follow. What is the common -coordinate shared on the remaining lines? nd line from the top: 6 rd line from the top: - th line from the top: slide to view Slide 7 / 67 Horizontal & Vertical Lines Slide 7 () / 67 Horizontal & Vertical Lines horizontal line has the equation = b, where b is the -intercept and the common -coordinate shared b all of the points on the line. Notice no is contained in the equation. = = 6 = - = horizontal line has the equation = b, where MP.6: b is ttend to precision = the -intercept and the common -coordinate Sometimes, students confuse = _ & shared b all of the points = 6 = _ for horizontal and vertical lines. on the line. wa to make sure that the get the equation correct is to have them select Notice no is contained - points on each line & find the in the equation. similarit between the points > When it's = _ on a vertical line, all of the -coordinates are the same = - > When it's = _ on a horizontal line, all of the -coordinates are the same = Math Practice Slide 8 / 67 9 Is the following equation that of a vertical line, a horizontal line, neither, or cannot be determined: Slide 8 () / 67 9 Is the following equation that of a vertical line, a horizontal line, neither, or cannot be determined: = = Vertical Horizontal Neither Vertical Horizontal Neither annot be determined annot be determined

15 Slide 9 / 67 Is the following equation that of a vertical line, a horizontal line, neither, or cannot determine: Vertical Horizontal Neither + = 9 annot be etermined Slide 9 () / 67 Is the following equation that of a vertical line, a horizontal line, neither, or cannot determine: Vertical Horizontal Neither + = 9 annot be etermined Slide 6 / 67 Is the following line that of a vertical, a horizontal, neither, or cannot be determined: Slide 6 () / 67 Is the following line that of a vertical, a horizontal, neither, or cannot be determined: = - Vertical Horizontal Neither annot be etermined Vertical Horizontal Neither = - annot be etermined Slide 6 / 67 Is the following equation that of a vertical line, a horizontal line,neither, or cannot be determined: Slide 6 () / 67 Is the following equation that of a vertical line, a horizontal line,neither, or cannot be determined: - = - = Vertical Horizontal Neither annot be etermined Vertical Horizontal Neither annot be etermined

16 Slide 6 / 67 Which statement describes the graph of =? Slide 6 () / 67 Which statement describes the graph of =? It passes through the point (, ) It is parallel to the -ais It is parallel to the -ais It passes through the point (, ) It is parallel to the -ais It is parallel to the -ais Slide 6 / 67 The intercepts method (cover-up method) of graphing could NOT have been used to graph which of the following graphs? Select all that appl. Slide 6 () / 67 The intercepts method (cover-up method) of graphing could NOT have been used to graph which of the following graphs? Select all that appl. and E E F E F Slide 6 / 67 Which of the following equations can't be graphed using the intercepts method? Select all that appl. Slide 6 () / 67 Which of the following equations can't be graphed using the intercepts method? Select all that appl. = - - = ( + 9) = - = - E = + 7 F - = G = -8 H = = - - = ( + 9) = - = - E = + 7 F - = G = -8 H =,,, H

17 Slide 6 / 67 Slide 66 / 67 "Steepness" of a Line Slope of a Line Return to Table of ontents Slide 67 / 67 Slide 68 / 67 Slopes and Points Slope It's possible, and often easier, to graph lines using a slope and a point as opposed to a table. lso, it's not difficult to write an equation for a line from finding the slope and a point from a graph. The slope of a line is a number that describes both the direction and steepness of a line. The letter m is tpicall used as the variable for slope. Let's first define slope, and then we can use that idea. The slope can have tpes of direction: - positive: rising from left to right - negative: falling from left to right - zero: horizontal - undefined: vertical To measure the steepness of a line, we use the ratio of "rise" over "run." Slide 69 / 67 Slide 7 / 67 Slope Slope The "rise" is the change in the value of the -coordinate while the "run" is the change in the value of the coordinate. The smbol for "change" is the Greek letter delta, "," which just means "change in." So the slope is equal to the change in divided b the change in, or divided b... delta over delta. m = Δ - Δ = - (, ) "run" (, ) "rise" (, ) m = Δ - Δ = - In this case: The rise is from to = - = 7 nd the run is from to 8, = 8 - = 6 So the slope is m = = 7 6 m = Δ - (, ) Δ = - "run" In this case: (8, ) "rise" The rise is from to = - = 7 (, ) nd the run is from to 8, = 8 - = 6 So the slope is m = = 7 6

18 Slide 7 / 67 Slide 7 / 67 Slope Slope n points on the line can be used to calculate its slope, since the slope of a line is the same everwhere. (, ) Slope is also referred to as a constant rate of change. Here is an application of slope: road might rise foot for ever feet of horizontal distance. The values of and ma be different for other points, but their ratio will be the same. You can check that with the red and green triangles shown here. m = Δ - Δ = - (, ) feet foot The ratio, /, which is called slope, is a measure of the steepness of the hill. Engineers call this use of slope grade and measure the grade with percentages. The grade of the road above is %. Slide 7 / 67 Slide 7 / 67 Horizontal and vertical lines have special slopes. horizontal line has a slope of, and a vertical line has an undefined slope. Let's see what makes these slopes special. points on the horizontal line are (, ) and (, ). If we look at the graph, the is and the is, so m = = Slope Slope of horizontal & vertical lines using Rise & Run (, ) (, ) (7, ) (7, ) Horizontal and vertical lines have special slopes. horizontal line has a slope of, and a vertical line has an undefined slope. Let's see what makes these slopes special. points on the vertical line are (7, ) and (7, ). If we look at the graph, the is and the is, so... m = = undefined because ou can't divide b. Slope Slope of horizontal & vertical lines using Rise & Run (, ) (, ) (7, ) (7, ) Slide 7 / 67 Slide 7 () / 67 6 The slope of the indicated line is: 6 The slope of the indicated line is: negative positive zero undefined negative positive zero undefined

19 Slide 76 / 67 Slide 76 () / 67 7 The slope of the indicated line is: negative 7 The slope of the indicated line is: negative positive zero undefined positive zero undefined Slide 77 / 67 Slide 77 () / 67 8 The slope of the indicated line is: 8 The slope of the indicated line is: negative positive negative positive zero zero undefined undefined Slide 78 / 67 Slide 78 () / 67 9 The slope of the indicated line is: 9 The slope of the indicated line is: negative positive zero undefined negative positive zero undefined

20 Slide 79 / 67 Slide 79 () / 67 The slope of the indicated line is: The slope of the indicated line is: negative positive zero undefined negative positive zero undefined Slide 8 / 67 Slide 8 () / 67 The slope of the indicated line is: The slope of the indicated line is: negative positive zero undefined negative positive zero undefined Slide 8 / 67 Slide 8 () / 67 What's the slope of this line? What's the slope of this line? m=

21 Slide 8 / 67 Slide 8 () / 67 What's the slope of this line? What's the slope of this line? m=-/ Slide 8 / 67 Slide 8 () / 67 What's the slope of this line? What's the slope of this line? m= Slide 8 / 67 Slide 8 () / 67 What's the slope of this line? What's the slope of this line? m=/

22 Slide 8 / 67 Slide 8 () / 67 6 What's the slope of this line? 6 What's the slope of this line? m=-7/ Slide 86 / 67 Slide 86 () / 67 7 What is the slope of this line? Students tpe their answers here 7 What is the slope of this line? Students tpe their answers here undefined Slide 87 / 67 Slide 87 () / 67 8 What is the slope of the line passing through the indicated points? 8 What is the slope of the line passing through the indicated points?

23 Slide 88 / 67 Slope Let's tr an eample that does not have a graph. Slide 89 / 67 9 What is the slope of the line through (-, ) and (, -)? alculate the slope of the line that passes through (, ) and (, ). First identif (, ) as our (, ) and (, ) as our (, ). Second, substitute our numbers into the slope formula for their assigned variables. - - () - = = - = + - Slide 89 () / 67 9 What is the slope of the line through (-, ) and (, -)? Slide 9 / 67 What is the slope of line MN if M(, 7) and N(, -)? - - m = = - (-) - Slide 9 () / 67 What is the slope of line MN if M(, 7) and N(, -)? Slide 9 / 67 What is the slope of the line containing (-, 7) and (, -7)? - -7 m = = - -

24 Slide 9 () / 67 What is the slope of the line containing (-, 7) and (, -7)? Slide 9 / 67 What is the slope of the line that passes through the points (, ) and (, )? m = = = - (-) -7 Slide 9 () / 67 What is the slope of the line that passes through the points (, ) and (, )? Slide 9 / 67 straight line with slope contains the points (, ) and (, k). Find the value of k m = = = - - Slide 9 () / 67 straight line with slope contains the points (, ) and (, k). Find the value of k. Slide 9 / 67 onstant Rate of hange Slope formula can be used to find the constant rate of change in a "real world" problem. k = When traveling on the highwa, drivers will set the cruise control and travel at a constant speed this means that the distance traveled is a constant increase. The graph at the right represents such a trip. The car passed mile-marker 6 at hour and mile-marker 8 at hours. Find the slope of the line and what it represents. 8 miles-6 miles miles m = = = hours- hours hours So the slope of the line is 6 and the rate of change of the car is 6 miles per hour. istance (miles) 6 miles hour (,6) Time (hours) (,8)

25 Slide 9 () / 67 onstant Rate of hange Slope formula can be used to find the constant rate of change in a "real world" problem. When traveling This problem on the addresses highwa, MP. drivers & MP.. will set the cruise control and travel at a constant dditional speed this Questions means that that the address MP's: distance What traveled information is a constant do ou increase. have? (MP.) What do ou need to find? (MP.) The graph What at connections the right represents do ou see such between a trip. The slope car and passed constant mile-marker rate of change? 6 at hour and (MP.) mile-marker 8 at hours. Find the slope of the line and what it represents. Math Practice 8 miles-6 miles miles m = = = hours- hours hours So the slope of the line is 6 and the rate of change of the car is 6 miles per hour. istance (miles) 6 miles hour (,6) Time (hours) (,8) Slide 9 / 67 If a car passes mile-marker in hours and mile-marker in hours, how man miles per hour is the car traveling? Use the information to write ordered pairs (, ) and (, ). slide to move onstant Rate of hange miles - miles miles miles m = = = hours - hours hours hour Slide 96 / 67 If a car passes mile-marker 9 in. hours and mile-marker in. hours, how man miles per hour is the car traveling? Slide 96 () / 67 If a car passes mile-marker 9 in. hours and mile-marker in. hours, how man miles per hour is the car traveling? = =. -. the car is traveling miles per hour Slide 97 / 67 How man meters per second is a person running if the are at meters in seconds and meters in seconds? Slide 97 () / 67 How man meters per second is a person running if the are at meters in seconds and meters in seconds? - 9 = = 9 - The person is running 9 meters per second

26 Slide 98 / 67 Slide 99 / 67 Point-Slope Form useful form of the equation of a line is the pointslope form. It's equation is - = m( - ) Point-Slope Form It's based on the use of the slope and an point that is on the line. This equation is the most effective when ou are given the slope and a point on the line because ou can use it to write the equation in multiple forms. Let's get started. Return to Table of ontents Slide 99 () / 67 Point-Slope Form useful form of the equation of a line is the pointslope form. It's equation is - = m( - ) It's based on the use of the slope and an point that is on the line. The net slides address MP. and This equation MP.. is the most effective when ou are given the slope and a point on the line because ou can use it to write the equation in multiple forms. Math Practice Let's get started. Slide / 67 Point-Slope Form m = - - Point-slope form starts using the definition of slope, in which two points on a line are given b the ordered pairs (, ) and (, ) s a first step, let's just name the coordinates for the second point (, ) rather than (, ). That will be true for an point on the line, not just one point, and will allow us to write an equation for all points on the line. Then our slope formula becomes: m = - - Slide / 67 Point-Slope Form m = - - Now let's solve that equation for using what we've learned about solving equations. Tr it ourself, before we show ou our answer. (Hint: Remember to treat the denominator ( - ) like it's in parentheses.) m = - - m( - ) = - - = m( - ) Slide / 67 Point-Slope Form Multipl both sides b ( - ) to get rid of the fraction. The last step is to switch the epressions to the opposite sides of the equals sign.

27 Point-Slope Form Where: Slide / 67 Point-Slope Form - = m( - ) m is the slope, (, ) is an of the infinite points that satisf the equation. Slide / 67 Point-Slope Form - = m( - ) If ou are provided a graph of a line, ou can calculate m and locate a point directl from the graph. That allows ou to write the equation of the line directl, which ou can then use to find an other needed points. Slide / 67 Using Point-Slope to raw a Line Slide 6 / 67 Using Point-Slope to raw a Line For instance, if I know that the equation of a line is - = ( - ), then one point on the line is (, ) and the slope of the line is. Using this information, I can find a second point, and then draw the line. I do this b recognizing that the slope of means that if I go up units on the -ais I have to go unit to the right on the -ais. Or if I go up, I have to go over units, etc. Slide 7 / 67 Using Point-Slope to raw a Line Slide 8 / 67 Using Point-Slope to raw a Line Then I draw the line through an two of those points. This method is the easiest to use if ou just have to draw a line given a point and slope. Graph the equation - = ( - 7) ased on the equation, we know that the line passes through (7, ) click and has a slope of. click Now the graph can be drawn. lick on the point to reveal it. lick slightl above it to show the rise & slightl to the right to show the run. lick on the new point. lick in between to reveal the line. (7, ) (7, )

28 Slide 9 / 67 Using Point-Slope to raw a Line Slide / 67 6 What is the slope of - = ( + )? Eample: Given the equation + = ( + ) etermine the point on the line and the slope. The point on the line is (-, -) The slope is click click Graph the line representing the equation. lick on the point to reveal it. lick slightl above it to show the slope. lick on the new point. lick in between to reveal the line. 6 (-, -) Slide () / 67 6 What is the slope of - = ( + )? Slide / 67 7 Which point is on the line - = ( + )? m = (-, ) (, -) (, -) (-, ) Slide () / 67 7 Which point is on the line - = ( + )? (-, ) (, -) (, -) (-, ) (-, ) Slide / 67 8 What is the slope and a point on the line + = -( - )? m = -; (, ) m = -; (-, ) m = ; (, ) m = ; (-, )

29 Slide () / 67 8 What is the slope and a point on the line + = -( - )? m = -; (, ) m = -; (-, ) m = ; (, ) m = ; (-, ) Slide / 67 9 Which is the slope and a point on the line - = ()? m = ; (-, ) m = - ; (, -) m = ; (, ) m is undefined; (, ) Slide () / 67 Slide / 67 9 Which is the slope and a point on the line - = ()? m = ; (-, ) m = - ; (, -) m = ; (, ) m is undefined; (, ) Which line represents + = -( - )? line line line line Slide () / 67 Slide / 67 Which line represents + = -( - )? line line line line Which line represents + = - ( + )? line line line line

30 Slide () / 67 Slide 6 / 67 Which line represents + = - ( + )? line line line line Which line represents - 6 = ( + )? line line line line Slide 6 () / 67 Slide 7 / 67 Which line represents - 6 = ( + )? line line line line Point-Slope Form - = m( - ) You can determine and graph equations in point-slope form even when ou are given limited information. For eample, if ou are given the slope (m) and an point (, ), then b substituting the point into the equation for and & the slope for m, ou have the equation of the line with no additional work required. Slide 8 / 67 Point-Slope Form - = m( - ) Or, if ou are given an two points, it's alwas possible to determine the slope, m. then substituting one of those points in for and, write the equation of a line from two points. Let's clarif the steps required b doing some eamples for both cases. Eample: Slide 9 / 67 Point-Slope Form - = m( - ) Write the equation of a line in point-slope form that has a slope of 7 and passes through the point (-9, ). We alread know that the slope is 7, or m = 7. We also know that a point that the line passes through is (-9, ), which represent (, ) respectfull. substituting these numbers into our equation, we have: - = 7( - (-9)) - = 7( + 9), which is our equation in point-slope form

31 Eample: Slide / 67 Write the equation of a line in point-slope form that passes through the points (, -9) and (, ). This time, we do not know the slope, so we need to calculate it. m = - (-9) = 9 = Point-Slope Form 9 - = m( - ) We also know that a point that the line passes through is (, -9), which represent (, ) respectfull. substituting these numbers into our equation, we have: - (-9) = - 9 ( - ) which simplifies to + 9 = - 9 ( - ) Eample: Slide / 67 Point-Slope Form - = m( - ) Write the equation of a line in point-slope form that passes through the points (, -9) and (, ). You might be asking if the equation on the previous slide is the onl answer. ctuall, ou can also write the equation in point-slope form using the other point (, ) as (, ). Therefore the equation could also be: - = - 9 ( - ) which simplifies to = - 9 ( - ) Eample: Slide / 67 Point-Slope Form - = m( - ) Write the equation of a line in point-slope form that passes through the points (, -9) and (, ). The reason that we have two possible answers is because the would both simplif into the same equation, in standard form. elow is the work for each case. + 9 = - 9 ( - ) = - 9 ( - ) + 8 = -9( - ) = -9( - ) + 8 = -9 + = = 9 + = = 7 Slide / 67 What is the equation of the line in point-slope form if its slope is - and it passes through the point (9, -7). - 7 = -( - 9) + 7 = -( + 9) + 7 = -( - 9) - 7 = -( + 9) Slide () / 67 What is the equation of the line in point-slope form if its slope is - and it passes through the point (9, -7). - 7 = -( - 9) Slide / 67 What is the equation of the line in point-slope form if its slope is and it passes through the point (-, -). - = ( - ) + 7 = -( + 9) + 7 = -( - 9) - 7 = -( + 9) + = ( + ) + = ( - ) - = ( + )

32 Slide () / 67 What is the equation of the line in point-slope form if its slope is and it passes through the point (-, -). - = ( - ) + = ( + ) + = ( - ) - = ( + ) Slide / 67 What is the equation of the line in point-slope form if its slope is - and it passes through the point (, 8). 9-8 = - ( - ) = - ( + ) = - ( - ) 9-8 = - ( + ) 9 Slide () / 67 What is the equation of the line in point-slope form if its slope is - and it passes through the point (, 8). 9-8 = - ( - ) = - ( + ) = - ( - ) 9-8 = - ( + ) 9 Slide 6 / 67 6 What is the equation of the line in point-slope form that passes through the points (, 8) and (-, -). + 8 = ( + ) - 8 = ( - ) + = ( + ) - = ( - ) Slide 6 () / 67 6 What is the equation of the line in point-slope form that passes through the points (, 8) and (-, -). + 8 = ( + ) - 8 = ( - ) + = ( + ) - = ( - ) Slide 7 / 67 7 What is the equation of the line in point-slope form that passes through the points (7, 9) and (, -). - 9 = ( - 7) - = ( - ) + = ( - ) - 9 = ( - 7)

33 Slide 7 () / 67 7 What is the equation of the line in point-slope form that passes through the points (7, 9) and (, -). Slide 8 / 67 8 What is the equation of the line in point-slope form that passes through the points (, -7) and (-, -). - 9 = ( - 7) - = ( - ) + = ( - ) - 9 = ( - 7) - 7 = - ( + ) - = - ( - ) + = - ( + ) - 7 = - ( - ) Slide 8 () / 67 Slide 9 / 67 8 What is the equation of the line in point-slope form that passes through the points (, -7) and (-, -). - 7 = - ( + ) - = - ( - ) + = - ( + ) - 7 = - ( - ) Slope-Intercept Form Return to Table of ontents Slide / 67 Slope-Intercept Form nother ver useful form of the equation of a line is the slope-intercept form. It's based on the use of the slope and the -intercept of a line. Similar to point-slope, we are going to start b showing a derivation proof of slope-intercept form. Slide () / 67 Slope-Intercept Form nother ver useful form of the equation of a line is the slope-intercept form. It's based on the use of the slope and the -intercept The net of slides a line. address MP. and MP.. Similar to point-slope, we are going to start b showing a derivation proof of slope-intercept form. Math Practice

34 Slide / 67 Slope-Intercept Form m = - - Let's start with the definition of slope, in which two points on a line are given b the ordered pairs (, ) and (, ) s a first step, let's just name the coordinates for the second point (,) rather than (, ). That will be true for an point on the line, not just one point, and will allow us to write an equation for all points on the line. Then our slope formula becomes: - m = - m = - - Slide / 67 Slope-Intercept Form Now let's solve that equation for using what we've learned about solving equations. Tr it ourself, before we show ou our answer. (Hint: Remember to treat the denominator ( - ) like it's in parentheses.) Slide / 67 Slope-Intercept Form Slide / 67 Slope-Intercept Form m = - - Multipl both sides b ( - ) to get rid of the fraction. = m( - ) + Use the -intercept for (, ). m( - ) = - m( - ) + = Now, add to both sides. Now switch the sides The -intercept is the point where the line crosses the -ais. The coordinates of that point are (, b). is zero anwhere on the -ais We name the -intercept "b." = m( - ) + This is fine, and allows us to graph a line given an point (, ). = m + b Then, this becomes: ut, one additional step is taken to get the most useful equation for a line. Slide / 67 Slide 6 / 67 Slope-Intercept Form Slope-Intercept Form = m + b = m + b Slope-Intercept Form is where: m is the slope, b is the -intercept (, b) (, ) is an of the infinite points that satisf the equation. This is the form of the equation of a line that is most often used. If ou are provided a graph of a line, ou can calculate m and read b directl from the graph. That allows ou to write the equation of the line directl. Which ou can then use to find an other needed points.

35 Slide 7 / 67 Slide 8 / 67 Slope-Intercept Form Slope-Intercept Form The line shown in the coordinate plane to the left has a -intercept at the point (, -) and has a slope of. Therefore, the equation of this line is = - (, -) Eample: Write the equation for the line shown in the graph to the right. The slope of the line (m) is - = - click The -intercept of the line (b) is - or (, -) click Therefore the equation of the line in slope intercept form is click = - - Slide 9 / 67 Slide / 67 Eample: Write the equation for the line shown in the graph to the right. The slope of the line (m) is = The -intercept of the line (b) is or (, ) click Therefore the equation of the line in slope intercept form is click click = + or = Slope-Intercept Form 9 Which equation is that of the graphed line? = - = = + = - E = F = + Slide () / 67 Slide / 67 9 Which equation is that of the graphed line? 6 Which equation is that of the graphed line? = - = = + = - E = F = + = - = = + = - E = F = +

36 Slide () / 67 Slide / 67 6 Which equation is that of the graphed line? 6 Which equation is that of the graphed line? = - = = + = - E = F = + = - = = + = - E = F = + Slide () / 67 Slide / 67 6 Which equation is that of the graphed line? = - = = + = - E = F = + 6 What is the equation of this line? = - = - - = - + = - + E = Slide () / 67 Slide / 67 6 What is the equation of this line? = - = - - = - + = - + E = Graphing a Line Using Slope-Intercept Form Having the equation of a line in slope-intercept form also allows us to quickl graph a line, using the -intercept and the slope. For instance, if we know that the equation of the line is = - + then we can graph the - intercept (, ) and use the slope of - to count the number of spaces, down unit and right units (or up one and left units), to get an additional point on the line. (, ) -

37 Then we draw the line through an two of those points. This method is the easiest to use if ou just have to draw a line given the -intercept and slope. Slide / 67 Graphing a Line Using Slope-Intercept Form (, ) - Slide 6 / 67 6 What is the slope of the linear equation = - -? - Slide 6 () / 67 6 What is the slope of the linear equation = - -? - Slide 7 / 67 6 What is the -intercept of the linear equation = - -? (, ) (, ) (, ) (, - ) Slide 7 () / 67 6 What is the -intercept of the linear equation = - -? (, ) (, ) (, ) (, - ) Slide 8 / 67 6 Which line in the graph below represents the linear equation = - -? Red lue Purple lack

38 Slide 8 () / 67 6 Which line in the graph below represents the linear equation = - -? Red lue Purple lack Slide 9 / What is the slope of the linear equation = +? - Slide 9 () / What is the slope of the linear equation = +? - Slide / What is the -intercept of the linear equation = +? (, ) (, ) (, ) (, - ) Slide () / What is the -intercept of the linear equation = +? (, ) (, ) (, ) (, - ) Slide / Which line in the graph below represents the linear equation = +? Red lue Purple lack

39 Slide () / Which line in the graph below represents the linear equation = +? Red lue Purple lack Slide / 67 Slope-Intercept Form = m + b You can also determine and graph equations in slopeintercept form even when ou are given limited information. For eample, if ou are given the slope (m) and an point (, ), then b substituting the point into the equation for and, ou can solve for b. Then, ou can write the equation of a line when given the slope and a point that the line passes through. Slide / 67 Slope-Intercept Form = m + b Or, if ou are given an two points, it's alwas possible to determine the slope, m. then substituting one of those points in for and, ou can solve for b. Then, ou can write the equation of a line from two points. Let's clarif the steps required b doing some eamples for both cases. Eample: Slide / 67 Slope-Intercept Form = m + b Write the equation of a line in slope-intercept form that has a slope of 8 and passes through the point (, ). We alread know that the slope is 8, or m = 8. We also know that a point that the line passes through is (, ), which represent (, ) respectfull. substituting these numbers into our equation, we have: = 8() + b Slide () / 67 Slope-Intercept Form = m + b Eample: This slide and the net 6 slides address MP., MP. & MP.6. Write the equation of a line in slope-intercept form that has a slope dditional of 8 and Questions passes through to address the point MP's: (, ). What is the problem asking? (MP.) We alread What know information that the slope do ou is 8, know? or m = (MP.) 8. What other resources could help ou We also know solve that this a problem? point that (MP.) the line passes through is id (, ), ou which use the represent most efficient (, ) wa respectfull. to solve the problem? (MP.6) substituting these numbers into our equation, we have: = 8() + b Math Practice Slide / 67 Slope-Intercept Form = m + b Now, we can solve for b. = 8() + b Multipl 8 and = + b Subtract from both sides - = b Switch the order b = - Our equation is then = 8 -.

40 Eample: Slide 6 / 67 Slope-Intercept Form Write the equation of a line in slope-intercept form that has a slope of 8 and passes through the point (, ). This problem can also be solved using point-slope and using the rules of algebra to get it into slope-intercept form. We have our point (, ) as (, ) and our slope of 8. If I substitute the numbers into pointslope, we will get this equation: - = 8( - ) - = 8 - = 8 - = m + b istribute the 8 to and - ddition of to both sides Eample: Slide 7 / 67 Slope-Intercept Form Write the equation of a line in slope-intercept form that passes through the points (, ) and (6, ). This time, we do not know the slope, so we need to calculate it. m = - = 6 - We also know that a point that the line passes through is (, ), which represent (, ) respectfull. substituting these numbers into our equation, we have: = () + b = m + b Slide 8 / 67 Slope-Intercept Form = m + b Now, we can solve for b. = () + b Multipl and = + b Subtract from both sides = b Switch the order b = Our equation is then = -. Eample: Slide 9 / 67 Slope-Intercept Form Write the equation of a line in slope-intercept form that passes through the points (, ) and (6, ). This problem can also be solved using point-slope and using the rules of algebra to get it into slope-intercept form. We can select one of our points as (, ) and our slope that we calculated was. Let's use (, ). If we substitute the numbers into point-slope, we will get this equation: - = ( - ) = - = m + b istribute the slope & drop the Slide 6 / What is the equation of the line that passes through the point (-, -7) and has a slope of? = + Slide 6 () / What is the equation of the line that passes through the point (-, -7) and has a slope of? = + = + = - + = + = - + = 7-7 = 7-7

41 Slide 6 / 67 7 What is the equation of the line that passes through the point (6, -) and has a slope of? 6 = = = 6-7 Slide 6 () / 67 7 What is the equation of the line that passes through the point (6, -) and has a slope of? 6 = = = 6-7 = = Slide 6 / 67 7 What is the equation of the line that passes through the point (-, -8) and has a slope of -? = = - + = - - = Slide 6 () / 67 7 What is the equation of the line that passes through the point (-, -8) and has a slope of -? = = - + = - - = Slide 6 / 67 7 What is the equation of the line that passes through the points (-, -8) and (, )? = - = + = - = - Slide 6 () / 67 7 What is the equation of the line that passes through the points (-, -8) and (, )? = - = + = - = -

42 Slide 6 / 67 7 What is the equation of the line that passes through the points (7, ) and (, )? = = = = Slide 6 () / 67 7 What is the equation of the line that passes through the points (7, ) and (, )? = = = = Slide 6 / 67 7 What is the equation of the line that passes through the points (-, ) and (, )? Slide 6 () / 67 7 What is the equation of the line that passes through the points (-, ) and (, )? = = = = -. 6 = = = = -. 6 Slide 66 / 67 Slide 67 / 67 Lab: Marble Masters Students use a linear equation, slope, and X-Y intercept to aim a marble launch tube so the marble will cross a specified set of artesian coordinates and hit the "target"! Proportional Relationships Lab: Proportional Relationships Return to Table of ontents

43 Math Practice Slide 67 () / 67 This lab addresses MP., MP., MP.6, MP.7 & MP.8. dditional Questions to address MP's: What information are ou given? (MP.) What is the problem asking? (MP.) How can ou represent the problem with Proportional Relationships smbols and numbers? (MP.) What connections do ou see? (MP.) What does proportional relationship mean? (MP.6) Lab: Proportional Relationships What labels could ou use? (MP.6) How are proportional relationships related to linear equations? (MP.7) What generalization can ou make? (MP.8) Slide 69 / 67 Proportional Relationships Return to Table of ontents Time (hr.) Famil Z istance (mi.) from home Slope (m) = 7 -intercept (b) = equation = 7 Slide 68 / 67 Proportional Relationships omplete the items below each table. click click click Time (hr.) Famil istance (mi.) from home 6 Slope (m) = 7 -intercept (b) = equation = 7 + click click click If this data from both tables were graphed on the same coordinate plane, what would ou notice? Slide 7 / 67 Proportional Relationships Proportional Relationship is a relationship in which the ratios of related terms are equal. Linear equations that have a proportional relationship are known as direct variation equations, which take the form = m. The graph to the right is the graph of = The proportional relationship is epressed in the slope and the ratio of ever -coordinate to ever -coordinate with the eception of the origin. Wh is the proportional relationship not shown with the origin? (-, -) (, ) (, 6) Slide 7 () / 67 Proportional Relationships Slide 7 / 67 Proportional Relationships The proportional relationship is epressed in the slope The proportional relationship and the ratio of ever -coordinate to is ever not shown in the origin -coordinate with because the the ratio would (, 6) be eception of the which origin. is not a real number. (, ) Wh is the proportional relationship not shown with the origin? The question on this slide addresses (-, -) [This object MP.. is a pull tab] Eample: What is the proportional relationship shown in the graph? click to reveal What is the equation of the line? = click to reveal

44 Slide 7 () / 67 Proportional Relationships Slide 7 / 67 Proportional Relationships Eample: Eample: What is the proportional relationship shown in This the slide and the net one address graph? MP.. click to reveal What is the equation of the line? = click to reveal Math Practice What is the proportional relationship shown in the table? click to reveal What is the equation of the line? = click to reveal 6 9 Slide 7 / 67 7 What is the proportional relationship shown in the graph? Slide 7 () / 67 7 What is the proportional relationship shown in the graph? Slide 7 / What is the proportional relationship shown in the table? Slide 7 / What is the proportional relationship shown in the graph?

45 Slide 7 () / What is the proportional relationship shown in the graph? Slide 76 / What is the proportional relationship shown in the equation = -? Slide 76 () / What is the proportional relationship shown in the equation = -? - Slide 77 / What is the proportional relationship shown in the table? - Slide 78 / 67 famil drove their car miles in hours. a) Write a direct variation equation that relates the distance traveled in respect to time. d = b) Graph the equation. c) click to reveal Proportional Relationships Predict about how long it will take the famil to drive 6 miles. 6 = ivide b = 8 hours slide to reveal istance (miles) t Time (hours) click on the line to reveal Slide 78 () / 67 Proportional Relationships famil drove their This car eample miles addresses in hours. MP., MP. & MP.7. a) Write a direct variation equation that relates the distance traveled in respect dditional to time. Questions to address MP's: d How can ou represent the problem = with smbols and numbers? (MP.) click to reveal What do ou alread know about b) Graph the equation. solving this problem? 6 (MP.) What connections 7do ou see? (MP.) c) Predict about how What long do ou know about proportional 8 it will take the famil relationships to that can appl to this 9 drive 6 miles. situation? (MP.7) 6 = ivide b = 8 hours slide to reveal Math Practice istance (miles) t Time (hours) click on the line to reveal

46 Slide 79 / 67 Proportional Relationships o all lines have proportional relationships? reate the equation of a line in the form = m + b, where b. Test out the relationship between the - and -coordinates in each ordered pair. ISUSS! Slide 79 () / 67 Proportional Relationships Sample : = - points on the line: (-, -9), (, -), (, ) o all lines The have ratios proportional of -coordinate/-coordinate relationships? for the points shown above are given below. reate the equation of a line in the form -9 - = m + b, where b -. Test out the relationship between the - and -coordinates in each ordered None of these ratios are equal. Therefore, pair. not all lines have proportional relationships. Onl those that pass through the origin have this relationship. ISUSS! This question addresses MP., MP. & MP. Eample: Slide 8 / 67 Proportional Relationships line passes through the points (, -), (8, -6), and (-, 8). What is the equation of the line? Use the equation of the line to eplain wh the ratio of the -coordinate to the -coordinate is the same for an point on the line ecept the -intercept. Eplain wh this is not true for the -intercept. Slide 8 () / 67 Proportional Relationships Eample: This eample addresses MP., MP., line passes MP. through & MP.7. the points (, -), (8, -6), and (-, 8). What is the dditional equation Questions of the line? to address MP's: How could ou start this problem? (MP.) Use the equation reate an of equation the line to to eplain represent wh the the ratio of the -coordinate problem? to the (MP.) -coordinate is the same for an point on the line What ecept eamples the -intercept. could prove Eplain or disprove wh this is not true for the our -intercept. argument? (MP.) Wh does the origin not work? (MP.7) Math Practice Eample: Slide 8 / 67 Proportional Relationships line passes through the points (, -), (8, -6), and (-, 8). What is the equation of the line? Using two of our points, we can calculate the slope of the line. m = -6 - (-) = -6 + = = Eample: Slide 8 / 67 Proportional Relationships line passes through the points (, -), (8, -6), and (-, 8). Use the equation of the line to eplain wh the ratio of the -coordinate to the -coordinate is the same for an point on the line ecept the -intercept. Eplain wh this is not true for the -intercept. Net, using the point and our slope we can find the equation of the line. - (-) = - ( - ) + = - + = -

47 Slide 8 () / 67 Proportional Relationships Sample : Eample: The graph passes through the origin, and the point (, -) is on the line. Therefore line passes through the calculated the points slope of (, the -), line (8, is -6), and (-, 8). Use the equation of the line - - to eplain wh the ratio of the - = - -coordinate to the -coordinate is the same for an point on the line ecept This is the -intercept. same ratio of the Eplain coordinates wh this is not true for the -intercept. / with the eception of the -intercept located at the origin (, ). The origin does not work because divided b is undefined. Slide 8 / 67 8 line passes through the points (, ), (, ), and (-, -7). What is the equation of the line? =. = =. -. = - 8 Slide 8 () / 67 8 line passes through the points (, ), (, ), and (-, -7). What is the equation of the line? =. = =. -. Slide 8 / 67 8 line passes through the points (, ), (, ), and (-, -7). oes this line represent a proportional relationship? Eplain. Yes No = - 8 Slide 8 () / 67 8 line passes through the points (, ), (, ), and (-, -7). oes this line represent a proportional relationship? Eplain. Yes No No, the equation has a -intercept of (, -.). Therefore it does not pass through the origin. Onl lines that pass through the origin have a proportional relationship. Slide 8 / 67 8 line passes through the points (, -8), (, -), and (-, ). What is the equation of the line? = -. = - = -. - = - +

48 Slide 8 () / 67 8 line passes through the points (, -8), (, -), and (-, ). What is the equation of the line? = -. = - = -. - Slide 86 / 67 8 line passes through the points (, -8), (, -), and (-, ). oes this line represent a proportional relationship? Eplain. Yes No = - + Slide 86 () / 67 8 line passes through the points (, -8), (, -), and (-, ). oes this line represent a proportional relationship? Yes. The Eplain. line passes through the origin, and all of Yes the points on the line have a ratio of -coordinate/coordinate that is equivalent No to the slope of the line, which is -. Slide 87 / 67 Solving Linear Equations Return to Table of ontents Slide 88 / 67 Solving Linear Equations Throughout this unit, ou have learned how to calculate the slope of a line and write the equation of a line in three different forms. What if ou are given the equation of a line, and the ask ou to find the slope? or an intercept? epending on what form the equation is in, the information might be eas to find. Sometimes, it might require some algebraic steps to manipulate the equation into the desired form. Eample: Slide 89 / 67 What is the slope of the equation 8 - =? This equation is in standard form, which can be used to find the intercepts. It doesn't tell ou the slope, though. So we have to manipulate it using what we know from algebra. Which form(s) of a linear equation can help us find the slope of this line? point-slope or slope-intercept click to reveal Solving Linear Equations Which form of a linear equation would be the most appropriate for this situation? Wh? slope-intercept It's easier to solve the equation 8 - = for, giving us the equation in slope-intercept form. click to reveal

49 Eample: Slide 89 () / 67 What is the slope of the equation 8 - =? This equation is in standard This form, eample which addresses can be used MP.7. to find the intercepts. It doesn't tell ou the slope, though. So we have to manipulate it using what we know from algebra. Which form(s) of a linear equation can help us find the slope of this line? point-slope or slope-intercept click to reveal Solving Linear Equations Math Practice Which form of a linear equation would be the most appropriate for this situation? Wh? Eample: Slide 9 / 67 Solving Linear Equations What is the slope of the equation 8 - =? 8 - = Subtract 8 from both sides = -8 + ivide both sides (or all terms) b = - Therefore, the slope of the equation is. click slope-intercept It's easier to solve the equation 8 - = for, giving us the equation in slope-intercept form. click to reveal Slide 9 / 67 Solving Linear Equations Eample: Write the equation of the line with an -intercept of and a -intercept of. Since two points are given, we can use the slope formula. The points are (, ) and (, ). m = - = = - click - click Since this problem does not specif a form for the equation, ou can select an form that ou desire. The equation of the line is in point-slope form, using the -intercept, is = -( - ) click to reveal Or ou could also use the -intercept to write the equation in pointslope form: - = - click to reveal Or since ou have the -intercept and ou calculated the slope, ou could have written the equation in slope-intercept form: = - +. click to reveal Eample: Write the equation - 7 = determine the - and -intercepts. Slide 9 / 67 Solving Linear Equations ( + ) in standard form. Then We need to get this equation into + =, where. Therefore, we need to use our inverse operations. - 7 = ( + ) Multipl both sides of the equation b the L ( - 7) = ( + ) istributive Propert - 8 = + Subtract from both sides = dd 8 to both sides - + = Multipl all terms b -, because - = - Slide 9 () / 67 Solving Linear Equations Eample: Write the equation of the line with an -intercept of and a -intercept of. This eample addresses MP., MP.6 & Since two points MP.7. are given, we can use the slope formula. The points are (, ) and (, ). dditional Questions to address MP's: m = - = = Which - equation would be best for this click - clickproblem? (MP.) Since this problem id does ou not use specif the most a form efficient for the wa equation, to ou can select an form solve that this ou problem? desire. The (MP.6) equation of the line is in point-slope form, Wh using is the it possible -intercept, to have is multiple = -( answers? - ) (MP.7) Math Practice click to reveal Or ou could also use the -intercept to write the equation in pointslope form: - = - click to reveal Or since ou have the -intercept and ou calculated the slope, ou could have written the equation in slope-intercept form: = - +. click to reveal Slide 9 () / 67 Solving Linear Equations This eample addresses MP. & MP.7. Eample: Write the equation - 7 = ( + ) in standard form. Then dditional Questions to address MP's: determine What the information - and -intercepts. are we given? (MP.) What do we have to find? (MP.) We need What to get do ou this know equation about into inverse + =, where. Therefore, we need operations to use our that inverse ou can operations. appl to this situation? (MP.7) - 7 = ( + ) Multipl both sides of the equation b the L ( - 7) = ( + ) istributive Propert - 8 = + Subtract from both sides = dd 8 to both sides - + = Multipl all terms b -, because - = - Math Practice

50 Eample: Write the equation - 7 = determine the - and -intercepts. Slide 9 / 67 ( + ) in standard form. Then We need to get this equation into + =, where. Therefore, we need to use our inverse operations. - = - Solving Linear Equations -intercept: = -intercept: = - () = - () - = - = - - = - = (, ) = - (-, ) move to reveal move to reveal Slide 9 / 67 8 What is the slope of the line whose equation is 7 = 9? Slide 9 () / 67 8 What is the slope of the line whose equation is 7 = 9? Slide 9 / 67 8 What is the slope of the line whose equation is + 9 = Slide 9 () / 67 8 What is the slope of the line whose equation is + 9 = Slide 96 / What is the equation of a line that has an -intercept of and a -intercept of 7? = - 7 ( - ) = = ll of the above

51 Slide 96 () / What is the equation of a line that has an -intercept of and a -intercept of 7? = - 7 ( - ) = = ll of the above Slide 97 / Which equation represents + 6 = ( - ) in standard form? - = = -8 = - 8 = Slide 97 () / Which equation represents + 6 = ( - ) in standard form? - = 8 Slide 98 / Which equation represents = in standard form? - = = -8 + = = - 8 = = = - ( - ) Slide 98 () / Which equation represents = in standard form? - = - + = = = - ( - ) Slide 99 / What is the equation of a line that has an -intercept of and a -intercept of -7? 7 = + = = 7 ll of the above

52 Slide 99 () / What is the equation of a line that has an -intercept of and a -intercept of -7? 7 = + = = 7 ll of the above For eample: Slide / 67 Function Notation Sometimes, ou will see an equation in the form f() = m + b instead of = m + b. This can occur because ever linear equation is a function. function is a relationship that eists when ever value has eactl one value. When this happens, nothing changes mathematicall. Yet, instead of writing mathematicians use f(), read "f of." f() given that is a function. = + 7 becomes f() = + 7 *f() = + 7 is still a line with a slope of and a -intercept of (, 7). Since we haven't et covered the material for functions, it would be easiest for ou, at this point, to switch the f() to and solve the equation from there. Slide / 67 Solving Linear Equations Slide () / 67 Solving Linear Equations Eample: The graph shown below represents the function f() = For what value of does f() =? If we start with the advice that we started with, substituting in for f(), we get the equation = Eample: The graph shown below represents the function This eample addresses MP. & MP.7. f() = dditional Questions to address MP's: For what What value information of are we given? (MP.) does What f() = do? we have to find? (MP.) How is f() related to? (MP.7) Math Practice If we start with the advice that we started with, substituting in for f(), we get the equation = Slide / 67 Slide / 67 Eample: The graph shown below represents the function f() = For what value of does f() =? Solving Linear Equations 9 The graph shown below represents the function g() = - 8. For what value of does g() =? -8 8 = If f() =, that means that =. This occurs when the graph crosses the -ais, giving us our -intercept. Looking at the graph, we see that our -intercept is (6, ), so the value of is 6. -

53 Slide () / 67 9 The graph shown below represents the function g() = - 8. For what value of does g() =? Slide / 67 9 The graph shown below represents the function h() = - +. For what value of does h() =? Slide () / 67 Slide / 67 9 The graph shown below represents the function 9 The graph shown below represents the function h() = - +. For what value of does h() =? j() = - -. For what value of does j() =? - Slide () / 67 Slide 6 / 67 9 The graph shown below represents the function j() = - -. For what value of does j() =? - Scatter Plots and the Line of est Fit Return to Table of ontents

54 Slide 7 / 67 scatter plot is a graph that shows a set of data that has two variables. Time Studing Test Score Scatter Plot Test Score (, 78) (, 88) (, 8) (, 8) (, 9) (, 9) (8, 87) 6 Time spent studing Slide 8 () / 67 Scatter Plot (6, 9) There are three tpes of Test Score vs. Time Spent linear association that Studing are possible for scatter plots. What are the? 9 (, 9) These concepts were taught in 8th Positive association Grade Math. fter the net slide, the (6, 9) Negative association remaining questions are a review of the 9 (, 9) click No to association reveal 8th grade lesson. If our class (, 88) needs more instruction, visit (8, 87) What tpe of linear the 8 (, 8) association does (, 8) graph have to the right? 8 Positive association (, 78) click to reveal Teacher Notes Test Score There are three tpes of linear association that are possible for scatter plots. What are the? Positive association Negative association No association click to reveal What tpe of linear association does the graph have to the right? Positive association click to reveal What connection do ou see between linear association and the slope of a line? The follow the same direction. Positive rises from left to right, and negative falls from left to right. click to reveal Slide 8 / 67 Scatter Plot Test Score Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) Slide 9 / 67 Scatter Plot Test Score (, 8) (, 9) (8, 87) 6 Time spent studing Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) (, 8) (, 9) (8, 87) (6, 9) (6, 9) 6 Time spent studing 6 Time spent studing Slide 9 () / 67 Scatter Plot What connection do ou Test Score vs. Time Spent see between linear Studing association and the slope of a line? 9 (, 9) The follow the same (6, 9) direction. Positive rises 9 (, 9) from left to right, and The question on this slide addresses (, 88) negative falls from left MP.. to (8, 87) 8 (, 8) right. click to reveal Math Practice Test Score 8 (, 78) (, 8) Slide / 67 9 What tpe of scatter plot is shown in the graph below? non-linear linear, positive association linear, negative association linear, no association Ice ream Sales ($) Ice ream Sales vs. Temperature 6 Time spent studing Temperature (ºF)

55 Slide () / 67 9 What tpe of scatter plot is shown in the graph below? non-linear linear, positive association linear, negative association linear, no association Ice ream Sales ($) Ice ream Sales vs. Temperature 6 Temperature (ºF) 7 8 Slide / 67 9 What kind of association is shown in the graph? non-linear linear, positive association linear, negative association linear, no association Number of clothing laers worn Number of lothing Laers Worn vs. Temperature Temperature (ºF) 6 8 Slide () / 67 Slide / 67 9 What kind of association is shown in the graph? non-linear linear, positive association linear, negative association linear, no association Number of clothing laers worn Number of lothing Laers Worn vs. Temperature Temperature (ºF) What kind of association is shown in the graph? non-linear linear, positive association linear, negative association linear, no association Height in Inches Shoe Size vs. Height Shoe Size 9 Slide () / 67 Slide / 67 9 What kind of association is shown in the graph? 96 What association is shown in this graph? non-linear linear, positive association linear, negative association linear, no association Height in Inches Shoe Size vs. Height non-linear linear, positive correlation linear, negative correlation linear, no correlation Weight (lbs.) 7 o's Height vs. Weight Shoe Size Height (in.)

56 Slide () / 67 Slide / What association is shown in this graph? non-linear linear, positive correlation linear, negative correlation linear, no correlation Weight (lbs.) 7 o's Height vs. Weight 97 Which of the following scenarios would produce a linear scatter plot with a positive association? Miles driven and mone spent on gas Number of pets and how man shoes ou own Work eperience and income Time spent studing and number of bad grades 6 6 Height (in.) Slide () / 67 Slide / Which of the following scenarios would produce a linear scatter plot with a positive association? Miles driven and mone spent on gas Number of pets and how man shoes ou own Work eperience and income Time spent studing and number of bad grades 98 Which of the following would have no association if plotted on a scatter plot? Number of tos and calories consumed in a da Number of books read and reading scores Length of hair and amount of shampoo used Person's weight and calories consumed in a da Slide () / Which of the following would have no association if plotted on a scatter plot? Number of tos and calories consumed in a da Number of books read and reading scores Length of hair and amount of shampoo used Person's weight and calories consumed in a da Notice that the points form a linear like pattern. To draw a line of best fit, use two points so that the line is as close as possible to the data points. Our line is drawn so that it fits as close as possible to the data points. The number of points above and below the line should be about the same. There are points above our line and points below our line. This line was drawn through (, 8) and (, 9). Slide 6 / 67 raw a Line Test Score Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) (, 8) (, 9) (8, 87) 6 Time spent studing (6, 9)

57 Using the line of best fit shown in the graph: Predict the test score of someone who spends minutes studing. Predict the test score of someone who spends 8 minutes studing. Slide 7 / 67 Scatter Plot Test Score Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) (, 8) (, 9) (8, 87) (6, 9) Slide 7 () / 67 Scatter Plot Test Score vs. Time Spent Using the line of best fit Studing minutes = 8 or 8 shown in the graph: 9 (, 9) Predict the test score 8 of minutes = 9 someone who spends minutes studing. 9 (, 9) & Math Practices (, 88) (8, 87) Predict the test score of 8 (, 8) someone who These spends questions 8 address MP., minutes studing. MP.7 and MP.8 (, 8) 8 Test Score (, 78) (6, 9) 6 Time spent studing 6 Time spent studing Use the two points that formed the line to write an equation for the line. Slide 8 / 67 Prediction Equation Test Score Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) (, 8) (, 9) (8, 87) (6, 9) Slide 8 () / 67 Prediction Equation Test Score vs. Time Spent Studing Use the two points This that slide and the net slide address formed the line to MP., write MP. an & MP.6. 9 (, 9) equation for the line. dditional Questions to address MP's: (6, 9) What is the problem 9 asking? (MP.) (, 9) What information do ou know? (, 88) (MP.) (8, 87) What other resources could help ou 8 (, 8) solve this problem? (MP.) id ou use the most efficient (, wa 8) to solve the problem? 8 (MP.6) Math Practice Test Score (, 78) 6 Time spent studing 6 Time spent studing Use the two points that formed the line to write an equation for the line. Find the slope 9 8 m = 8 m = Slide 9 / 67 Prediction Equation Test Score Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) (, 8) (, 9) (8, 87) 6 Time spent studing (6, 9) Use the two points that formed the line to write an equation for the line. Find the equation of the line - 9 = 8 ( - ) - 9 = 8 - = Where is the score for minutes of studing. This equation is called the Prediction Equation. Slide / 67 Prediction Equation Test Score Test Score vs. Time Spent Studing 9 (, 9) (, 88) (, 78) (, 8) (, 8) (, 9) (8, 87) 6 Time spent studing (6, 9)

58 Slide / 67 Etrapolation Prediction Equations can be used to predict other related values. Slide / 67 Interpolation If a person studies minutes, what would be the predicted score? = If a person studies minutes, what would be the predicted score? = 8 () 9 7. S = 8 9 () 8.7 This is an interpolation, because the time was inside the range of the original times. This is an etrapolation, because the time was outside the range of the original times. Slide / 67 What is Wrong? Interpolations are more accurate because the are within the set. The farther points are awa from the data set the less reliable the prediction. Using the same prediction equation, consider: If a person studies minutes, what will be there score? Slide () / 67 What is Wrong? Interpolations are more accurate because the are within the set. The farther points are awa from the data set the less reliable the prediction. score cannot be more Using the same prediction equation, than %, consider: so 7.% is not a possible score. If a person studies minutes, what will be there score? S = 8 () What is wrong with this prediction? S = 8 () What is wrong with this prediction? Slide / 67 What is the Prediction? If a student got an 8 on the test, What would be the predicted length of their stud time? 8 = = 8. = The student studied about minutes. raw the line of best fit for our data. etermine the equation for the line of best fit. Predict the height of a person who wears a size 8 shoe. Predict the shoe size of a person who is inches tall. Slide / 67 Shoe Size vs. Height Height in Inches Shoe Size vs. Height (6, 6) (6., 67) (., ) 6 (6., 9) 7 8 Shoe Size 9 (9., 77) (8., 66) (9., 7) (8., 7)

59 Slide () / 67 Shoe Size vs. Height Shoe Size vs. Height raw the line This of slide best and fit for the net slides address (9., 77) our data. MP., MP. & MP.6, MP.7 & MP.8. 7 (9., 7) etermine dditional the equation Questions to address MP's: for the line What of best is the fit. problem asking? 7 (MP.) (8., 7) (6., 67) What information do ou know? (MP.) Predict the What height other of a resources could 6 help ou (8., 66) person who solve wears this a problem? size (MP.) (6, 6) 8 shoe. id ou use the most efficient 6 wa to solve the problem? (MP.6) (6., 9) Predict the o shoe ou size see of a pattern? a (MP.7) (., ) person who an is ou inches use the pattern to predict the net value? (MP.8) tall Math Practice Height in Inches Shoe Size raw the line of best fit for our data. - lick on the graph to reveal the line of best fit. etermine the equation for the line of best fit. Find the slope 7 - = = slide to reveal Find the equation of the line - = ( -.) - = - 7. = + 7. slide to reveal Slide 6 / 67 Shoe Size vs. Height Height in Inches Shoe Size vs. Height (6, 6) (6., 67) (., ) 6 (6., 9) 7 8 Shoe Size (8., 7) (8., 66) 9 (9., 77) (9., 7) Predict the height of a person who wears a size 8 shoe. = (8) + 7. = 67. inches slide to reveal Predict the shoe size of a person who is inches tall. = =. = slide to reveal Slide 7 / 67 Shoe Size vs. Height Height in Inches Shoe Size vs. Height (6, 6) (6., 67) (., ) 6 (6., 9) 7 8 Shoe Size (8., 7) (8., 66) 9 (9., 77) (9., 7) Slide 8 / onsider the scatter graph to answer the following: Which points would give the best line of fit? and and and there is no pattern 8 6 (, 9) (., 8) 6 8 X Y 9. 8 (, 7) 7 (6, ) 6 (8, ) 8 (9, ) 9 (, ) Slide 9 / 67 onsider the scatter graph to answer the following: Which points would give the best line of fit? and and and there is no pattern X Y Slide / 67 onsider the scatter graph to answer the following: What is the slope of the line of best fit going through and? 8 6 (, 9) 6 8 (9, ) X Y

60 Slide / 67 onsider the scatter graph to answer the following: What is the -intercept of the line of best fit going through and? (, 9) (9, ) X Y Slide / 67 onsider the scatter plot to answer the following: Using the line of best fit shown in the graph below, what would the prediction be if = 7? Is this an interpolation or etrapolation?, interpolation, etrapolation 6, interpolation 6, etrapolation 8 6 (, 9) (9, ) Slide () / 67 onsider the scatter plot to answer the following: Using the line of best fit shown in the graph below, what would the prediction be if = 7? Is this an interpolation or etrapolation?, interpolation, etrapolation 6, interpolation 6, etrapolation 8 6 (, 9) 6 8 (9, ) Slide / 67 onsider the scatter graph to answer the following: Using the line of best fit shown in the graph below, what would the prediction be if =? Is this an interpolation or etrapolation?, interpolation, etrapolation, interpolation, etrapolation 8 6 (, 9) 6 8 (9, ) Slide () / 67 onsider the scatter graph to answer the following: Using the line of best fit shown in the graph below, what would the prediction be if =? Is this an interpolation or etrapolation?, interpolation, etrapolation, interpolation 8 6 (, 9) Slide / 67 onsider the scatter graph to answer the following: Using the line of best fit shown in the graph below, what would the prediction be if =? Is this an interpolation or etrapolation?, interpolation, etrapolation, interpolation 8 6 (, 9), etrapolation (9, ), etrapolation (9, )

61 Slide () / 67 onsider the scatter graph to answer the following: Using the line of best fit shown in the graph below, what would the prediction be if =? Is this an interpolation or etrapolation?, interpolation, etrapolation, interpolation, etrapolation 8 6 (, 9) (9, ) Slide / 67 6 In the previous questions, we began b using the table at the right. Which of the predicted values (7,) or (, ) will be more accurate and wh? (7, ); it is an interpolation (7, ); there alread is a and a 7 in the table (, -) it is an etrapolation (, -); the line is going down and will become negative X Y Slide 6 / 67 Slide 7 / 67 PR Sample Questions andle Lab Students measure the height of a burning candle, graph the data and find the line of best fit. The remaining slides in this presentation contain questions from the PR Sample Test. fter finishing this unit, ou should be able to answer these questions. Good Luck! Return to Table of ontents Slide 8 / 67 7 Which points are on the graph of the equation = -7? Select all that appl. (-, 6) (-, ) Slide 8 () / 67 7 Which points are on the graph of the equation = -7? Select all that appl. (-, 6) (-, ) (, -) (6, -) (, -) (6, -) and E E (8, ) E (8, ) From PR P sample test non-calculator #7 From PR P sample test non-calculator #7

62 Slide 9 / 67 8 The graph of the function f() = - +. is shown on the coordinate plane. For what value of does f() =? Slide 9 () / 67 8 The graph of the function f() = - +. is shown on the coordinate plane. For what value of does f() =? From PR P sample test non-calculator # From PR P sample test non-calculator # Slide / 67 9 Graph the equation 6 - = on the -coordinate plane. Identif the -intercept of the graph and the -intercept of the graph. When ou finish, tpe in the number that represents the -intercept. Slide () / 67 9 Graph the equation 6 - = on the -coordinate plane. Identif the -intercept of the graph and the -intercept of the graph. When ou finish, tpe in the number that represents the -intercept. From PR P sample test non-calculator #7 From PR P sample test non-calculator #7 Slide / 67 onsider the three points (-, -), (, ), and (8, 6). Slide () / 67 onsider the three points (-, -), (, ), and (8, 6). Part Part Graph the line that passes through these three points on the coordinate plane. When ou finish graphing, tpe in the number that represents the -intercept. Graph the line that passes through these three points on the coordinate plane. When ou finish graphing, tpe in the number that represents the -intercept. From PR P sample test calculator # From PR P sample test calculator #

63 Slide / 67 onsider the three points (-, -), (, ), and (8, 6). Part Use the graph in Part to eplain wh the ratio of the -coordinate to the -coordinate is the same for an point on the line ecept the -intercept. Eplain wh this is not true for the -intercept. When ou finish writing our answer, tpe in the number "" using our SMRT Responder. Slide () / 67 onsider the three points (-, -), (, ), and (8, 6). Part ample Student Response: Use the graph in Part to eplain wh the ratio he graph of passes the -coordinate through the origin, to the so if-coordinate is the same, ) is a point on the line, then the slope can e represented for an b point - which on the is the line same ecept as the -intercept. - he ratio of Eplain the coordinate wh this values. is not ecause true for the -intercept. he slope is constant, the ratio is the same for When ou finish writing our answer, tpe in the ll points on the line, with the eception of he -intercept number which "" is (, using ). The our -intercept SMRT Responder. he origin) does not work because divided is undefined. From PR P sample test calculator # From PR P sample test calculator # Slide / 67 onsider the three points (-, -), (, ), and (8, 6). Part o the points on the line = - have a constant ratio of the -coordinate to the -coordinate for an point on the line ecept the -intercept? Eplain our answer. When ou finish writing our answer, tpe in the number "" using our SMRT Responder. Slide () / 67 onsider the three points (-, -), (, ), and (8, 6). Part Sample Student Response: o the points on the line = - have a constant ratio of The the equation -coordinate = to - the has -coordinate a for an point on -intercept the line ecept of -, so the the -intercept? line Eplain our answer. will not pass When through ou finish the writing our answer, tpe in origin. the number s a result, "" using the our line SMRT Responder. will not have the same propert as in Part. From PR P sample test calculator # From PR P sample test calculator # Slide / 67 The ordered pairs (, -9.), (, -) and (, -.) are points on the graph of a linear equation. Graph the line that shows all of the ordered pairs in the solution set of this linear equation. fter ou finish graphing the line, tpe in the number "" in our SMRT Responder. Slide () / 67 The ordered pairs (, -9.), (, -) and (, -.) are points on the graph of a linear equation. Graph the line that shows all of the ordered pairs in the solution set of this linear equation. fter ou finish graphing the line, tpe in the number "" in our SMRT Responder. m = -. [This object = is a -. pull tab] +. n points in the plane could be used, e: (, -) & (, -) From PR EOY sample test calculator # From PR EOY sample test calculator #

64 Slide / 67 Slide () / 67 Glossar and Standards Teacher Notes Vocabular Words are bolded in the presentation. The tet Glossar bo the and word Standards is in is then linked to the page at the end of the presentation with the word defined on it. Return to Table of ontents Return to Table of ontents Slide 6 / 67 onstant Rate of hange rate that describes how one quantit changes in relation to another. This rate never changes. 6 miles hour $. gallon Slide 7 / 67 irect Variation relationship between two variables in which one is a constant multiple of the other. When one variable changes the other changes in proportion to the first. = m Goes through the origin. (,) ack to Instruction ack to Instruction Slide 8 / 67 Etrapolation data point that is outside the range of data. Slide 9 / 67 Grade unit engineers use to measure the steepness of a hill. Test Score (, 9) (, 8) 6 Time spent studing If ou stud for minutes, what would be our predicted score? = 8 () + = < 8 9 If ou stud for 6 minutes, what would be our predicted score? = 8 (6) + 9 = > 9 feet feet =. = % grade of hill is %. meters meters grade of hill is = % The sign warns cars the hill has a grade of 7%. ack to Instruction ack to Instruction

65 Slide / 67 Horizontal Line line whose direction is left and right. ll of the -coordinates on the line are equal. Slide / 67 Interpolation data point that is inside the range of data. = (, ) (, ) (-, ) = = - = Test Score (, 9) (, 8) 6 Time spent studing If ou stud for minutes, what would be our predicted score? = 8 () + = < 8.6 < 9 If ou stud for minutes, what would be our predicted score? = 8 () + = < 9.6 < 9 ack to Instruction ack to Instruction Slide / 67 Line of est Fit line on a graph showing the general direction that a group of points seem to be heading. Trend line. Slide / 67 Negative Slope When a line falls down from left to right negative slope m = - a b = - = a b a -b ack to Instruction ack to Instruction Slide / 67 Point-Slope Form The point-slope equation for a line is - = m ( - ) where m is the slope and (, ) is a point on the line. ) Plot (, ) - - = ( - ) (, ) - - = ( - ) m = - : down right Slide / 67 Positive Slope When a line rises from left to right. positive slope m = a b = -a -b ack to Instruction ack to Instruction

66 Slide 6 / 67 Prediction Equation n equation that is created using the line of best fit. line that can predict outcomes using the given data. Slide 7 / 67 Proportional Relationship When two quantities have the same relative size. Test Score (, 9) (, 8) 6 Time spent studing 9 8 m = 8 m = - 9 = 8 ( - ) - 9 = 8-8 = If ou stud for minutes, what would be our predicted score? = 8 () + = if weight is proportional to age, then a weight of kg on the st da means it will weigh 6kg on the nd da, 9kg on the rd da, kg on the th da, etc. ack to Instruction ack to Instruction Slide 8 / 67 Scatter Plot graph of plotted points that show the relationship between two sets of data. Slide 9 / 67 Slope How much a line rises or falls. Steepness of a line. The ratio of a line's rise over its run. 7 = m + b "m" = slope or how the line "moves" "Steepness" and "Position" of a Line formula for slope: m = - - ack to Instruction ack to Instruction Slide 6 / 67 Slope-Intercept Form One tpe of straight line equation that utilizes the slope and -intercept to graph. Slide 6 / 67 Standard Form Standard form looks like + =, where,, and are integers and >. = m + b slope "moves" -int: (,b) where the line "begins" = m = -int: (,) "down, right " - + = -6 - = -6 = (,) + = -6 = -6 = - (,-) (,) (,-) ack to Instruction ack to Instruction