Function Transformations

How move and resize the graphs of functions?

Example

f(x) = x2

Function Transformations

How move the graphs of functions?

  • Add a constant to the y-value will move it up or down
  • Add a constant to the x-value will move it left or right

Examples

Functionc > 0c < 0
f(x) = x2 g(x) = x2 + c   Function Transformations Example 1Function Transformations Example 2  
f(x) = x2 g(x) = (x + c)2   Function Transformations Example 3 Function Transformations Example 4

To put it simply:
  • add to y to go high
  • add to x to go left

How resize the graphs of functions?

  • multiplying the whole function by a constant will stretch or compress it in the y-direction
  • multiplying x by a constant will stretch or compress it in the x-direction

Examples

Functionc > 10 < c < 1
f(x) = x2 g(x) = cx2   Function Transformations Example 5   Function Transformations Example 6
f(x) = x2 g(x) = (cx)2   Function Transformations Example 7   Function Transformations Example 8

Note:
  • for the y-direction, bigger values cause more stretch.
  • for the x-direction, bigger c cause more compression.

How flip the graphs of functions?

  • Multiplying the whole function by −1 will flip it upside down
  • Multiplying the x-value by −1 will flip it upside down
FunctionGraph
f(x) = x2g(x) = -x2   Function Transformations Example 9
f(x) = x2g(x) = (-x)2   Function Transformations Example 10

Summary

y = f(x) + cc > 0 moves it upc < 0 moves it down
y = f(x + c)c > 0 moves it leftc < 0 moves it right
y = cf(x)c > 1 stretches it in the y-direction0 < c < 1 compresses it
y = f(cx)c > 1 stretches it in the x-direction0 < c < 1 stretches it
y = -f(x)Reflects it about x-axis
y = f(-x)Reflects it about y-axis

All transformation in one go

To put it simply, use

a f(b(x + c)) + d

SymbolDetail
a|a| > 1 stretches |a| < 1 compresses a < 0 flips the graph upside down
b|b| > 1 compresses |b| < 1 stretches b < 0 flips the graph left-right
cc < 0 shifts to the right c > 0 shifts to the left
dd > 0 shifts upward d < 0 shifts downward

Learn More

Injective, Surjective & Bijective

Intervals

Set-Builder Notation

Inverse Functions

Sets Index

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