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 Business Families 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 lines have in common?

Family of Graphs y = ⅓x + 1 y = -x + 1 Same y-intercept What do these lines have in common?

Not a Family of Graphs y = ⅓x y = x + 2 Different y-intercept and slope What do these lines have in common?

Example 1 Graph each pair of equations. Describe any similarities or differences. Explain why they are a family of graphs. y = 3x + 4 y = 3x – 2 The 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 2 Graph each pair of equations. Describe any similarities or differences. Explain why they are a family of graphs. y = x + 3 y = -½x + 3 Each 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 y = 5x y = 3x

Your Turn Compare and contrast the graphs of the equations. Verify by graphing the equation. y = -3x + 4 y = -x + 4

Answer y = -3x + 4 y = -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 + 3 y = ⅔x -1

Answer y = ⅔x + 3 y = 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 = x y = 3x y = ¼x As the value of m Increases, the line Gets steeper

Parent: y = -x y = -x y = -3x y = -¼x As the value of m Decreases, the line Gets steeper.

Parent: y = 2x y = 2x y = 2x + 3 y = 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 slope The y-intercept is 3, and the slope is -½. The new equation will also have a y-intercept of 3. In order for the slope to be steeper and still be negative, its value must be less than -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 units The slope of the new equation will be -½. Since the current y-intercept will be 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, shifted down 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