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Contents Lesson 8-1Functions Lesson 8-2Linear Equations in Two Variables Lesson 8-3Graphing Linear Equations Using Intercepts

Lesson 1 Contents Example 1Ordered Pairs and Tables as Functions Example 2Use a Graph to Identify Functions Example 3Use a Function to Describe Data

Example 1-1a Determine whether the relation is a function. Explain. {(–3, –3), (–1, –1), (0, 0), (–1, 1), (3, 3)} Answer:No; –1 in the domain is paired with both –1 and 1 in the range.

Example 1-1a Determine whether the relation is a function. Explain. Answer:Yes; each x value is paired with only one y value. x –3 –6 y –2

Example 1-1b x 3 1 –1 –3 1 –5 y –4 2 1 Determine whether each relation is a function. Explain. a. {(2, 5), (4, –1), (3, 1), (6, 0), (–2, –2)} b. Answer:Yes; each x value is paired with only one y value. Answer:No; 1 in the domain is paired with 4 and 2 in the range.

Example 1-2a Determine whether the graph is a function. Explain. Answer:Yes; it passes the vertical line test.

Example 1-2b Determine whether the graph is a function. Explain. Answer:No; it does not pass the vertical line test.

Example 1-3a Business The table shows the number of boxes made. Answer:Yes; for each 10 hours, only one amount of boxes is made. Number of HoursNumber of Boxes Do these data represent a function? Explain.

Example 1-3a Describe how box production is related to hours of operation. Answer:As the number of hours increases, the number of boxes produced increases. Number of HoursNumber of Boxes

Example 1-3b Answer:Yes; for each 5 hours, only one amount of chairs is made. Number of HoursNumber of Chairs a. Do these data represent a function? Explain. Assembly Line The table shows the number of chairs made.

Example 1-3b Answer:As hours increase, the number of chairs produced increases. Number of HoursNumber of Chairs b.Describe how chair production is related to hours of operation. Assembly Line The table shows the number of chairs made.

End of Lesson 1

Lesson 2 Contents Example 1Find Solutions Example 2Solve an Equation for y Example 3Graph a Linear Equation

x (x, y)y Example 2-1a Find four solutions of. Choose four values for x. Then substitute each value into the equation to solve for y. There are many possible solutions. The solutions you find depend on which x values you choose. 3 (0, 3) 7 (1, 7) 11 (2, 11) 15 (3, 15)

Example 2-1a Sample Answer:Four possible solutions are (0, 3), (1, 7), (2, 11), and (3, 15).

Find four solutions of. Example 2-1b Sample Answer: (0, –4), (1, –2), (2, 0), and (3, 2).

Example 2-2a Business At a local software company, Level 1 employees x earn $48, 000 and Level 2 employees y earn $24, 000. Find four solutions of to determine how many employees at each level the company can hire for $216, 000. First, rewrite the equation by solving for y.

Example 2-2a Write the equation. Subtract 48,000x from each side. Divide each side by 24,000. Simplify.

x(x, y)y 9(0, 9) 7(1, 7) 5 (2, 5) 3 (3, 3) Example 2-2a Sample Answer: (0, 9), (1, 7), (2, 5), and (3, 3) 0 Level 1, 9 Level 2 1 Level 1, 7 Level 2 2 Level 1, 5 Level 2 3 Level 1, 3 Level 2 Choose four x values and substitute them into

Example 2-2b Sample Answer: (0, 14), (1, 12), (2, 10), and (3, 8) 0 hardbacks, 14 paperbacks 1 hardbacks, 12 paperbacks 2 hardbacks, 10 paperbacks 3 hardbacks, 8 paperbacks Bookstore At a local bookstore, hardbacks are on sale for $6 and paperbacks are on sale for $3. Bob has $42 to spend on books. Find four solutions to determine how many books of each type Bob can buy with his $42.

Example 2-3a Graph by plotting ordered pairs. First, find ordered pair solutions –1 x (x, y)y –4 (–1, –4) –3 (0, –3) –2 (1, –2) –1 (2, –1) Four solutions are (–1, –4), (0, –3), (1, –2), and (2, –1).

Example 2-3a Plot these ordered pairs and draw a line through them. Note that the ordered pair for any point on this line is a solution of. The line is a complete graph of the function. Answer:

Example 2-3a CheckIt appears from the graph that (4, 1) is also a solution. Check this by substitution. Write the equation. Simplify. Replace x with 4 and y with 1.

Example 2-3b Answer: Graph by plotting ordered pairs.

End of Lesson 2

Lesson 3 Contents Example 1Find Intercepts from Graphs Example 2Find Intercepts from Equations Example 3Use Intercepts to Graph Equations Example 4Intercepts of Real-World Data Example 5Horizontal and Vertical Lines

Example 3-1a State the x -intercept and the y -intercept of the line. The graph crosses the x -axis at (–2, 0). The x -intercept is –2. The graph crosses the y -axis at (0, –2). The y -intercept is –2. Answer: –2 ; –2

Example 3-1a State the x -intercept and the y -intercept of the line. The graph does not cross the x -axis. There is no x -intercept. The graph crosses the y -axis at (0, 2). The y -intercept is 2. Answer: none; 2

Answer: –1 ; none Example 3-1b State the x -intercept and the y -intercept of each line. Answer: 1 ; 3 a. b.

Example 3-2a Find the x -intercept and the y -intercept for the graph of. To find the x -intercept, let. Write the equation. Simplify. Replace y with 0. The x -intercept is 4. So, the graph crosses the x -axis at (4, 0).

Example 3-2a The y -intercept is 2. So, the graph crosses the y -axis at (0, 2). To find the y -intercept, let. Answer: x -intercept, 4 ; y -intercept, 2 Write the equation. Replace x with 0. Simplify. Divide each side by 2.

Find the x -intercept and the y -intercept for the graph of. Example 3-2b Answer: x -intercept, 9 ; y -intercept, 3

Example 3-3a Graph using the x - and y -intercepts. Step 1Find the x -intercept. Write the equation. Replace y with 0. Add 6 to each side. Divide each side by 3. The x -intercept is 2, so the graph passes through (2, 0).

Example 3-3a Step 2Find the y -intercept. The y -intercept is –6, so the graph passes through (0, –6). Write the equation. Replace x with 0. Simplify. Step 3Graph the points (2, 0) and (0, –6) and draw a line through them.

Example 3-3a CheckChoose some other point on the line and determine whether its ordered pair is a solution of. Answer:

Example 3-3b Graph using the x - and y -intercepts. Answer:

Example 3-4a Home Repair Renata Jones has $300 for home repairs. A plumber charges her $50 an hour. The equation represents the amount of money left in her budget after x number of plumbing hours. Use the intercepts to graph the equation. Step 1Find the x -intercept. Write the equation. Replace y with 0. Subtract 300 from each side. Simplify. Divide each side by –50. The x -intercept is 6.

Example 3-4a Step 2Find the y -intercept. The y -intercept is 300. Write the equation. Replace x with 0. Simplify. Step 3Plot the points with coordinates (6, 0) and (0, 300). Then draw a line through the points.

Example 3-4a Answer: (6, 0) (0, 300)

Example 3-4a Describe what the intercepts mean. Answer: The y -intercept shows how much money Renata has in her budget before any work is done. The x -intercept means that she can afford only 6 hours of plumbing repair.

Example 3-4b Shopping Suzanne has $125 to spend on new shoes. The shoe store is having a sale in which every pair of shoes is priced at $25. The equation represents the amount of money Suzanne has left after purchasing x pairs of shoes. a.Use the intercepts to graph the equation. Answer:

Example 3-4b b. Describe what the intercepts mean. Answer: The y -intercept shows how much money Suzanne has before buying any shoes. The x -intercept means that she can afford only 5 pairs of shoes.

Example 3-5a Graph using the x - and y -intercepts. The y -intercept is –4, and there is no x -intercept. Answer: Note that is the same as

Example 3-5a The x -intercept is 5, and there is no y -intercept. Graph using the x - and y -intercepts. Answer: Note that is the same as

Example 3-5b Graph each equation using the x - and y -intercepts. a.b. Answer:

End of Lesson 3

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