1. 3.2 Graph Linear Equations You will graph linear equations in a coordinate plane. Essential Question: How do you graph linear equations? You will learn how to answer this question by using tables to graph linear equations.

2. Warm-Up Exercises 1. 3x + 4y = 16 Rewrite the equation so y is a function of x. ANSWER 3 4 y = – x + 4 2. –6x – 2y = –12 ANSWER y = –3x + 6

3. Warm-Up Exercises Substitute 3 for x and 4 for y. Simplify. Write original equation. Check whether each ordered pair is a solution of the equation. SOLUTION Which ordered pair is a solution of 3x – y = 7? EXAMPLE 1 Standardized Test Practice (3, 4) A (1, –4) B (5, –3) C (–1, –2) D Test (3, 4) : 3(3) – 4 = ? 7 3x – y =3x – y =7 5 =7

4. Warm-Up Exercises Simplify. Write original equation. Standardized Test Practice EXAMPLE 1 Test (1, –4): 3x – y =3x – y =7 3(1) – (–4) = ? 7 Substitute 1 for x and –4 for y. So, (3, 4) is not a solution, but (1, – 4) is a solution of 3x – y = 7. ANSWER The correct answer is B. AB DC 7 =7

5. Warm-Up Exercises GUIDED PRACTICE for Example 1 Tell whether 4, – is a solution of x + 2y = 5. 1 2 not a solutionANSWER

6. Warm-Up Exercises Solve the equation for y. SOLUTION EXAMPLE 2 Graph an equation Graph the equation –2x + y = –3. –2x + y = –3 y = 2x –3 STEP 1

7. Warm-Up Exercises Graph an equation EXAMPLE 2 Plot the points. Notice that the points appear to lie on a line. STEP 3 Make a table by choosing a few values for x and finding the values of y. x–2–1012 y–7–5–3–11 STEP 2

8. Warm-Up Exercises Graph an equation EXAMPLE 2 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end. STEP 4

9. Warm-Up Exercises Graph (a) y = 2 and (b) x = –1. Graph y = b and x = a EXAMPLE 3 SOLUTION For every value of x, the value of y is 2. The graph of the equation y = 2 is a horizontal line 2 units above the x -axis. a.

10. Warm-Up Exercises Graph y = b and x = a EXAMPLE 3 For every value of y, the value of x is –1. The graph of the equation x = –1 is a vertical line 1 unit to the left of the y -axis. b.

11. Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 Graph the equation. 2. y + 3x = –2 ANSWER

12. Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 3. y = 2.5 Graph the equation. ANSWER

13. Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 4. x = –4 Graph the equation. ANSWER

14. Warm-Up Exercises SOLUTION STEP 1 Make a table. x 02468 y43210 EXAMPLE 4 Graph a linear function 1 2 Graph the function y = with domain x  0. – x + 4 Then identify the range of the function.

15. Warm-Up Exercises STEP 2 STEP 3 Connect the points with a ray because the domain is restricted. STEP 4 Identify the range. From the graph, you can see that all points have a y- coordinate of 4 or less, so the range of the function is y ≤ 4. EXAMPLE 4 Graph a linear function Plot the points.

16. Warm-Up Exercises GUIDED PRACTICE for Example 4 5. Graph the function y = –3x + 1 with domain x  0. Then identify the range of the function. ANSWER y  1

17. Warm-Up Exercises SOLUTION RUNNING The distance d (in miles) that a runner travels is given by the function d = 6t where t is the time (in hours) spent running. The runner plans to go for a 1.5 hour run. Graph the function and identify its domain and range. STEP 1 Identify whether the problem specifies the domain or the range. You know the amount of time the runner plans to spend running. Because time is the independent variable, the domain is specified in this problem. The domain of the function is 0 ≤ t ≤ 1.5. Solve a multi-step problem EXAMPLE 5

18. Warm-Up Exercises EXAMPLE 5 Solve a multi-step problem STEP 2 Graph the function. Make a table of values. Then plot and connect the points. t ( hours ) 00.511.5 d ( miles ) 0369 STEP 3 Identify the unspecified domain or range. From the table or graph, you can see that the range of the function is 0 ≤ d ≤ 9.

19. Warm-Up Exercises SOLUTION EXAMPLE 6 Solve a related problem WHAT IF? Suppose the runner in Example 5 instead plans to run 12 miles. Graph the function and identify its domain and range. STEP 1 Identify whether the problem specifies the domain or the range. You are given the distance that the runner plans to travel. Because distance is the dependent variable, the range is specified in this problem. The range of the function is 0 ≤ d ≤ 12.

20. Warm-Up Exercises EXAMPLE 6 Solve a related problem STEP 2 Graph the function. To make a table, you can substitute d -values ( be sure to include 0 and 12) into the function d = 6t and solve for t. t ( hours ) 012 d ( miles ) 0612 STEP 3 Identify the unspecified domain or range. From the table or graph, you can see that the domain of the function is 0 ≤ t ≤ 2.

21. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 GAS COSTS For gas that costs $2 per gallon, the equation C = 2g gives the cost C (in dollars) of pumping g gallons of gas. You plan to pump $10 worth of gas. Graph the function and identify its domain and range. domain: 0 ≤ g ≤ 5, range: 0 ≤ C ≤ 10 6. ANSWER

22. Warm-Up Exercises Daily Homework Quiz 1. Graph y + 2x = 4. ANSWER

23. Warm-Up Exercises Daily Homework Quiz 2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range. domain : 0  t  2.5 range : 0  d  12.5 ANSWER

24. You will graph linear equations in a coordinate plane. Essential Question: How do you graph linear equations? All points on the graph of a linear equation are solutions of the equation. Use a line to connect points when the domain is unrestricted, a ray when it is restricted, and a segment when both domain and range are restricted. Make a table of appropriate x-values, determine corresponding y-values, plot the points from the table, and connect the points with a line.