# Transformations of Exponential and Logarithmic Functions

The transformation of functions includes the shifting, stretching, and reflecting of their graph. The same rules apply when transforming logarithmic and exponential functions.

## Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph of:

y = f(x) + c: shift the graph of y= f(x) up by c units

y = f(x) - c: shift the graph of y= f(x) down by c units

y = f(x - c): shift the graph of y= f(x) to the right by c units

y = f(x + c): shift the graph of y= f(x) to the left by c units

Example: The graph below depicts g(x) = ln(x) and a function, f(x), that is the result of a transformation on ln(x). Which of the following functions represents the transformed function (blue line) on the graph?

A. $$f(x)=\ln(x)+2$$
B. $$f(x)=\ln(x)-2$$
C. $$f(x)=\ln(x-2)$$
D. $$f(x)=\ln(x+2)$$

Solution: The correct answer is C. The curve is shifted to the right.

## Vertical and Horizontal Stretches/Compressions

Suppose c > 1. To obtain the graph of:

y = cf(x): stretch the graph of y = f(x) vertically by a factor of c.

y = 1/c f(x): compress the graph of y = f(x) vertically by a factor of c

y = f(cx): compress the graph of y = f(x) horizontally by a factor of c

y = f(x/c): stretch the graph of y = f(x) horizontally by a factor of c

Example: Which curves do the following functions correspond to if black curve represents $$e^x$$?

A. $$f(x)=e^{5x}$$

B. $$f(x)=e^{\frac{x}{5}}$$

Solution: The blue curve matches function A. It is compressed horizontally by a factor of 5. The green curve matches function B. It is stretched horizontally by a factor of 5.

## Reflections

To obtain the graph of

y= -f(x): reflect the graph of y = f(x) about the x-axis

y= f(-x): reflect the graph of y = f(x) about the y-axis

Example: Sketch the graphs of f(x) = ln(x), g(x) = ln(-x), and h(x)= -ln(x).

Solution:

The black curve is the original, $$f(x)=\ln(x)$$.

The blue curve is the reflection about the y-axis, $$g(x)=\ln(-x)$$.

The red curve is the reflection about the x-axis, $$h(x)=-\ln(x)$$.

Transformation of Exponential Functions Example:

Transformation of Logarithmic Functions Example: