Chapter 6.3, Problem 1E

To determine

To graph: An exponential function that has a $y$ -intercept of 2.

Explanation of Solution

Given information:

A function has $y$ -intercept of 2.

Graph:

The exponential function is the continuous function that is defined as $f\left(x\right)=a{b}^x$ where $b$ is the base such that $a\ne 0,b>0,b\ne 1$ and $x$ is the independent variable.

If the $y$ -intercept is 2 that means the value of $y=2\text{ at }x=0$ .

Substitute the value $x=0,y=2$ in the exponential function $f\left(x\right)=a{b}^x$ to obtain the value of $y$ .

  $\begin{array}{l}2=a{\left(b\right)}^0\\ a=2\end{array}$

If the value of $a=2$ then the function has $y$ -intercept of 2 that means the value of $b$ is free to choose.

Choose the value of $b=3$ to graph the function.

Now substitute $a=2$ and $b=3$ in the function $f\left(x\right)=a{b}^x$ .

  $f\left(x\right)=2{\left(3\right)}^x$

Now find the coordinates to draw the graph of the function $f\left(x\right)=2{\left(3\right)}^x$ .

Substitute $x=1$ in the function $f\left(x\right)=2{\left(3\right)}^x$ to obtain the $y$ -coordinate.

  $\begin{array}{c}y=2{\left(3\right)}^1\\ =6\end{array}$

Substitute $x=2$ in the function $f\left(x\right)=2{\left(3\right)}^x$ to obtain the $y$ -coordinate.

  $\begin{array}{c}y=2{\left(3\right)}^2\\ =18\end{array}$

Now draw the graph with the help of coordinates $\left(0,2\right)\left(1,6\right)\text{ and }\left(2,18\right)$ as shown in Figure 1.

  

Interpretation:

It can be observed from Figure 1 that the function converges to 0 when $x$ tends to $-\infty$ .