Algebra 1 - Exponential Functions

Introduction:

The general exponential function is a function written in the form \(f(x)=a(b)^x\), where the base \(b>0\) and \(b\ne1\).   It is called an exponential function because the variable \(x\) is in the exponent position.   The value of \(a\) is the \(y\)-coordinate of the \(y\)-intercept:   when \(x=0\), \(y=a(b)^0\implies y=a(1)\implies y=a\).   There are two types of graphs for an exponential function:

Exponential Growth

When \(b>1\), exponential growth occurs.   The graph below shows a growth curve.

  • Domain:   \((-\infty,\infty)\)   and   Range:   \((0,\infty)\)
  • \(y\)-intercept:   \((0,a)\);   always INCREASING
growth

Exponential Decay

When \(0< b< 1\), exponential decay occurs.   The graph below shows a decay curve.

  • Domain:   \((-\infty,\infty)\)   and   Range:   \((0,\infty)\)
  • \(y\)-intercept:   \((0,a)\);   always DECREASING
growth

Video on Graphing an Exponential Function

Video on Write an Equation from a Graph

Summary Keys from the Videos

When Graphing . . .

  • Start at \(y\)-intercept \((0,a)\)
  • One unit to the right: MULTIPLY BY \(b\)
  • One unit to the left: DIVIDE BY \(b\)
  • Find at least three points for shape
  • Graph shows growth or decay depending on \(b\)

When Creating an Equation . . .

  • Find the \(y\)-intercept for \(a\)
  • Find two grid points ONE unit apart
  • Moving RIGHT:   solve \(y_1\cdot b=y_2\)
  • Moving LEFT:   solve \(\displaystyle{\frac{y_1}{b}=y_2}\)
  • Equation:   \(y=a\left(b\right)^x\)