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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

Alg2B 4.5 Stretch


  • Stretching and Compressing Examples

Alg2B 4.5 Stretch Compress

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.

Alg2B 4.5 Stretch Compress Flip

 


  •  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

Alg2B 4.5 Flip y-axis


  • 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
Alg2B 4.5 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

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