Name Class Date. Using Graphs to Relate Two Quantities

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1 4-1 Reteaching Using Graphs to Relate Two Quantities An important life skill is to be able to a read graph. When looking at a graph, you should check the title, the labels on the axes, and the general shape of the graph. What information can you determine from the graph? The title tells you that the graph describes Trina s trip. The axes tell you that the graph relates the variable of time to the variable of distance to the destination. In general, the more time that has elapsed, the closer Trina gets to her destination. In the middle of the trip, the distance does not change, showing she stops for a while. What are the variables in each graph? Describe how the variables are related at various points on the graph

2 4-1 Reteaching (continued) Using Graphs to Relate Two Quantities A graph can show the relationship described in a table. Which graph shown below represents the information in the table at the right? Notice that for each additional CD purchased, the total cost increases by $15. The points on the graph should be in a straight line that goes up from left to right. The graph that shows this trend is Graph B. A. B. C. Match each graph with its related table. Explain your answers A. B. C. 10

3 4-2 Reteaching A relationship can be represented in a table, as ordered pairs, in a graph, in words, or in an equation. Patterns and Linear Functions Consider the relationship between the number of squares in the pattern and the perimeter of the figure. How can you represent this relationship in a table, as ordered pairs, in a graph, in words, and in an equation? Table For each number of squares determine the perimeter of the figure. Write the values in the table. Remember to focus on the perimeter of the figure, not the squares. Ordered Pairs Let x represent the number of squares and y represent the perimeter. Use the numbers in the table to write the ordered pairs. Graph Use the ordered pairs to draw the graph. (1, 20), (2, 30), (3, 40), (4, 50), (5, 60) Words The pattern shows the perimeter is the number of squares times 10 plus 10. Equation Write an equation for the words. y = 10x

4 4-2 Reteaching (continued) Patterns and Linear Functions Consider each pattern a. Make a table to show the relationship between the number of trapezoids and the perimeter. a. Make a table to show the relationship between the number of cubes and the surface area. b. Write the ordered pairs for the relationship. c. Make a graph for the relationship. b. Write the ordered pairs for the relationship. c. Make a graph for the relationship. d. Use words to describe the relationship. d. Use words to describe the relationship. e. Write an equation for the relationship. e. Write an equation for the relationship. 20

5 4-4 Reteaching Graphing a Function Rule By finding values that satisfy a function rule, you can graph points and discover the shape of its graph. What is the graph of the function rule y = 3x + 5? First, choose any values for x and find the corresponding values of y. Make a table of your values. Then, graph the points from your table. In this case, the points are in a line. Draw the line. What is the graph of the function rule y = x 2? First, choose any values for x and find the corresponding values of y. Make a table of your values. Then, graph the points from your table. In this case, the points make a V shape. Draw the V. 39

6 4-4 Reteaching (continued) Graphing a Function Rule Graph each function rule. 1. x y y = x 3 3. y = x y = \x\ +1 40

7 Extra Practice Chapter 4 Lesson 4-1 Match each graph with its related table. Explain your answers. 1. A. 2. B. 3. C. Sketch a graph to describe each situation. Label each section of the graph. 4. the number of apples on a tree over one year 5. the amount of milk in your bowl as you eat cereal 6. the energy you use in a 24-h period 7. your distance from home plate after your home run Prentice Hall Algebra 1 Extra Practice 13

8 Extra Practice (continued) Chapter 4 Lesson For the diagram below, find the relationship between the number of shapes and the perimeter of the figure they form. Represent this relationship using a table, words, an equation, and a graph. For each table, determine whether the relationship is a function. Then represent the relationship using words, an equation, and a graph. 9. Rainfall 10. Paint in Can 11 Grocery Bill Lesson 4-4 Graph each function. 17. y = 2x y = 4 x Make a table of values and graph each function. 19. The function f(x) = x represents the amount of money in a savings account that started with $175 after a deposit of x dollars. Prentice Hall Algebra 1 Extra Practice 14

9 5-1 Reteaching Rate of Change and Slope The rate of the vertical change to the horizontal change between two points on a line is called the slope of the line. There are two special cases for slopes. A horizontal line has a slope of 0. vertical change rise slope horizontal change run A vertical line has an undefined slope. What is the slope of the line? slope vertical change rise horizontal change run The slope of the line is 3. In general, a line that slants upward from left to right has a positive slope. What is the slope of the line? vertical change slope horizontal change rise run The slope of the line is 2. In general, a line that slants downward from left to right has a negative slope. 9

10 5-1 Reteaching (continued) Rate of Change and Slope Find the slope of each line Suppose one point on a line has the coordinates (x 1, y 1 ) and another point on the same line has the coordinates (x 2, y 2 ). You can use the following formula to find the slope of the line. slope = rise run = y 2 y 1 x 2 x 1, where x 2 x 1 0 What is the slope of the line through R(2, 5) and S( 1, 7)? y slope x y x Let y = 7 and y = Let x = -1 and x = Find the slope of the line that passes through each pair of points. 4. (0, 0), (4, 5) 5. (2, 4), (7, 8) 6. ( 2, 0), ( 3, 2) 7. ( 2, 3), (1, 1) 8. (1, 4), (2, 3) 9. (3, 2), ( 5, 3) 10

11 5-2 Reteaching Direct Variation A direct variation is a relationship that can be represented by a function in the form y = kx where k 0. The constant of variation for a direct variation k is the coefficient of x. The equation y = kx can also be written as y x 5. Does the equation 6x + 3y = 9 represent a direct variation? If so, find the constant of variation. If the equation represents a direct variation, the equation can be rewritten in the form y = kx. So, solve the equation for y to determine whether the equation can be written in this form. 6x + 3y = 9 3y = 9 6x Subtract 6x from each side. y = 3 2x Divide each side by 3. You cannot write the equation in the form y = kx. So 6x + 3y = 9 does not represent a direct variation. Does the equation 5y = 3x represent a direct variation? If so, find the constant of variation. Again, if the equation represents a direct variation, the equation can be rewritten in the form y = kx. So, solve the equation for y to determine whether the equation can be written in this form. 5y 3x y 3 x 5 Divide each side by 5. The equation has the form y = kx, so the equation represents a direct variation. The coefficient of x is 3 5, so the constant of variation is 3 5. Determine whether each equation represents a direct variation. If it does, find the constant of variation. 1. 2y = x 2. 3x + 2y = y = 8x 4. 2x = y x 3y = x = 2y 19

12 5-2 Reteaching (continued) Direct Variation To write an equation for direct variation, find the constant of variation k using an ordered pair. Then use the value of k to write an equation. Suppose y varies directly with x, and y = 24 when x = 8. What direct variation equation relates x and y? What is the value of y when x = 10? You are given that x and y vary directly. This means that the relationship between x and y can be written in the form y = kx, where k is a constant. y = kx Start with the direct variation equation. 24 = k(8) Substitute the given values: 8 for x and 24 for y. 3 = k Divide each side by 8 to solve for k. y = 3x Write the direct variation equation that relates x and y by substituting 3 for k in y = kx. The equation y = 3x relates x and y. When x = 10, y = 3(10) or 30. Suppose y varies directly with x. Write a direct variation equation that relates x and y. Then find the value of y when x = y = 14 when x = y = 3 when x = y = 12 when x = y = 81 when x = y = 16 when x = y = 5 when x = Consider the direct variation y = 3x. a. List three ordered pairs that satisfy the equation. b. Plot your three ordered pairs from part (a) on a coordinate grid. c. Complete the graph of y = 3x on the grid. 20

13 5-3 Reteaching Slope-Intercept Form The slope-intercept form of a linear equation is y = mx + b. In this equation, m is the slope and b is the y-intercept. What are the slope and y-intercept of the graph of y = 2x 3? The equation is solved for y, but it is easier to determine the y-intercept if the right side is written as a sum instead of a difference. y = 2x 3 y = 2x + ( 3) Write the subtraction as addition. The slope is 2 and the y-intercept is 3. What is an equation for the line with slope 2 3 and y-intercept 9? When the slope and y-intercept are given, substitute the values into the slopeintercept form of a linear equation. y = mx + b y= 2 3 x + 9 Substitute 2 3 for m and 9 for b. What is an equation in slope-intercept form for the line that passes through the points (1, 3) and (3, 1)? Substitute the two given points into the slope formula to find the slope of the line. 1 ( 3) 4 m Then substitute the slope and the coordinates of one of the points into the slope-intercept form to find b. y = mx + b Use slope-intercept form. 3 = 2(1) + b Substitute 2 for m, 1 for x, and 3 for y. 5 = b Solve for b. Substitute the slope and y-intercept into the slope-intercept form. y = mx + b Use slope-intercept form. y = 2x + ( 5) Substitute 2 for m and 5 for b. 29

14 5-3 Reteaching (continued) Slope-Intercept Form Find the slope and y-intercept of the graph of each equation y x 7 2. y = 5x y x y = x y x 2 6. y = 4x 6 Write an equation for the line with the given slope m and y-intercept b m = 3, b = 7 8. m, b 8 9. m = 4, b = m, b m, b m = 7, b = 2 Write an equation in slope-intercept form for the line that passes through the given points. 13. (1, 3) and (2, 5) 14. (2, 1) and (4, 0) 15. (1, 2) and (2, 1) 16. (1, 5) and (3, 3) 17. (3, 3) and (6, 5) 18. (4, 3) and (8, 4) 19. Consider the equation y = 2x + 4. a. What is the y-intercept of the graph of the equation? b. Graph the y-intercept. c. What is the slope of the graph of the equation? d. Use the point you graphed in part (b) and the slope to find another point on the graph of the equation. e. Graph the equation. 30

15 5-4 Reteaching Point-Slope Form The point-slope form of a nonvertical linear equation is y y 1 = m(x x 1 ). In this equation, m is the slope and (x 1, y 1 ) is a point on the graph of the equation. A line passes through (5, 2) and has a slope 3. What is an equation for this line in point-slope form? y y 1 = m(x x 1 ) Use point-slope form. y ( 2) = 3(x 5) Substitute (5, 2) for (x 1, y 1 ) and 3 for m. y + 2 = 3(x 5) Simplify. A line passes through (1, 4) and (2, 9). What is an equation for this line in point-slope form? What is an equation for this line in slope-intercept form? First use the two given points to find the slope m Use the slope and one point to write an equation in point-slope form. y y 1 = m(x x 1 ) Use point-slope form y 4 = 5(x 1) Substitute (1, 4) for (x 1, y 1 ) and 5 for m. y 4 = 5x 5 Distributive Property y = 5x 1 Add 4 to each side. An equation in point-slope form is y 4 = 5(x 1). An equation in slope-intercept form is y = 5x 1. Write an equation for the line through the given point and with the given slope m. 1. ( 1, 3); m = (7, 5); m = 4 3. ( 2, 5); m = 2 3 Write an equation in point-slope form of the line through the given points. Then write the equation in slope-intercept form. 4. (1, 4) and (2, 7) 5. (2, 0) and (3, 2) 6. (4, 5) and ( 2, 2) 39

16 5-4 Reteaching (continued) Point-Slope Form You can use the point-slope form of an equation to help graph the equation. The point given by the point-slope form provides a place to start on the graph. Plot a point there. Then use the slope from the point-slope form to locate another point in either direction. Then draw a line through the points you have plotted. 1 What is the graph of the equation y 2 ( x 1)? 3 The equation is in point-slope form, so the line passes through (1, 2) and has a slope of 1 3. Plot the point (1, 2). Use the slope, 1. From (1, 2), go up 1 unit 3 and then right 3 units. Draw a point. Draw a line through the two points. Because 1 1, you can start at (1, 2) and go down 1 unit and to the left 3 units to 3 3 locate a third point on the line. Graph each equation. 7. y 3 = 2(x + 1) 8. y + 2 = 2 3 (x 2) 9. y 4 = 1 (x + 1) 2 40

17 5-5 Reteaching T e standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. You can easily determine the x- and y-intercepts of the graph from this form of the equation. Each intercept occurs when one coordinate is 0. When substituting 0 for either of x or y, one of the terms on the left side of the standard form equation disappears. This leaves a linear equation in one variable, with a variable term on the left and a constant on the right. Determining the other coordinate of the intercept requires only multiplication or division. Standard Form What are the x- and y-intercepts of the graph of 6x 9y = 18? First find the x-intercept. 6x 9y = 18 6x 9(0) = 18 Substitute 0 for y. 6x = 18 Simplify. x=3 Divide each side by 6. Then find the y-intercept. 6x 9y = 18 6(0) 9y = 18 Substitute 0 for x. y = 2 Divide each side by 9. The x-intercept is 3 and the y-intercept is 2. 9y = 18 Simplify. Find the x- and y-intercepts of the graph of each equation. 1. x y = x + 2y = x + 3y = x 6y = x 4y = x + y = x + 3y = x 2y = x + 2y = Write an equation in standard form with an x-intercept of 5 and a y-intercept of 4. 49

18 5-5 Reteaching (continued) Standard Form You can graph linear equations in standard form by plotting the x- and y-intercepts. What is the graph of 2x y = 4? Find the intercepts. 2x y = 4 2x (0) = 4 2x = 4 x = 2 2x y = 4 2(0) y = 4 y=4 y= 4 The x-intercept is 2, and the y-intercept is 4. Plot the x- and y-intercepts and draw a line through the points. Graph each equation using x- and y-intercepts. 11. x + y = x 3y = x + 2y = x + 4y = x 3y = x + 2y = 10 50

19 5-6 Reteaching Parallel and Perpendicular Lines Nonvertical lines are parallel if they have the same slope and different y-intercepts. The graphs of y = 2x 6 and y = 2x + 3 are parallel because they have the same slope, 2, but different y-intercepts, 6 and 3. What is an equation in slope-intercept form of the line that passes through (8, 7) and is parallel to the graph of The slope of Because the desired equation is for a line parallel to a line with slope, the slope of the parallel line must also be. Use the slope and the given point in the point-slope form of a linear equation and then solve for y to write the equation in slope-intercept form. y y 1 = m (x x 1 ) Start with the point-slope form Substitute (8, 7) for (x 1, y 1 ) and for m. Distributive Property Add 7 to each side. The graph of passes through (8, 7) and is parallel to the graph of 1. Writing Are the graphs of parallel? Explain how you know. Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation. 2. (3, 1); y = 2x (1, 3); y = 7x (1, 6); y = 9x

20 Reteaching (continued) 5-6 Parallel and Perpendicular Lines Two lines that are neither horizontal nor vertical are perpendicular if the product of their slopes is 21. Th e graphs of and are perpendicular because What is an equation in slope-intercept form of the line that passes through (2, 11) and is perpendicular to the graph of The slope of is Since the slope of the line perpendicular to the given line is 4. Use this slope and the given point to write an equation in point-slope form. Then solve for y to write the equation in slope-intercept form. y y 1 = m(x x 1 ) Start with the point-slope form y 11 = ( 4x 2) Substitute (2, 11) for (x 1, y 1 ) and 4 for m. y 11 = 4x + 8 Distributive Property y = 4x + 19 Add 11 to each side. The graph of y = 4x + 19 passes through (2, 11) and is perpendicular to the graph of 8. Writing Are the graphs of and parallel, perpendicular, or neither? Explain how you know. Write an equation in slope-intercept form of the line that passes through the given point and is perpendicular to the graph of the given equation. Write an equation in slope-intercept form of the line that passes through the given point and is perpendicular to the graph of the given equation. 9. (5, 3); y = 5x , (4, 8); y = 2x (-2, -5); y = x (5, 3); y = 5x

21 5-7 Reteaching Scatter Plots and Trend Lines A scatter plot is a graph that relates two different sets of data by displaying them as ordered pairs. A scatter plot can show a trend or correlation, which may be either positive or negative. Or the scatter plot may show no trend or correlation. It is often easier to determine whether there is a correlation by looking at a scatter plot than it is to determine by looking at the numerical data. If the points on a scatter plot generally slope up to the right, the two sets of data have a positive correlation. If the points on a scatter plot generally slope down to the right, the two sets of data have a negative correlation. If the points on a scatter plot do not seem to generally rise or fall in the same direction, the two sets of data have no correlation. The table below compares the average height of girls at different ages. Make a scatter plot of the data. What type of correlation does the scatter plot indicate? Treat the data as ordered pairs. The average height of a 2-year old girl is 34 inches, so one ordered pair is (2, 34). Plot this point. Then plot (3, 37), (4, 40), (5, 42), (6, 45), (7, 48), (8, 50), (9, 52), and (10, 54). Notice that the height increases as the age increases. There is a positive correlation for this data. A trend line is a line on a scatter plot that is drawn near the points. You can use a trend line to estimate other values. 69

22 5-7 Reteaching (continued) Scatter Plots and Trend Lines Draw a trend line for the scatter plot in the previous problem. What is the equation for your trend line? What would you estimate to be the average height of a girl who is 12 years old? Draw a line that seems to fit the data. The line drawn for this data goes through (4, 40) and (8, 50). Use these points to write an equation m Use the point-slope form of the line. y y 1 = m(x x 1 ) y 40 = 2.5(x 4) y 40 = 2.5x 10 y = 2.5x + 30 Use this equation to estimate the average height of 12-year old girls. y = 2.5(12) + 30 y = 60 Ryan practices throwing darts. From each distance listed below, he throws 10 darts and records how many times he hits the center. 1. Use the space at the right to make a scatter plot of the data. 2. Describe the type of correlation that is shown in the scatter plot. 3. Draw a trend line. 4. What equation represents your trend line? 5. How many hits do you estimate Ryan would make from 6 feet? 70

23 Extra Practice Chapter 5 Lesson 5-1 Find the slope of each line Find the rate of change for each situation. 4. growing from 1.4 m to 1.6 m in one year 5. bicycling 3 mi in 15 min and 7 mi in 55 min 6. growing 22.4 mm in 14 s 7. reading 8 pages in 9 min and 22 pages in 30 min 8. The cost of four movie tickets is $30 and the cost of seven tickets is $ Five seconds after jumping out of the plane, a sky diver is 10,000 ft above the ground. After 30 seconds, the sky diver is 3750 ft above the ground. 10.Find the slope of the line that includes the points (1, 4) and ( 3, 2). Prentice Hall Algebra 1 Extra Practice 17

24 Extra Practice (continued) Chapter 5 Lesson 5-3 Find the slope and y-intercept. 20. y = 6x x + 4y = y = y x y= 3x x 5y = 2 4 A line passes through the given points. Write an equation for the line in slope-intercept form. 26. ( 2, 4) and (3, 9) 27. (1, 6) and (9, 4) 28. (0, 7) and ( 1, 0) 29. (7, 0) and (3, 4) 30. (0, 0) and ( 7, 1) 31. (10, 0) and (0, 7) Graph each equation. 32. y = 2x y = 2 3 x 4 Write an equation in slope-intercept form for each situation. 34. A skateboard ramp is 5 ft high and 12 ft long from end to end. 35. An airplane with no fuel weighs 2575 lbs. Each gallon of gasoline added to the fuel tanks weighs 6 lbs. Lessons 5-4 and 5-5 Write an equation in point-slope form for the line through the given point with the given slope. 36. (4, 6); m = (3, 1); m = (8, 5); m = 1 2 Prentice Hall Algebra 1 Extra Practice 18

25 Extra Practice (continued) Chapter 5 Find the x- and y-intercepts for each equation. 39. y = 7x 40. y = 1 x y = 5x 12 2 Graph each equation. 42. x + 4y = y 5 = 2(x + 1) 44. x + 3 = x 3y = y = y + 1 = 1 2 (x + 2) Write an equation in point-slope form for each situation. 48. A train travels at a rate of 70 mi/h. Two hours after leaving the station it is 210 miles from its destination. 49. An escalator has a slope of 3. After traveling forward 32 feet, the escalator is 24 feet 4 above the floor. Write an equation in standard form for each situation. 50. Juan can ride his bike at 12 mi/h and walk at 4 mi/h. Write an equation that relates the amount of time he can spend riding or walking combined, to travel 20 miles. 51. You have $25 to buy supplies for a class party. Juice costs $3 per bottle and chips cost $2 per bag. Write an equation that relates the amount of juice and chips you can buy using $25. Lesson 5-6 Write an equation in standard form that satisfies the given conditions. 52. parallel to y = 4x + 1, 53. perpendicular to y = x 3, through ( 3, 5) through (0, 0) Prentice Hall Algebra 1 Extra Practice 19

26 Extra Practice (continued) Chapter perpendicular to 3x + 4y = 12, 55. parallel to 2x y = 6, through (7, 1) through ( 6, 9) 56. parallel to the x-axis and through (4, 1) 57. through (4, 44) and parallel to the y-axis Tell whether each statement is true or false. Explain your choice. 58. Two airplanes traveling at the same rate leave an airport 1 hour apart. The graphs of the distance each plane travels will be parallel. 59. Two lines with negative slopes can be perpendicular. Lesson a. Graph the (ages, grades) data of some students in a school. b. Draw a trend line. c. Find the equation of the line of best fit. 61. Use a calculator to find a line of best fit for the data in the chart at the right. Find the value of the correlation coefficient r. Let x = 0 correspond to Prentice Hall Algebra 1 Extra Practice 20