High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize AP Calculus (BC) Textbook > Review: Functions/Pre-Calculus

Transformations of Functions

Even and Odd FunctionsPrevious Section
PolynomialsNext Section
Popular Courses
Find My Course
0:00 / 0:00

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!

0:00 / 0:00

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

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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