# The Difference Between Linear and Exponential Functions | Mathematics

October 8, 2015

From graphing the domain and range, to finding the minimum and maximum, to calculating the average range of change, your knowledge of functions continues to grow. You probably recall learning the most about quadratic functions. Are you ready to learn about two additional types of functions?

Understanding linear and exponential functions is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest. Let’s look at the difference between the two:

## Linear Functions

A linear function grows or shrinks at a constant rate defined by its slope. Linear functions are in the form of f(x)=ax, where:

• The variables have no exponents or powers.
• The variable a is any real number.
• The graph of f is a straight line through the origin.
• The variable a is the slope of the function.

True to linear functions, f(0)=a×0=0. Because of this, the graph will always pass through the origin (0,0). In the graph below, a = 1 = the slope of the function. This is shown by the shaded triangle:

Graph image via MathInsight.org.

## Exponential Functions

An exponential function describes growth or decay at a rate that increases or decreases over time. Exponential functions are in the form of f(x)=ax, where:

• The independent variable is the exponent x.
• The variable a is any real number.

In exponential growth, the function f(x)=ax increases according to the variable a every time you add to its exponent x. For example, in the exponential function f(x)=2x, the function doubles every time you add one to its exponent x. This is shown in the graph below:

Graph image via MathInsight.org.

In exponential decay, the function g(x)=ax decreases according to the variable a every time you add to its exponent x. For example, in the exponential function g(x)=(1/2)x, the function shrinks in half every time you add one to its exponent x. This is shown in the graph below:

Graph image via MathInsight.org.

Stay tuned for future TASC Mathematics subtest posts to learn more about these two function types, including how to graph them.

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