Question

Comparing Exponential Functions In these exercises we compare the graphs of two exponential functions.

(a) Sketch the graphs ofand

(b) How are the graphs related?

Short answer

(a) The graph is:

Green curve is f(x) and purple curve is g(x).

(b) The relationship is $g\left(x\right)=3f\left(x\right)$. g(x) is a vertical stretch of f(x).

Step-by-step solution

a. Step-1 – Given

The given functions are $f\left(x\right)=2^x\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}g\left(x\right)=3\left(2^x\right)$.

Step-2 – To determine

We have to sketch the graphs of the given functions.

Step-3 – Calculation

We make tables for $f\left(x\right)=2^x\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}g\left(x\right)=3\left(2^x\right)$.

$x$	-1	0	1
$f\left(x\right)$	0.5	1	2

$x$	-1	0	1
$g\left(x\right)$	1.5	3	6

Then we plot the points and join using curves.

b. Step-1 – Given 

The given functions are $f\left(x\right)=2^x\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}g\left(x\right)=3\left(2^x\right)$.

Step-2 – To determine

We have to find how are the graphs related.

Step-3 – Calculation

From the tables in part a, we see that the g(x) values are 3 times the f(x) values.

Or,$g\left(x\right)=3f\left(x\right)$

So, the graph f(x) got vertically stretched to make the graph g(x).

Hence, g(x) is a vertical stretch of f(x).