Lesson 3.2 Graph Linear EquationsEssential Question: How do you graph linear equations in the coordinate plane?

10-2-14

Warm-up:

CommonCore CC.9-12.F.IF.7aGraph linear and quadraticfunctions and show intercepts,maxima, andminima.

1. Graph y = -x – 2 with the domain -2, -1, 0, 1, and 2

Rewrite the equation so y is aFunction of x.

2. 3x + 4y = 16

3. -6x – 2y = -12

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

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

GUIDED PRACTICE for Example 1

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

not a solutionANSWER

Solve the equation for y.

SOLUTION

EXAMPLE 2 Graph an equation

Graph the equation –2x + y = –3.

–2x + y = –3y = 2x –3

STEP 1

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 –1 0 1 2

y –7 –5 –3 –1 1

STEP 2

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

Standard Form of a Linear Equations• The standard form of a linear equations is:

• Ax + By = C• where A, B, and C are real numbers and A and

B are not both zero.

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.

Graph 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.

GUIDED PRACTICE for Examples 2 and 3

Graph the equation.

2. y + 3x = –2

ANSWER

GUIDED PRACTICE for Examples 2 and 3

3. y = 2.5

Graph the equation.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

4. x = –4

Graph the equation.

ANSWER

Equations of Horizontal and Vertical Lines

• The graph of y = b is a horizontal line. The point passes through the point (0, b)

• The graph of x = a is vertical line that passes through the point (a,0)

Horizontal Line

It is a function!

Vertical Line• “y” values can be any numbers.

However the line is not a function!

Linear Function

What is a Linear Function?

Linear Function

•  

The graph of x = 5 is not a function.

• Why?

The graph of x = 5 is not a function.

• Why?• Hint: Lookat the graph!

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.

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.

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

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 problemEXAMPLE 5

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

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.

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.

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) 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.

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

Classwork/Homework

3.2 Practice B