# 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 $f\left(x\right)=b^x$:

• If $0 \lt b \lt 1$, the function is decreasing.
• If $b>1$, the function 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 $\left(1,b\right)$.
• Its domain is all real numbers.
• Its range is $y>0$.

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 decreasing increasing increasing decreasing
• It has a horizontal asymptote at the $x$-axis.
• It has a $y$-intercept at $\left(0,a\right)$.
• It has another identifiable point at $\left(1,ab\right)$.
• Its domain is all real numbers.
• Its range is $y>0$ if $a$ is positive, and $y \lt 0$ if $a$ is negative.