Unit 6: Exponents and Exponential Functions

6.3: Graphs of Exponential Functions

Graphs of Exponential Functions $f\left(x\right)=b^x$

Consider the following graphs:

$f\left(x\right)=2^x$ $f\left(x\right)=\left(\frac{1}{3}\right)^x$
  • This function has base $2$, and it is increasing.
  • It has a horizontal asymptote at the $x$-axis.
  • It has a $y$-intercept at $\left(0,1\right)$.
  • It has another identifiable point at $\left(1,2\right)$.
  • Its domain is all real numbers.
  • Its range is $y>0$.
  • This function has base $\frac{1}{3}$, and it is decreasing.
  • It has a horizontal asymptote at the $x$-axis.
  • It has a $y$-intercept at $\left(0,1\right)$.
  • It has another identifiable point at $\left(1,\frac{1}{3}\right)$.
  • Its domain is all real numbers.
  • Its range is $y>0$.

We could look at more examples, and end up with the following rules, for the graph of an exponential function $f\left(x\right)=b^x$:

Graphs of Exponential Functions $f\left(x\right)=a \cdot b^x$

Consider the following graphs:

$f\left(x\right)=3 \cdot 2^x$ $f\left(x\right)=2 \cdot \left(\frac{1}{3}\right)^x$
  • This function has base $2$, and it is increasing.
  • It has a horizontal asymptote at the $x$-axis.
  • It has a $y$-intercept at $\left(0,3\right)$.
  • It has another identifiable point at $\left(1,6\right)$.
  • Its domain is all real numbers.
  • Its range is $y>0$.
  • This function has base $\frac{1}{3}$, and it is decreasing.
  • It has a horizontal asymptote at the $x$-axis.
  • It has a $y$-intercept at $\left(0,2\right)$.
  • It has another identifiable point at $\left(1,\frac{2}{3}\right)$.
  • Its domain is all real numbers.
  • Its range is $y>0$.

So far so good, but what if $a<0$?

$f\left(x\right)=-2 \cdot 3^x$
  • This function has base $3$, but it is decreasing because $a$ is negative.
  • It has a horizontal asymptote at the $x$-axis.
  • It has a $y$-intercept at $\left(0,-2\right)$.
  • It has another identifiable point at $\left(1,-6\right)$.
  • Its domain is all real numbers.
  • Its range is $y<0$ (again, because $a$ is negative).

Graphs of Exponential Functions $f\left(x\right)=a \cdot b^x$

$a$ positive$a$ negative
$0 \lt b \lt 1$decreasingincreasing
$b>1$increasingdecreasing