CHAPTER 5: LINEAR EQUATIONS AND THEIR GRAPHS Notes#26: Section 5-1: Rate of Change and Slope

Transcription

1 Name: Date: Period: CHAPTER : LINEAR EQUATIONS AND THEIR GRAPHS Notes#: Section -: Rate of Change and Slope A. Finding rates of change vertical change Rate of change change in x The rate of change is constant in each table and graph. Find the rate of change. Explain what the rate of change means for each situation..).) Cost of Renting a Computer Number of Das Rental Charge $0 $ $0 $0 $0 The slope of a line describes how and in what B. Finding slope Keep in mind that slope of a line is its of change Choose and mark two points on the line Count how man steps up (this is the rise of the hill) Count how man steps across (this is the run of the hill) rise slope m run Lines with a positive slope Lines with a negative slope Lines with undefined slope Lines with zero slope, Chapter

2 Find slope of the line. (Double check for +/- ).) m =.) m =.) m =.) m =.) m =.) m = Without using a graph and given two points:, x, x and Slope = m = x x 0 n n 0 0 undefined Use the formula to find the slope of the lines containing these points..) (, ) (, ) 0.) (-, -) (, -).) (-, ) (, ).) (, -) (-, ).) (, ) (, ).) (, -) (, 0)

3 Each pair of points lies on a line with the given slope. Find x or..) (, ), (x, ) slope = -.) (, ), (, ) slope = Section -: Slope-Intercept Form A. Writing linear equations A linear function is a function that graphs a. -intercept is the coordinate of the point where a line crosses the. Slope-intercept form of a linear equation is, where m stands for and b stands for the. Example: x Slope = m = -intercept = b = Find the slope (m) and -intercept (b) of each line..) x =.) x + =.) -x + = -.) x =.) x = - Write an equation of a line with the given slope and -intercept. Plug m and b into.) m ; b.) m ; b.) m ; b

4 Write the slope-intercept form of the equation for each line. Find the point where the line crosses the -axis. This is the -intercept of the line, or Count rise to the next marked point to find the slope of the line, or run Plug these values into.) 0.).) B. Verifing solutions and graphing linear equations Determine whether the ordered pair lies on the graph of the given equation. Label our point as (x, ) Plug in both values into the equation. If it is true, then the point is a solution..) (, ) x =.) (-, ) x + =.) (-, -) x =.) (, 0) x

5 Lines can be written in either Slope-Intercept form ( = mx + b) or Standard Form (Ax + B = C). You need to know how to convert from one to the other. Converting to Slope-Intercept Form Goal: = mx + b (where m and b are integers or fractions) Converting to Standard Form Goal: Ax + B = C (where A, B, and C are integers and where A is positive) Get alone Reduce all fractions Get x and terms on the left side and the constant term on the right side of the equation Multipl ALL terms b the common denominator to eliminate the fractions If necessar, change ALL signs so that the x term is positive.) Convert to slope-intercept form: x =.) Convert to standard form: x.) Convert to both slope-intercept form and standard form: a) = -(x + ) b.) ( x )

6 Notes #: Sections. and. A. Graphing Lines using the slope and -intercept: - Get alone so the equation is in = mx + b form (m =, b = ) - Graph b first. This point goes on the axis. - Use slope and count rise over run to the next point(s). When ou have at least three points on our graph, then connect the points with a ruler to make a straight line. - Label our graphed line with the original equation Most common errors: Graphing b on the x-axis instead of the -axis Graphing the slope in the wrong direction (e.g. forgetting a negative).) x ( I m alread in slope-intercept form!) m = ( graph me second! Watch the negative!) b = ( graph me first! I go on the -axis!).) x = ( Get me in slope-intercept form first) m = b = x x

7 .) x + = - ( Get me in slope-intercept form first) m = b =.) x = 0 ( Get me in slope-intercept form first) m = b = x Section -: Standard Form A. Graphing Equations Using Intercepts x-intercept is the x-coordinate of the point where a line crosses the. To find the x-intercept, make = 0 and solve for x. -intercept is the -coordinate of the point where a line crosses the. To find the -intercept, make x = 0 and solve for. Find the x- and -intercepts..) x.) x

8 Graphing Lines using the x- and - intercepts. The intercepts are the point(s) where a line intersects the axes of the coordinate plane. - Find the x and intercepts (b setting the opposite variable to zero) - Write these answers as two different points - Graph and connect these points to graph the line - Label the graphed line with the original equation Most common error: Forgetting that the intercepts are two different points and graphing as just one.) x + = x-intercept -intercept (set = 0) (set x = 0) x-int: (, 0) -int: (0, ) x ) x = x-int: (, ) -int: (, ) x

9 .) x = x-int: (, ) -int: (, ) x Special Cases: Graphing Horizontal and Vertical Lines.) x = (This equation describes the line for which ALL points have an x-coordinate of. There are no restrictions on the value of )..) = - (This equation describes the line for which ALL points have a -coordinate of -. There are no restrictions on the value of x). Use the pattern ou found above to complete these sentences: An line in the form x = is a line because it intersects the An line in the form = is a line because it intersects the

10 Use this pattern to graph these lines without a table of solutions..) =.) x = - 0.) = - Write an equation in standard form to describe each situation. Be sure to define our variables..) Two apples and three bananas cost a total of $.0. Seven apples and four bananas cost $.0..) A free lance photographer makes $0 per photograph that is published in the newspaper and $00 per photograph that is published in a magazine. The photographer needs to earn $00. Write an equation describing the photographer s next week..) Write an equation in standard form to find the number of minutes someone who weights 0 lb would need to biccle and swim laps in order to burn 00 Calories. Use the fact that a 0 lb person burns 0 Calories per minute riding a bike and Calories per minute swimming laps. 0

11 Notes #: Section -: Point-Slope and Writing Linear Equations A. Point Slope Form Point-Slope form of the equation of a non-vertical line that passes through point (x, ) and has slope m is: Your textbook emphasizes point-slope form, but we will not use it in this class. If an equation is given in point slope form, alwas convert it into = mx + b form. Graph each equation..) (x ).) (x ) x x

12 B. Writing linear equations given the slope and -intercept - Find the slope (m) and -intercept (b) [If the given information is a graph, then ou will have to count b hand to find these values.] - Fill in m and b so ou have an equation of the line in = mx + b form. = x + ( Put m here!) ( Put b here!).) Find the equation of the line with slope of and - intercept of -. Write in standard form..) Find the equation of the given line in slope-intercept form..) Write the equation of a line that has the same slope as x and has a - intercept of. Write in standard form. C. Writing linear equations given the slope and a point plug slope = m into = mx + b name our point (x, ) and plug these values in for x and solve for b plug m and b back into = mx + b convert to standard form, if necessar ** Remember to leave x and as variables! **.) Find the equation of the line with slope of - and going through (-, ) in slope-intercept form..) Find the equation of the line with slope of and going through (, -) in standard form.

13 .) Find the equation of the line in slopeintercept form with slope and passing through the point (-, )..) Find the equation of the line with the slope of zero going through the point (-, ) in standard form. D. Writing linear equations given two points find the slope pick one of our points to be x and plug m, x, into = mx + b solve for b; plug m and b into = mx + b convert to standard form, if necessar ** Remember to leave x and as variables! ** 0.) Find the equation of the line going through (-, ) and (, ) in slope-intercept form..) Find the equation of the line with x-intercept and -intercept - in standard form..) Find the equation of the line going through (, ) and (-, ) in standard form..) Find the equation of the line with x- intercept and -intercept - in slope-intercept form.

14 Notes #: Review Graphing Lines using the x- and - intercepts Find the x and intercepts (b setting the opposite variable to zero). These answers represent two different points. Use these points to graph the line.. x + =. x = x-int: (, ) -int: (, ) x-int: (, ) -int: (, ) Graphing Lines using the slope and -intercept:. Get the equation into = mx + b form. Graph b first. This point goes on the axis.. Use slope and count rise over run to the next point(s).. x =. x + = -

15 Vertical and Horizontal Lines Graph each line.. x =. = - Writing Linear Equations Find the slope and -intercept of each line and write an equation of the line in = mx + b form. = x +... Find the equation of the line with slope of and -intercept of -. Write in slopeintercept form. 0.) Find the equation of the line with slope of and -intercept of -. Write in standard form.

16 . Find the equation of the line with slope of - and going through (-, ). Write in slopeintercept form.. Find the equation of the line with slope of and going through (, -).. Find the equation of the line with slope of and going through (, ).. Find the equation of the line going through (-, 0) and (, ) in slope-intercept form.. Find the equation of the line with x-intercept - and -intercept in standard form.. Find the equation of the line going through (-, ) and (-, ) in standard form.

17 Notes#0: Section -: Perpendicular and Parallel Lines A. Determining if lines are parallel or perpendicular Draw two parallel lines: Draw two perpendicular lines: What can ou sa about their slopes? What can ou sa about their slopes? Parallel lines have slope. Examples: Perpendicular lines have slope. Examples: The slope of a line is given. Find the slope of a line parallel to it and a line perpendicular to it..) m.) m.) m =.) m = 0 Determine whether the lines are parallel, perpendicular, or neither. (Hint: find their slopes first).) = x.) x + =.) x = -x + = -½x + x

18 Writing an equation of a line given a parallel or perpendicular line Find Im from the given line b getting alone first If the line is parallel, If the line is perpendicular, Plug m, x, into = mx + b Solve for b Plug m and b into = mx + b (leave x and as variables!) Write the equation of the line that is parallel or perpendicular to the given line and that passes through the given point..) Parallel: = x + ; (, ).) Perpendicular: = x ; (, -) 0.) Parallel: x = ; (, ).) Perpendicular: x = ; (-, )

19 .) Parallel x + = ; (-, ).) Perpendicular: x = ; (-, 0)

20 Notes #: Section -:Solving Sstems b Graphing The solution to a sstem of equations represents the where the. A. Naming the solution from a graph Find the where the lines Name the solution of each sstem of equations:.).) x x B. Verifing whether a point is a solution Plug the point into. (Remember, points come in alphabetical order!) If it works for both lines, then If it does not work for both lines, then Determine whether the given ordered pair (point) is a solution of the sstem of equations..) (, ) for = x +.) (-, ) for a + b = - x + = b + a = 0

21 C. Finding solutions of linear sstems b graphing If the equation is not in terms of x and, rewrite with x s and s Get alone into = mx + b form Graph each line using the -intercept and slope Name the solution (where the lines ) Sketch the three possibilities for our graphed sstem: Lines intersect at Lines do not intersect Lines overlap one point Solve b graphing:.) + x = = -x x

22 .) + x = x = 0.) x = = x x x

23 .) x x x D. Applications.) Suppose ou plan to start taking an aerobics class. Non-members pa $ per class while members pa a $ membership fee plus an additional $ per class. Which sstem of equations models the cost, C(a), as a function of classes, a? A C(a) = a B C(a) = a + C C(a) = a D C(a) = a + C(a) = + a C(a) = + a C(a) = a + C(a) = a + 0.) The sstem below models the cost of taking an aerobics class as a function of the number of classes. Find the solution of the sstem b graphing. What does the solution mean in terms of the situation? C(a) = a C(a) = + a x

24 .) Suppose ou have $00 in our bank account. You start saving $ each week. Your friend has $0 in her bank account but saves $0 each week. After how man weeks will ou and our friend have the same account balance?.) Budget Rent-A-Car charges a dail rental fee of $0 plus $0.0 per mile driven. Avis Rent-A-Car charges a dail rental fee of $ plus $0.0 per mile driven. After how man miles driven would the cost of the two companies be the same?