# Function Transformations

How move and resize the graphs of functions?

Example

f(x) = x2

## 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

 Function c > 0 c < 0 f(x) = x2 g(x) = x2 + c    f(x) = x2 g(x) = (x + c)2    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

 Function c > 1 0 < c < 1 f(x) = x2 g(x) = cx2    f(x) = x2 g(x) = (cx)2    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
 Function Graph f(x) = x2g(x) = -x2  f(x) = x2g(x) = (-x)2  ## Summary

 y = f(x) + c c > 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

 Symbol Detail 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 c c < 0 shifts to the right c > 0 shifts to the left d d > 0 shifts upward d < 0 shifts downward