Algebra I Notes Unit Six: Graphing Linear Equations and Inequalities in Two Variables, Absolute Value Functions

Transcription

1 Sllabus Objective.4 The student will graph linear equations and find possible solutions to those equations using coordinate geometr. Coordinate Plane a plane formed b two real number lines (axes) that intersect at a right angle Horizontal Axis x-axis Vertical Axis -axis Ordered Pair the coordinates of a point x, Quadrants the four sections of the plane formed b the axes Quadrant II Quadrant I The Cartesian (coordinate) Plane: - - x - Origin the point 0,0 Quadrant III Quadrant IV - Plotting Points on the Coordinate Plane Ex: Plot the points A,, B 4,, and 0,6 C. Note: Point A is in Quadrant I, B is in Quadrant II, and C is on the -axis. - - x - Solution of a Linear Equation in Two Variables an ordered pair, x,, that makes the equation true. - Graph of a Linear Equations in Two Variables the set of all points, x,, that are solutions to the equation. The graph will be a line. Ex: Is the point 4, a solution to the linear equation x? Substitute the x- and -coordinates of the point into the equation true Because the values satisf the equation, the ordered pair is a solution. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

2 Graphing a Linear Equation To graph a linear equation, we will choose values for the independent variable, x, and substitute these values into the equation to find the corresponding values for the dependent variable,. It is helpful to organize the ordered pairs in a table (or t-chart ). Ex: Graph the equation x 4. Step One: Solve the equation for. x 4 x x x4 Step Two: Choose at least values for x. Include 0, two positive integers, and two negative integers. We will choose,, 0,,. Step Three: Make a table and substitute each x-value into the equation to find the corresponding -value. x x 4 4 Step Four: Plot each ordered pair from the table and connect to form a line. - - x Ex: Graph the linear function 4 x. Step One: Solve the equation for. 4 x x Step Two: Choose at least values for x. Include 0, two positive integers, and two negative integers. Because we will be multipling x b the fraction, we will choose x-values that are multiples of. 4,,0,,4 Page of 9 McDougal Littell: , 4.6, 4.7, 6.

3 Teacher Note: Explain to students that it is easier to evaluate when we use x-values that are multiples of the denominator. Step Three: Make a table and substitute each x-value into the equation to find the corresponding -value. Step Four: Plot each ordered pair from the table and connect to form a line. - - x - x x Ex: Graph the line. This equation is alread solved for, so we will simpl make a table. x 0 Note that the value is alwas, regardless of the value of x. Plotting the points and drawing the line, we see that the result is a horizontal line. - x - Ex: Graph the line x 4. This equation has no, and the x-coordinate must equal. Because is not restricted as part of the equation, we can choose an value for in our table. Plotting the points and drawing the line, we see that the result is a vertical line. x Note: This is the graph of a line, but it is not a function. - x - Page of 9 McDougal Littell: , 4.6, 4.7, 6.

4 Equation of a Vertical Line: x a, where a is a real number Equation of a Horizontal Line: b, where b is a real number You Tr:. Use the linear equation x. Is the ordered pair, 4 a solution to the equation? Find five points that are solutions to the equation and graph the line.. Graph the lines x and. At what point do these two lines intersect? QOD: Wh isn t a vertical line a function? Page 4 of 9 McDougal Littell: , 4.6, 4.7, 6.

5 Sllabus Objective. The student will determine the x- and -intercepts of a line. x-intercept the x-coordinate of the point where the graph intersects the x-axis (Note: The -coordinate of the x-intercept is 0.) -Intercept the -coordinate of the point where the graph intersects the -axis (Note: The x-coordinate of the -intercept is 0.) Ex: Find the x- and -intercept of the graph of the equation x 9. Finding the x-intercept: Let 0 and solve for x. Finding the -intercept. Let x 0 and solve for. x 0 9 x 9 9 x 09 9 Note: The line crosses the x-axis at the point Sketching the Graph of a Line Using Intercepts 9,0 and the -axis at the point 0,. Ex: Sketch the graph of the line x. To sketch a line, we onl need two points. We will use the intercepts as the two points. Step One: Find the x-intercept. Step Two: Find the -intercept. x 0 x x Step Three: Plot the intercepts on the axes and draw the line. - - Ex: Sketch the graph of the line f x x. 4 To sketch a line, we onl need two points. We will use the intercepts as the two points. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

6 Step One: Find the x-intercept. 0 x 4 x 4 8 x Step Two: Find the -intercept x Step Three: Plot the intercepts on the axes and draw the line. Note: Use a friendl scale to view the graph Application Problem: Using a Linear Model Ex: Adult tickets to a football game cost $, and student tickets cost $. A school collects $7 at Frida night s game. Write an equation and draw a line that represents the possible number of adult tickets, a, and student tickets, s, that were sold at the game. Write the equation: as 7 Find the intercepts: Draw the graph: 0 s 7 s a 70 a 0 7 a s Note: Ever whole-number point on the graph of the line are ordered pairs that represent the possible number of adults and students who purchased a ticket for the football game. You Tr: Find the x- and -intercepts of the graph of the equation 0.x 6. Graph the line. QOD: Describe the graph of a line that has no x-intercept. Write an equation of a line with no x-intercept. Page 6 of 9 McDougal Littell: , 4.6, 4.7, 6.

7 Sample CCSD Common Exam Practice Question(s):. What are the x- and -intercepts of 9x? A. x-intercept = 9; -intercept = B. x-intercept = ; -intercept = C. x-intercept = ; -intercept = D. x-intercept = ; -intercept = 9. Graph the equation: 6x. Page 7 of 9 McDougal Littell: , 4.6, 4.7, 6.

8 Sllabus Objective. The student will determine the slope of lines using coordinate geometr and algebraic techniques. Slope of a Line: the number of units the line rises or falls for each unit of horizontal change. horizontal change rise change in m vertical change run change in x x x x x two points on the line Note: Slope is a rate of change. It determines the steepness of a line. where, and, x x are Finding the Slope Given the Graph of a Line Ex: Find the slope of the line shown in the graph. Step One: Find two points on the line. - - x x - - Step Two: Start at one of the points. Count how man units up or down ou would have to step to get to the other point. This is our rise. Note: If ou go down, the rise is negative; if ou go up, the rise is positive. Starting at the point on the left, we would have to step down units, therefore the rise is. Step Three: Now determine how man units right or left ou would have to step to get to the other point. This is our run. Note: If ou go left, the run is negative; if ou go right, the run is positive. We would have to step right 4 units, therefore the run is +4. Step Four: Write the slope as a fraction, rise run. The slope is 4 On Your Own: If ou start at the point on the right, will ou still get the same slope? Tr it! Page 8 of 9 McDougal Littell: , 4.6, 4.7, 6.

9 Ex: Find the slope of the line shown in the graph. Step One: Find two points on the line. - - x x - - Step Two: Start at one of the points. Count how man units up or down ou would have to step to get to the other point. This is our rise. Note: If ou go down, the rise is negative; if ou go up, the rise is positive. Starting at the point on the left, we would have to step up units, therefore the rise is+. Step Three: Now determine how man units right or left ou would have to step to get to the other point. This is our run. Note: If ou go left, the run is negative; if ou go right, the run is positive. We would have to step right 4 units, therefore the run is +. Step Four: Write the slope as a fraction, rise run. The slope is On Your Own: If ou start at the point on the right, will ou still get the same slope? Tr it! Note: If a car was driving from left to right on the lines in the examples above, it would be going downhill on the line with the negative slope and uphill on the line with the positive slope. Finding the Slope of a Line Given Two Points Ex: Find the slope of the line that passes through the points, and, 4. We will use the slope formula m x x, where x, =, and x,, 4. Substituting into the formula: m 7 On Your Own: If ou change the order of the points, will ou get the same slope? Tr it! Page 9 of 9 McDougal Littell: , 4.6, 4.7, 6.

10 Review: Find the quotients. 0? 4? 0 Note: because So to find 0, we need to find a number,?, such that 0?. This is impossible, so we sa that 0 is undefined or does not exist. Special Slopes Slopes of Horizontal and Vertical Lines Ex: Find the slope of the line in the graph. - - x - Step One: Find two points on the line. (See graph.) - Step Two: Start at one of the points. Count how man units up or down ou would have to step to get to the other point. This is our rise. Note: If ou go down, the rise is negative; if ou go up, the rise is positive. Starting at the point on the right, we would have to step down 0 units, therefore the rise is 0. Step Three: Now determine how man units right or left ou would have to step to get to the other point. This is our run. Note: If ou go left, the run is negative; if ou go right, the run is positive. We would have to step right 4 units, therefore the run is 4. Step Four: Write the slope as a fraction, rise run. The slope is Note: All horizontal lines have a slope of 0. If a car was driving from left to right, it would be going neither uphill nor downhill. Therefore, the slope is neither positive nor negative. Ex: Find the slope of the line passing through the points,6 and,. We will use the slope formula m x x, where, =,6 and,, x x. Substituting into the formula: 6 4 m, so the slope is undefined. 0 Note: The points in the example above will graph a vertical line. The slope of a vertical line is undefined. If a car tried to drive on a vertical line it would crash! Page of 9 McDougal Littell: , 4.6, 4.7, 6.

11 Application of Slope: Rates of Change Ex: In 970, the price of a movie ticket at a particular theater was $.0. In 990 the price of a movie ticket at the same theater was $6.00. What is the rate of change of the cost of a movie ticket? Use correct units in our answer. Using the slope formula, we will find the change in the cost of a ticket over the change in time. You Tr: change in cost ($) dollars per ear change in time (ears) Find the slope of the line passing through the points 4, 9 and 8,. Find the slope of the line shown in the graph. - - x - - QOD: Show algebraicall using the slope formula wh the slope of a horizontal line is 0 and the slope of a vertical line is undefined. Challenge: Find the value of x if the slope of the line passing through the points 4, and x, is. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

12 Sample CCSD Common Exam Practice Question(s):. Which statement is true about the characteristics of the linear functions in the graph below? A. The are the same function. B. The have the same slope. C. The have the same x-intercepts. D. The have the same -intercepts.. Find the slope of the line in the graph. A. B. 4 4 C. 4 D. 4 Page of 9 McDougal Littell: , 4.6, 4.7, 6.

13 . Find the slope of the line that contains the points (7, 6) and (4, ). A. B. C. D. 4. Find the slope of the line that contains the points (8, ) and (8, 8). A. B. 0 C. 6 D. undefined. A job pas a base salar of $0,000. Each ear an emploee will earn an additional $,000. The graph of an emploee s salar over ears is shown below. Emploee Salar over Time Annual Salar ($) Number of Years What would happen to the graph if the base salar was changed from $0,000 to $40,000? A. The graph would translate down. B. The graph would translate up. C. The graph would rise less steepl from left to right. D. The graph would rise more steepl from left to right. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

14 Sllabus Objective:. The student will compare characteristics of a given famil of linear functions..4 The student will graph linear equations and find possible solutions to those equations using coordinate geometr. Activit: Graph the following lines on the graphing calculator. Write down the slope and - intercept of each line. Then compare the steepness of the line to the parent function, x. To graph on the calculator, enter the function into the Y= screen. Use a Zoom Standard window.. x Slope: m -intercept = 0. x Slope: m -intercept = steeper than x. 4. x 4 Slope: x Slope: m -intercept = 4 x is steeper m -intercept = x is steeper. 4x Slope: m 4 -intercept = steeper than x 6. x Slope: m -intercept = 0 same steepness as x Write down our conclusions: The coefficient of x is the slope, and the constant term is the -intercept. Slope-Intercept Form of a Linear Equation: mx b, m slope, b = -intercept Ex: Write the equation of the line x in slope-intercept form. Identif the slope and - intercept. To write in slope-intercept form, solve for. x The slope is, and the -intercept is. x Graphing a Line in Slope-Intercept Form:. Plot the -intercept.. Starting at the -intercept, step to the next point on the line using the slope, rise run. Page 4 of 9 McDougal Littell: , 4.6, 4.7, 6.

15 Note: If the slope is POSITIVE ou ma rise up and run right, or ou ma rise down and run left. If the slope is NEGATIVE ou ma rise up and run left, or ou ma rise down and run right. Ex: Graph the line x. 4 Step One: Identif the slope and -intercept. Step Two: Plot a point at the -intercept. m -intercept = 4 Step Three: Step out the slope to plot another point. Do this a couple of times so that ou can draw an accurate line. - - x Note: Because the slope is positive, we can rise up and run right 4, or rise down and run left 4. Draw the line connecting the points x - - Ex: Graph the line f xx. Step One: Identif the slope and -intercept. Step Two: Plot a point at the -intercept. m -intercept = Step Three: Step out the slope to plot another point. Do this a couple of times so that ou can draw an accurate line. - - x - Note: Because the slope is negative, we can rise up and run left, or rise - down and run right. Draw the line connecting the points. - - x - Page of 9 McDougal Littell: , 4.6, 4.7, 6. -

16 Note: Sometimes ou must put the equation in slope-intercept form first. 4 x Ex: Graph the equation of the line using slope-intercept form. Step One: Write the equation in slope-intercept form. x x Step Two: Identif the slope and -intercept. m -intercept = Step Three: Graph the line b plotting the -intercept and then stepping out the slope. - - x - Ex: Graph the equation of the line 0.x using slope-intercept form. - Step One: Write the equation in slope-intercept form. Step Two: Identif the slope and -intercept. 0.x x m -intercept = Step Three: Graph the line b plotting the -intercept and then stepping out the slope. - - x - Ex: Graph the equation of the line x 48x using slope-intercept form. - Step One: Write the equation in slope-intercept form. 6x 9 x Step Two: Identif the slope and -intercept. m -intercept = - - x - Step Three: Graph the line b plotting the -intercept and then stepping out the slope. - Page 6 of 9 McDougal Littell: , 4.6, 4.7, 6.

17 Ex: Find the slope and -intercept of the line. Describe the graph. Slope = 0, -intercept = The graph will be a horizontal line that intersects the -axis at. Application Problems Using Slope-Intercept Form Ex: Jack has $0 in his savings account, and plans to save $ per week. Write an equation in slope-intercept form that represents the amount, A, Jack will have in his account after w weeks. Graph the equation in an appropriate window. Amount = Number of Weeks + 0 A w 0 A w On Your Own: What does the slope and -intercept represent on the graph in relation to the problem? You Tr: Graph the linear equation 8 x 6 using slope-intercept form. QOD: Can ou write the equation of a vertical line in slope-intercept form? Explain. Page 7 of 9 McDougal Littell: , 4.6, 4.7, 6.

18 Sample CCSD Common Exam Practice Question(s):. Use the graph below. What is the equation of the line in the graph? A. x B. x C. x D. x Page 8 of 9 McDougal Littell: , 4.6, 4.7, 6.

19 Sllabus Objective.4 The student will graph linear equations and find possible solutions to those equations using coordinate geometr. Zero (Root) of a Linear Equation: the value of x where the line crosses the x-axis (where = 0). Solving a Linear Equation Graphicall: Rewrite the equation in the form ax b 0. Graph the line ax b. The solution of the equation is the zero of the line. (Note: We are substituting a in place of the zero, so the solution of the equation will be the x-intercept, or the point on the graph when = 0.) Ex: Solve the equation x 8 4graphicall. Check our answer algebraicall. Step One: Rewrite the equation in the form ax b 0. x 84 x 60 Step Two: Graph the line ax b. Graph x 6. Step Three: Find the zero. x - - x - Step Four: Check the answer. x True - Ex: Solve the equation x 4 x graphicall. Check our answer algebraicall. Step One: Rewrite the equation in the form ax b 0. Step Two: Graph the line ax b. Graph Step Three: Find the zero. x 9 x4 x 4 6 True Step Four: Check the answer x 4 x x 0 x. - - x - - Page 9 of 9 McDougal Littell: , 4.6, 4.7, 6.

20 Graphing Calculator Activit: Solving linear equations on the graphing calculator. Ex: Solve the equation 0.8x. graphicall on the calculator. Step One: Rewrite the equation in the form ax b x x.7 Step Two: Graph the line 0.8x.7. Enter the line in the Y= screen.. Step Three: Find the zero using the CALC Menu. Kestrokes: Note: You ma enter in values for the bounds and the guess, or use the left and right arrows. Solution: x. Step Four: Check our answer on the home screen. Your calculator has the value of the zero stored as x. Alternate Method: Ex: Solve the equation x7 x Because it is tedious to rewrite this equation in the form ax b 0 b hand, we will use an alternate method on the graphing calculator. Step One: Graph each side of the equation as two separate lines. x x 7 and. Step Two: Find the point of intersection. This is the solution to the equation x 7 x Kestrokes: Note: The guess should be near the point of intersection. The x- coordinate is the solution. x Page 0 of 9 McDougal Littell: , 4.6, 4.7, 6.

21 Checking this b hand algebraicall, we have Application Problem x7 x True 8 Ex: A small business makes and delivers box lunches. The calculate their average weekl cost C of delivering b lunches using the function C.b 7. Last week their cost was $600. How man lunches did the make last week? Solve algebraicall and graphicall. Because their cost was $600, we will substitute this value in for C. 600.b 7 Solve for b..b 0 b Solving graphicall, we will rewrite the equation as 0.b, graph the line.x, and find the zero. Solution : The business made 0 lunch boxes last week. You Tr: Solve the equation 96x graphicall and check our solution algebraicall. QOD: What are three different terms to describe the graphical solution to a linear equation in the form ax b 0? Page of 9 McDougal Littell: , 4.6, 4.7, 6.

22 Sllabus Objective. The student will graph linear inequalities in two variables and find possible solution sets to those inequalities using coordinate geometr. Linear Inequalit in Two Variables: an inequalit that can be written in one of the following forms ax b c, ax b c, ax b c, ax b c Solutions of a Linear Inequalit: the ordered pairs, x, that make the inequalit true Checking a Solution to a Linear Inequalit Ex: Is the ordered pair, a solution to the linear inequalit x 8? Substitute the ordered pair in for x and in the inequalit. Evaluate and determine if it makes the inequalit true. x False The ordered pair does not make the inequalit true, so it is NOT a solution. Half-Plane: one of the two planes that a line separates the coordinate plane into Graphing a Linear Inequalit ax b c. Graph the line ax b c. Use a dashed line for < or >, a solid line for or.. Choose a test point in one of the half-planes and substitute the ordered pair into the inequalit for x and. If the ordered pair makes the equation true, shade the half-plane that contains our test point. If it does not, shade the other half-plane. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

23 Ex: Graph the linear inequalit x. Step One: Graph the line x. x-intercept =, -intercept = - - x - Note: We will use a solid line, because the inequalit is. - Step Two: Pick a test point in one of the half-planes and test it in the linear inequalit. We will choose 0,0. x 00 False Step Three: Because 0,0is NOT a solution to the inequalit, we will shade the other half-plane. Note: All of the solutions of the linear inequalit are all points on the line and in the shaded region. - - x - - Ex: Graph the linear inequalit x. Step One: Graph the line x. Slope =, -intercept = 0 Note: We will use a dashed line, because the inequalit is <. Step Two: Pick a test point in one of the half-planes and test it in the linear inequalit. We will choose the point 0, x 0 True. 0 Step Three: Because 0, IS a solution to the inequalit, we will shade the half-plane that the point lies in. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

24 Ex: Sketch the graph of x in the coordinate plane. Step One: Graph the line x. Note: We will use a solid line, because the inequalit is. Step Two: Pick a test point in one of the half-planes and test it in the linear - - x - - inequalit. We will choose the point 0,0. x 0 False Step Three: Because 0,0IS NOT a solution to the inequalit, we will shade the other half-plane. - - x - - Application Problem Ex: A basketball team is 8 points behind with minutes left in the game. Write an inequalit that represents the number of -point and -point shots the team could score to earn at least 8 points. Graph the inequalit and give two different numbers of -point and -point shots the team could score. Assign labels: w = number of -point shots, r = number of -point shots r Write an inequalit: wr 8 Graph the inequalit: w-intercept = 9, r-intercept = 6 Test point 0,0 : False w Two possible - and -point combinations: Three -point shots and four -point shots (This corresponds to the ordered pair, 4, which lies on the line.) Six -point shots and three -point shots (This corresponds to the ordered pair 6,, which lies in the shaded region.) You Tr: Graph the linear inequalities.. 9. x 6 QOD: Explain how to determine if an ordered pair is a solution of a linear inequalit. Page 4 of 9 McDougal Littell: , 4.6, 4.7, 6.

25 Sample CCSD Common Exam Practice Question(s): Graph the linear inequalit x. Page of 9 McDougal Littell: , 4.6, 4.7, 6.

26 Sllabus Objective. The student will graph absolute value equations and find possible solutions to those equations using coordinate geometr. Absolute Value Function: a function that can be written in the form a xb c Ex: Use a table of values to graph the function x. x x - - Ex: Use a table of values to graph the function x. x x - Graph of an Absolute Value Function: V-shaped graph of an absolute value function (ma open up or down). Note: The corner point is called the vertex. - Exploration: Graph the following functions on the graphing calculator and compare each graph to the graph of x. Note: On the graphing calculator, use the command ABS for absolute value. This is found in the MATH, NUM menu. To enter the first function below, use the kestrokes: Comparison to x. x Shift left ; Vertex:, 0. x Shift right ; Vertex:,0. x 4 Shift up 4; Vertex: 0,4 4. x Shift down ; Vertex: 0,. x Narrower, slopes are and on each side of the vertex Page 6 of 9 McDougal Littell: , 4.6, 4.7, 6.

27 6. x Wider; slopes are and on each side of the vertex 7. 4 x Narrower and flipped 8. x Wider and flipped Conclusions: Describe the effects of a, b, and c in the equation a xb c on the graph of x (the Parent Function ). a: a Upside-down V (reflected across x-axis), narrower than x a 0 Upside-down V (reflected across x-axis), wider than x 0a Wider than x a Narrower than x b: b 0 Shifts the graph b units to the right (x-coordinate of vertex = b) b 0 Shifts the graph b units to the left (x-coordinate of vertex = b) c: c 0 Shifts the graph c units down (-coordinate of vertex = c) c 0 Shifts the graph c units up (-coordinate of vertex = c) Ex: Without graphing, find the vertex of the absolute value function x. b, so the x-coordinate of the vertex is. c, so the -coordinate of the vertex is. The vertex is,. Ex: Graph the absolute value function x. Transformations on x : reflect over x-axis, wider, shifted up Find the vertex: b 0 and c 0,., so the vertex is a, so the function opens down with a slope of and to the left and right of the vertex. - - x - - Page 7 of 9 McDougal Littell: , 4.6, 4.7, 6.

28 Ex: Graph the absolute value function x. Find the vertex: b and c, so the vertex is,. a, so the function opens up with a slope of and to the left and right of the vertex. - - x - Solving Absolute Value Equations Graphicall - Ex: Solve the equation x 9 graphicall and algebraicall. Graphicall: Graph both sides of the equation on the same coordinate grid. x and 9 Find the point(s) of intersection. The x-coordinates of the points of intersection are the solutions to the equation. - - x - Solutions: x, - Algebraicall: x 9 x x x x x You Tr: Graph the absolute value function f x x4. QOD: Describe graphicall wh an absolute value equation can have two, one, or no solutions. Page 8 of 9 McDougal Littell: , 4.6, 4.7, 6.

29 Sample CCSD Common Exam Practice Question(s): Use the graph below. What is the equation of the function? A. x B. x C. x D. x Page 9 of 9 McDougal Littell: , 4.6, 4.7, 6.