o Checkpoint Plot the points in the same coordinate plane.

Transcription

1 The ~ al Coordinate Plane Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin x-axls -axis Ordered pair x-coordinate -coordinate Quadrant Scatter plot ~. Lesson 4.. Algebra. Concepts and Skills Notetaking Guide 77

2 Example c Plot Points in a Coordinate Plane Plot the points A( -2, ), 8(, -4), and C(O, -2) in a coordinate plane. To plot the point A( - 2, ), start at the Move 2 units to the and units -, To plot the point 8(, -4), start at the Move units to the and 4 units To plot the point C(O, - 2), start at the Move units to the and 2 units - - J x f Example 2 dentif Quadrants Name the quadrants the points D( -2, -9) and E(2, 4) are in. The point D(-2, 9 ) is in Quadrant because its x- and -coordinates are both The point (2,4) is in Quadrant because its x- and -coordinates are both o Checkpoint Plot the points in the same coordinate plane.. A(-, -2) 2.8(4,0) -t-r-r-t-r-t-r. C(, 4) 4. D(-,2) - - x r Name the quadrant the point is in. 5. (-6, 7) 7. (6, - 7) 6. (~6, -7) 8.(6, 7) 78 A<gebra Concepts and Skills Notetaking Guide. Chapter 4.-~- - _

3 Example Make a Scatter Plot NCAA Basketball Teams The number of NCAA men's college basketball teams is shown in the table. Year Men's teams 868, a. Make a scatter plot of the data. b. Describe the pattern of the number of men's basketball teams. Solution a. Let M represent. Let t represent - Because ou want to see how the number of teams changed over time, put t on the axis and M on the -- axis. Choose a scale. Use a break in the scale for the number of teams to focus on the values between and NCAA Men's Basketball Teams b. From the scatter plot, ou can see that the number of men's teams in the NCAA was for three ears and then began to Lesson 4.. Algebra Concepts and Skills Notetaking Guide 79

4 Grap ing Linear quat ions Goal Graph a linear equation using a table of values. VOCABULARY Linear equation Solution of an equation Function form Graph of an equation Example Check Solutions of Linear Equations Determine whether the ordered pair is a solution of 2x + = - 6. a. (, -4) b. (-4, ) Solution a. 2x + = - 6 Write original equation. 2( ) + ( ) :b -6 Substitute for x and for. - 6 Simplif. statement. Answer (, -4) a solution of the equation 2x + = -6. b. 2x + = -6 Write original equation. 2( ) + (_) :b - 6 Substitute for x and for. - 6 Simplif. statement. Answer (-4,) a solution of the equation 2x + = Algebra Concepts and Skills Notetaking Guide. Chapter 4

5 o Checkpoint Determine whether the ordered pair is a solution of -2x + Y =.. (0, ) 2. (, ). (, 5) Example 2 Find Solutions of Linear Equations Find three ordered pairs that are solutions of -5x + = -2.. Rewrite the equation in function form to make it easier to substitute values into the equation. - 5x + Y = - 2 Write original equation. = Add to each side. 2. Choose an value for x and substitute it into the, equation to find the corresponding -value. The easiest x-value is = 5( ) - 2 Substitute for x. = Simplif. The solution is -. Select a few more values of x and make a table to record the solutions. Answer, and are three solutions -- of - 5x + Y = - 2. Lesson 4.2. Algebra Concepts and Skills Notetaking Guide 8

6 GRAPHNG A LNEAR EQUATON Step Rewrite the equation in form, if necessar. Step 2 Choose a few values of x and make a - Step Plot the points 'from the table of values. A line through these points is the of the equation. Example Graph a Linear Equation Use a table of values to graph the equation x + 4 = 4.. Rewrite the equation in function form b solving for. x + 4 = 4 Write original equation. 4 = + 4 Subtract from each side. = Divide each side b When graphing a linearequation, tr choosing values of x that include negative values, zero, and positive values to see how the graph behaves to the left and right of the -axis. 2. Choose a few values of x and make a table of values. : : You have found three solutions. (-4, ), (0, ), (4,. Plot the points and draw a line through them. J - - J x r,--:- o Checkpoint Complete the following exercise. 4. Use a table of values to graph the equation x - 2 =. - - J x r 82 Algebra Concepts and Skills Notetaking Guide. Chapter 4 -._... -._. -..._- - _

7 Graphing Horizo tal and Vertical Lines Goal Graph horizontal and vertical lines. VOCABULARY Constant function EQUATONS OF HORZONTAL AND VERTCAL LNES ' = b ~. x x =a ~ n the coordinate plane, n the coordinate plane, the graph of = b is a the graph of x = a is a line. line. -- ' t Example Graph the Equation = b Graph the equation = -. The equation does not have x as a variable. The -coordinate is alwas _._' regardless of the value of x. Some points th at are solutions of the equation are: (-, ), (0, _), and (, _) The graph of = - is a line units the _ ] - -] ] x r], Lesson 4.. Algebra Concepts and Skills Notetaking Guide 8

8 Example 2 Graph the Equation x = a Graph the equation x = 2. Solution The equation does not have as a variable. The x-coordinate is alwas regardless of the value of. Some points that are solutions of the equation are:, -), (, 0), and (, ) Answer The graph of x = 2 is a - line units to the of the - - x r. ~ Example Write an Equation of a Line Write the equation of the line in the graph. a. b. - - x - - x Solut ion a. The graph is a line. The x-coordinate is alwas. The equation of the line is b. The graph is a line. The -coordinate is alwas The equation of the line is - 84 Algebra Concepts and Skills Notetaking Guide. Chapte r 4

9 Checkpoint Complete the following exercises.. Graph the equation x = -.-t f f- f l 2 2. Write the equation of the line in the graph. - -] ] x - -] ] x f--i--+-+,- fi i--+--l-..':-. ~~-l-+--l '" Example 4 Write a Constant Function Tree Trunks The graph shows the diameter of a tree trunk over a 6-week period. Write an equation to represent the diameter of the tree trunk for this period. What is the domain of the function? What is the range? Diameter of a Tree Trunk D 8 7,g 6 a; 5 'ai 4 is E 2 o ~-~ - c r Number of Weeks Solution From the graph, ou can see that the diameter was about inches throughout the 6-week period. Therefore, the diameter D during this time t is D =. The domain is. The range is Lesson 4.. Algebra Concepts and Skills Notetaking Guide 85

10 Graphing Lines Using ntercepts Goal Find the intercepts of the graph of a linear equation and then use them to make a quick graph of the equation. VOCABULARY x-intercept -intercept Example Find x- and -ntercepts Find the x- and-intercepts of the graph of the equation -x + 4 = 2. Solution To find the x-intercept, substitute for and solve for x. - x + 4 = 2 Write original equation. - x + 4( ) = 2 Substitute for. = 2 Simplif. x = Solve for x. To find the -intercept, substitute for x and solve for. -x + 4 = 2 Write original equation. -( ) + 4 = 2 Substitute for x. = 2 Simplif. = Solve for. Answer The x-intercept is. The -intercept is 86 Algebra Concepts and Skills Notetaking Guide. Chapter 4

11 o Checkpoint Complete the following exercise.. Find the x-intercept and the -intercept of the graph of the equation 2x - 5 = 0. The Quick Graph process works because onl two points are needed to determine a line. MAKNG A QUCK GRAPH Step Find the -- Step 2 Draw a coordinate plane that includes the -- Step Plot the and draw a line through them. Example 2 Make a Quick Graph Graph the equation 9x + 6 = 8. Solution. Find the intercepts. 9x + 6 = 8 Write original equation. 9x + 6( ) = 8 Substitute for. x = The x-intercept is.._. 9x + 6 = 8 Write original equation. 9( ) + 6 = 8 Substitute for x. = The -intercept is 2. Draw a coordinate plane that includes the points (_, _) and (_, _).. Plot the points (, ) and - - (_, _) and draw a line through them. [ - - [.l )' x Less on 4.4. Algebra. Concepts and Skills Notetaking Guide 87

12 o Checkpoint Complete the following exercise. 2. Graph the equation - 4x + 5 = 20,, x f2 'J 6 Choose Appropriate Scale.s. When ou make a quick graph, find the intercepts before ou draw the coordinate plane. This will help ou find an appropriate scale on each axis. Solution. Find the intercepts. = 5x + 5 Write original equation. = 5x + 5 Substitute _ for. = 5x Subtract from each side. =x Divide each side b. The x-intercept is = 5x + 5 Write original equation. = 5(_) + 5 Substitute for x. = 2. Draw a coordinate plane that includes the points (, _) and (_, ). With these values, it is reasonable to use tick marks at intervals.. Plot the points (, _) and (_, ) and draw a line through them. Simplif. The -intercept is L: i - - x 88 Algebra Concepts and Skills Notetaking Guide. Chapter 4 - _.. _-_ _

13 T e Slope of a Line Goal Find the slope of a line. VOCABULARY Slope Example The Slope Ratio Find the slope of a ramp that has a vertical rise of feet and a horizontal run of 8 feet. Let m represent the slope. ====:::::::J Vertical Rise = feet Horizontal Run = 8 feet Solution m= Answer The slope of the ramp is n the slope formula, xl is read THE SLOPE OF A LNE The slope m of a line that passes through the points (xi' Yi) and (x2' Y2) is rise as "x sub one" and m run Y2 as " sub two:' change in D change in D Xz - Xl (Xz. z) x Lesson 4.5. Algebra Concepts and Skills Notetaklng Guide 89

14 Example 2 Positive Slope. Find the slope of the line that passes through the points (, 2) and (-2, -).,--. Solution Let (xi' Yi) = (,2) and (x2' Y2) = (-2, -). m= 0-0 Formula for slope = Substitute values. 0-0 = Simplif. - J / " ~ -5 ; - -~ x -, - ) / / v ' Slope is - Answer The slope of the line is. The line from left to -- right. The slope is - Example Zero Slope Find the slope of the line passing through the points ("':"2, -) and (4, -). Solution Let (xi' Yi) = (-2, -) and (x2' Y2) = (4, -). m= 0-0 Formula for slope 0-0 = 0-(0) Substitute values. 0-(0) Simplif. [ - -[ 5 x ( 2, ) (4, - ) 5 Slope is Answer The slope of the line is. The line is Algebra Concepts and Skills Notetaking Guide. Chapter 4

15 Example 4 Undefined Slope Find the slope of the line passing through the points (-, -4) and (-, -2). Solution Let (xi' Yi) = (-, -4) and (x2' Y2) = (-, -2). 0-0 m= Formula for slope (0) Substitute values. 0-(0) D Division b is D Answer Because division b is. The line is ~ x (- -2) (- -4) f-- - 5, the slope is i o Checkpoint Find the slope of the line passing through the points. Then state whether the slope of the line is positive, negative, zero, or undefined.. (-5,2), (7, -2) 2. (0, 0), (-9, 0). (-7, -8), (-7, 8) 4. (2, -4), (8, 6) Lesson 4.5. Algebra Concepts and Skills Notetaklng Guide 9

16 Direct Variation Goal Write and graph equations that represent direct variation. VOCABULARY Direct variation Constant of variation Write a Direct Variation Model The model for directvariation = kx is read as " varies directl with x:' The variables x and var directl. One pair of values is x = 7 and = 2. a. Write an equation that relates x and. b. Find the value of when x = 4. Solut ion a. Because x and var, the equation is of the form - = kx Write model for direct variation. = k( ) Substitute for x and for. = k Divide each side b Answer An equation that relates x and is b. Y = (_) Substitute for x. = Simplif. Answer When x = 4, Y = ~., 92 Algebra Concepts and Skills Notetaking Guide. Chapter

17 PROPERTES OF GRAPHS OF DRECT VARATON MODELS The graph of = kx is a line through the The slope of the graph of = kx is x k is negative. k is positive. Example 2 Graph a Direct Variation Model Graph the equation = -x.. Plot a point at the 2. Find a second point b choosing an value of x and substituting it into the equation to find the corresponding -value. Use the value - for x. = - x Write original equation. = -( Substitute for x. = - Simplif. The second point is (, _).. Plot the second point and draw a line. through the and the second point [ [ x o Checkpoint The variables x and var directl. Use the given values to write an equation that relates x and.. x = 6, = 0 2. x = 8, Y = x =.6, Y =.8.~ Lesson 4.6. Algebra conceptsand Skills Notetaking Guide 9 --_._~

18 Graphing Lines Using Slope-ntercept Form Graph a linear equation in slope-intercept form. VOCABULARY Slope-intercept form Parallel lines SLOPE-NTERCEPT FORM OF THE EQUATON OF A LNE The linear equation = mx + b is written in slope-intercept form, where is the slope and is the -intercept. slope t = x + -intercept t Example Find the Slope and-ntercept Find the slope and -intercept of -x - = 2. Solution Rewrite the equation in slope-intercept form. - x - = 2 Write original equation. - = + 2 -Add to each side. Answer The slope is = Divide each side b m = and b =. The -intercept is 94 Algebra Concepts and Skills Notetaking Guide. Chapter 4 --_. _._

19 Example 2 Graph an Equation in Slope-ntercept Form Graph the equation =2x -.. Find the slope, _' and the -intercept, 2. Plot the point (0, b) when b is. Use the slope to locate a second point on the line. [ - -[ x [ o m= D = --7 rise run move D units up move D unit right 4. Draw a line through the two points. o Checkpoint Find the slope and -intercept of the equation.. = 4 - x 2.2x + Y = -.4 = x 8 Graph the equation in slope-intercept form. 4. = x tj+-+l f t[ f--f ~ S. = --x + 4 ft+-+-+ t--l , - -[ f [ x x -t-+--t--.., Lesson 4.7. Algebra Concepts and Skills Notetaking Guide 95 --~--~._-~~-- -_.. - _._

20 Example dentif Parallel Lines Which of the following lines are parallel? line a: - 2x + = line b: 2x + Y = - line c: 2x - = Solution. Rewrite each equation in slope-intercept form. line, a: = line b: = line c: = - 2. dentif the slope of each equation. The slope of line a is. The slope of line b is. The slope of line c is. Compare the slopes. Lines and are parallel because each has a slope of Line is not parallel to either of the other two lines because it has a slope ~ Check The graph gives ou a visual check. t shows that line b -- each of the two parallel lines. Answer Lines and- are parallel. l..,-_..z- r \ )' -jri-. \ b 78 e - f - \c-- / J \ V 0 J - -/\ / x J ~/ / f\ f--ll /"~ l-- t \ / " o Checkpoint Which of the following lines are parallel? 6. line a: 4x - = 6 line b: -8x + 6 = 8 line e: 4x + = 8 96 Algebra Concepts and Skills Notetaking Guide. Chapter 4

21 VOCABULARY Relation Function Function notation Linear function Example dentif Functions Decide whether the relation is a function. f it is a function, give the domain and the range. a. nput Output b. nput Output Solution a. The relation a function because - b. The relation a function. For each there is. The domain of the function is. The range is -- Lesson 4.8. Algebra Concepts and Skills Notetaking Guide 97

22 VERTCAL LNE TEST FOR FUNCTONS A graph is a function if no line intersects the graph at ': point. - rlf - " i x +! x! i Example 2 Use the Vertical Line Test Use the vertical line test to determine whether the graph represents a function. a. b. x x Solution a. vertical line can be drawn to intersect the graph more than once. The graph -- b. vertical line can be drawn to intersect the graph more than once. The graph Algebra Concepts and Skills Notetaking Guide. Chapter

23 o Checkpoint Decide whether the relation is a function. f it is a function, give the domain and the range.. nput Output 2. nput Output Use the vertical line test to determine whether the graph represents a function.. 4. x x Evaluate a Function You don't have to use f to name a function. Just as ou can use an letter as a variable, ou can use an letterto name a function. Evaluate g(x) = -2x + when x = 4. Solution. g(x) = - 2x + Write original function. g( ) = -2( ) + Substitute for x. Simplif. Answer When x = 4, g(x) = Lesson 4.8. Algebra Concepts and Skills Notetaking Guide 99

24 Example 4 Graph f(x) = 4x - 2. Graph a Linear Function. Rewrite the function as = 2. Find the slope and -intercept. m = and b =. Use the to locate a second.point. 4. Draw a line through the two points. - - j x o Checkpoint Evaluate the function for the given value of the variable. 5. f(x) = - 7x + when x =- 6. f(x) = x 2-5 when x = 2 Graph the linear function. 7. f(x) = -2x + 8. f(x) = 4x - f--t ! x x '- t--r i i 00 Algebra Concepts and Skills Notetaking Guide. Chapter 4

25 Words to Review Give an example of the vocabular word. Coordinate plane Origin x-axis, -axis Ordered pair x-coordinate, -coordinate Linear equation Quadrants Scatter plot Chapter 4. Algebra. Concepts and Skills Notetaking Guide 0

26 Solution of an equation Graph of an equation x-intercept, -intercept Slope Direct variation, constant of variation Slope-intercept form Parallel lines Relation ~ Function notation Linear function Review our notes and Chapter 4 b using the Chapter Review on pages of our textbook. 02 Algebra Concepts and Skills Notetaklng Guide. Chapter 4. _ _. _ - - -