# 4.5 – Stretching Functions Vertically

## Objectives

• Determine the equation of a function G(x) whose graph has been vertically stretched or compressed from the graph of F(x) when you are given the equation of F(x).
• Describe how the graph of a function changes when the function is multiplied by a positive or negative number.
• Determine the equation of a function G(x) whose graph has been either vertically stretched and shifted or vertically compressed and shifted from the graph of F(x) when you are given the equation of F(x).

## Key Terms

• Compress – To make something shorter (flatter & wider).
• Stretch – To make something longer (taller & more thin).

## Review

• To move functions vertically, add or subtract the y-value at the end of the equation
• Up: + #
• Down: – #
• To move functions horizontally, subtract a positive (right) or subtract a negative (left) x-value from the x-part of the equation
• Left: (x – (-#)), which becomes (x + #)
• Right: (x – (+#)), which becomes (x – #)

## Notes

Vertically Stretching and Compressing
• Vertically stretching and compressing a function changes the y-value for each coordinate
• Vertically stretching or compressing functions changes the SHAPE of the graph
• Stretching: larger y-values (positive or negative numbers larger than 1)
• # > 1
• # < -1
• Compressing: smaller y-values (positive or negative numbers less than 1: decimals and fractions)
• 0 < # < 1
• -1 > # > 0
• Remember: parent functions have y-values that equal 1
• 1y can be written as just “y”
• # = 1

• Stretching Example (Blue to Red)
• The Red function, G(x), is stretched by 2 from the Blue function, F(x), below
• Notice that all y-values are 2 times larger (doubled)
• Blue, F(x): $F(x)=x^3-3x$
• Red, G(x): $2[F(x)]=2(x^3-3x)$
• Simplied to: $G(x)-2x^3-6x$

• Stretching and Compressing Examples

Flipping Over Axes
• Explanation of Stretching, Compressing, and Flipping Over the x-axis
• When you multiply a function by -1, what is the effect on its graph?
• Answer: the graph flips over the x-axis

• Examples of Stretched and Compressed Functions Flipped Over the x-axis
• – 0.8 is compressed from the parent function because its value is between zero and 1.
• The negative means the function is flipped over the x-axis.
• – 3 is stretched from the parent function because its value is greater than 1.
• The negative means the function is flipped over the x-axis.

•  Flipping Over the y-axis
• Not every part of the function is affected (such as the case with flipping over the x-axis).
• ONLY the x-values are affected, just like when you shift to the left or to the right
• Notice that the x (inside the parenthesis) has changed signs

• Example: Flipping Over the y-axis
• Blue, F(x): $F(x)=x^3-x$
• Red, G(x): $G(x)=(-x)^3-(-x)$
• Simplify: $G(x)=-x^3+x$

More Examples

• Quiz Example 1
• Parent: $F(x)=x^2$
• Shift 7 units Up
• Shift 2 units to the Left
• Stretch vertically by 3
• Answer: $G(x)=3(x+2)^2+7$

• Quiz Example 2
• Parent: $F(x)=x^2$
• Shift 1 unit Down
• Shift 4 units to the Right
• Compress Vertically by 0.2
• Answer: $H(x)=0.2(x-4)^2-1$

• Quiz Example 3
• Parent: $F(x)=x^2$
• Shift 9 Units Up
• Shift 3 Units to the Right
• Stretch Vertically by -8 (it flipped over the x-axis)
• Answer: $J(x)=-8(x-3)^2+9$

• Quiz Example 4
• Parent: $F(x)=x^2$
• Shift 2 Units Down
• Shift 4 Units to the Right
• Compress Vertically by -0.6 (it flipped over the x-axis)
• Answer: $K(x)=-0.6(x-4)^2+2$

• Quiz Example 5
• Parent: $F(x)=x^2$
• Shift 3 Units Up
• Shift 6 Units to the Right
• Compress Vertically by -0.2 (flipped over x-axis)
• Flip Over the y-axis
• Answer: $L(x)=-0.2(-x-6)^2+3$