7.4.3 Comparing Exponential and Quadratic Functions

Exponential versus Quadratic Functions

For each of the two functions, do the following:

1. Complete the table of values.
2. Sketch a graph.
3. Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know.

1. $f(x)=3·2^x$

This function is exponential. It has a growth factor of $2$.

2. $g(x)=3·x^2$

This function is not linear because it does not have a constant rate of change, and it is not exponential because it does not have a constant growth factor. The function is quadratic because it can be represented by $g(x)=3x^2$, which has $x$ to the second power, and that is the greatest power.

Note: Between $f(x)$ and $g(x)$, $f(x)$ will have the greater value as $x$ increases.

Try it

Exponential versus Quadratic Functions

Which of the following has the greater value as $x$ increases, $f(x)$ or $g(x)$?

$f(x)=3x^2$

$g(x)=3^x$

Here is how to determine which function has the greater value as $x$ increases:

Make a table of values for each function, and then graph the function:

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$f(x)=3x^2$	$12$	$3$	$0$	$3$	$12$	$27$	$48$

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$g(x)=3^x$	$\frac{1}{9}$	$\frac{1}{3}$	$1$	$3$	$9$	$27$	$81$

Looking at both the table of values and the graphs, $g(x)$ will have a greater value than $f(x)$ as $x$ increases. This is because $g(x)$ is exponential and has a constant growth factor.