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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Transformations of Functions

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Transformation of Graphs

The transformations of the graph of y=f(x)y=f\left(x\right) are
  • Vertical Reflection: y=−f(x)y=-f\left(x\right)

  • Horizontal Reflection: y=f(−x)y=f\left(-x\right)

  • Vertical stretch/shrink: y=a[f(x)]y=a\left[f\left(x\right)\right]

  • Horizontal stretch/shrink: y=f(ax)y=f\left(ax\right)

  • Vertical shift: y=f(x)+ky=f\left(x\right)+k

  • Horizontal shift: y=f(x−k)y=f\left(x-k\right)

Wize Tip
All transformations done to the yy component (the entire function) changes it in the vertical direction.
All transformations done to the xx component changes it in the horizontal direction, most of the time in the opposite way!

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Example: Transformations of Functions

Given f(x)=1xf\left(x\right)=\dfrac{1}{x} , state the equation of the transformed graph that has been:
  • translated 2 units left
  • reflected in the y-axis
  • stretched vertically by a factor of 4
  • translated 5 units up
Stretches and reflections first (order doesn't matter), then translations:

Reflection in the x-axis:
f(x)= −1xf\left(x\right)=\ -\dfrac{1}{x}

Vertical stretch by a factor of 4:
f(x)= −4xf\left(x\right)=\ -\dfrac{4}{x}

Translation 2 units left:
f(x)= −4x+2f\left(x\right)=\ -\dfrac{4}{x+2}

Translation 5 units up:
f(x)= −4x+2+5f\left(x\right)=\ -\dfrac{4}{x+2}+5

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State the transformations that have been applied to y=x52y=x^{\frac{5}{2}} to get: y=2(−3x+4)52−1y=2\left(-3x+4\right)^{\frac{5}{2}}-1

Extra Practice
Transforming a Log
Given f(x)=log⁡3(2x+4)f\left(x\right)=\log_3\left(2x+4\right) , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Reflecting along the xx axis
Practice: Transforming a Trig Function
Graph the function: y=3sin⁡(2x+π2)+1\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1
Transforming a Log
Given f(x)=log⁡3(2x+4)f\left(x\right)=\log_3\left(2x+4\right) , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Q:\bf{Q:} State the transformations that have been applied to y=x52y=x^{\frac{5}{2}} to get: y=2(−3x+4)52−1y=2\left(-3x+4\right)^{\frac{5}{2}}-1
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Reflecting along the xx axis
Transformations
Which of the following is the equation of the function produced by vertically shifting the graph of y=exy=e^x up 3 units and then vertically stretching it by a factor of 4?
Transforming a Log
Given f(x)=log⁡3(2x+4)f\left(x\right)=\log_3\left(2x+4\right) , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Q:\bf{Q:} State the transformations that have been applied to y=x52y=x^{\frac{5}{2}} to get: y=2(−3x+4)52−1y=2\left(-3x+4\right)^{\frac{5}{2}}-1
Practice: Transforming a Trig Function
Graph the function: y=3sin⁡(2x+π2)+1\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1