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Review Videos

• Graphing the x and y intercept• Graphing the x and y intercepts• Graphing a line in slope intercept form• Converting into slope intercept form

Chapter 7 Section 6

Families of Linear Graphs

What You’ll Learn

You’ll learn to explore the effects of changing the slopes and y-intercepts of linear

functions.

Why It’s Important

BusinessFamilies of graphs can display different fees.

Families of linear graphs often fall into two categories-

1.Those with the same slope

2. Those with the same y-intercept.

Family of Graphs

y = ½x + 3

y = ½x - 1

Same Slope

What do these lineshave in common?

Family of Graphs

y = ⅓x + 1y = -x + 1

Same y-intercept

What do these lineshave in common?

Not a Family of Graphs

y = ⅓x

y = x + 2

Differenty-intercept and slope

What do these lineshave in common?

Example 1Graph each pair of equations. Describe any similarities or

differences. Explain why they are a family of graphs.y = 3x + 4y = 3x – 2The graphs have y-intercepts of 4 and -2, respectively.

They are a family of graphs because the slope of each line is 3.

y = 3x + 4

y = 3x - 2

Example 2Graph each pair of equations. Describe any similarities or

differences. Explain why they are a family of graphs.y = x + 3y = -½x + 3Each graph has a different slope.

Each graph has a y-intercept of 3.Thus, they are a family of graphs. y = x + 3

y = -½x + 3

Hint:

You can compare graphs of lines by looking at their equations.

Example 3• Matthew and Juan are starting their own pet care business. Juan wants to

charge $5 an hour. Matthew thinks they should charge $3 an hour. Suppose x represents the number of hours. Then y = 5x and y = 3x represents how much they would charge, respectively. Compare and contrast the graphs of the equations.

The equations have the same y-intercept, but the graph of y = 5x is steeper. This is because its slope, which represents $5 per hour, is greater that the slope of the graph of y = 3x.

6

5

4

3

2

01 1.5.5 2

y = 5x

y = 3x

Your Turn

Compare and contrast the graphs of the equations. Verify by graphing the equation.

y = -3x + 4y = -x + 4

Answer

y = -3x + 4y = -x + 4

Same y-intercept Different slope y = -3x + 4

y = -x + 4

Try This One

Compare and contrast the graphs of the equations. Verify by graphing the equation.

y = ⅔x + 3y = ⅔x -1

Answer

y = ⅔x + 3y = ⅔x -1

Same slope Different y-intercept y = ⅔x - 1

y = ⅔x + 3

Parent Graph

A parent graph is the simplest of the graphs in a family. Let’s summarize how changing the m or b in y = mx + b affects the

graph of the equation. Parent: y = x

y = xy = 3x

y = ¼x

As the value of m Increases, the line Gets steeper

Parent: y = -x

y = -xy = -3x

y = -¼xAs the value of m Decreases, the line Gets steeper.

Parent: y = 2x

y = 2x

y = 2x + 3y = 2x - 4

As the value of b Increases, the graph shifts Up on the y-axis. As the value of b decreases,the graph shifts down on the y-axis

You can change a graph by changing the slope or y-intercept.

Example 4

Change y = -½x + 3 so that the graph of the new equation fits each description.

Same y-intercept, steeper negative slopeThe y-intercept is 3, and the slope is -½. The new equation will also have a y-interceptof 3. In order for the slope to be steeper and still be negative, its value must be lessthan -½, such as -2. The new equation is y = -2x + 3.

y = -2x + 3

y = -½x + 3

Example 5

Change y = -½x + 3 so that the graph of the new equation fits each description.

Same slope, y-intercept is shifted up 4 unitsThe slope of the new equation will be -½. Since the current y-intercept will be 3 + 4 or 7.The new equation is y = -½x + 7. Always check by graphing

y = -½x + 7

y = -½x + 3

Your Turn

Change y = 2x + 1 so that the graph of the new equation fits each description.

Same slope, shifteddown 1 unit.

y = 2x + 0 Simplified to y = 2x

Your Turn

Change y = 2x + 1 so that the graph of the new equation fits each description.

Same y-intercept, less steep positive slope

y = x + 1