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 howto answer this question by usingtables to graph linear equations.

Warm-Up Exercises

1. 3x + 4y = 16

Rewrite the equation so y is a function of x.

ANSWER 34

y = – x + 4

2. –6x – 2y = –12

ANSWER y = –3x + 6

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 = 7

5 = 7

Warm-Up Exercises

Simplify.

Write original equation.

Standardized Test PracticeEXAMPLE 1

Test (1, –4): 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. A B DC

7 = 7

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Tell whether 4, – is a solution of x + 2y = 5.12

not a solutionANSWER

Warm-Up Exercises

Solve the equation for y.

SOLUTION

EXAMPLE 2 Graph an equation

Graph the equation –2x + y = –3.

–2x + y = –3y = 2x –3

STEP 1

Warm-Up ExercisesGraph an equationEXAMPLE 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 –1 0 1 2

y –7 –5 –3 –1 1

STEP 2

Warm-Up ExercisesGraph an equationEXAMPLE 2

Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

STEP 4

Warm-Up Exercises

Graph (a) y = 2 and (b) x = –1.

Graph y = b and x = aEXAMPLE 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.

Warm-Up ExercisesGraph y = b and x = aEXAMPLE 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.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

Graph the equation.

2. y + 3x = –2

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

3. y = 2.5

Graph the equation.

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

4. x = –4

Graph the equation.

ANSWER

Warm-Up Exercises

SOLUTION

STEP 1Make a table.

x 0 2 4 6 8

y 4 3 2 1 0

EXAMPLE 4 Graph a linear function

12Graph the function y = with domain x 0. – x + 4

Then identify the range of the function.

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.

Warm-Up ExercisesGUIDED 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

Warm-Up Exercises

SOLUTION

RUNNINGThe 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 1Identify 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 problemEXAMPLE 5

Warm-Up ExercisesEXAMPLE 5 Solve a multi-step problem

STEP 2Graph the function. Make a table of values. Then plot and connect the points.

t (hours) 0 0.5 1 1.5d (miles) 0 3 6 9

STEP 3Identify the unspecified domain or range. From the table or graph, you can see that the range of the function is 0 ≤ d ≤ 9.

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 1Identify 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.

Warm-Up ExercisesEXAMPLE 6 Solve a related problem

STEP 2Graph 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) 0 1 2

d (miles) 0 6 12

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.

Warm-Up ExercisesGUIDED 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

Warm-Up ExercisesDaily Homework Quiz

1. Graph y + 2x = 4.

ANSWER

Warm-Up ExercisesDaily 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 12.5ANSWER

• 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 pointswhen the domain is unrestricted, a ray when it is restricted, and a segment when both domain andrange are restricted.

Make a table of appropriatex-values, determine corresponding y-values, plot the points from the table, and connect the points with a line.