Comparing Exponential and Quadratic Functions

7.4.3 Comparing Exponential and Quadratic Functions

7.4.3 • Comparing Exponential and Quadratic Functions

Activity

Here are two functions: $p (x) = 6 x^{2}$ and $q (x) = 3^{x}$ .

Investigate the output of $p$ and $q$ for different values of $x$ . For large enough values of $x$ , one function will have a greater value than the other.

Which function will have a greater value as $x$ increases?

Support your answer with tables, graphs, or other representations.

Compare your answer:

$q$ is larger when $x = 0$ and again when $x = 5$ . Once $x$ is larger than $5$ , $q$ is growing much more quickly, as shown in the table.

	$p (x)$	$q (x)$
$0$	$0$	$1$
$1$	$6$	$3$
$2$	$24$	$9$
$3$	$54$	$27$
$4$	$96$	$81$
$5$	$150$	$243$
$6$	$216$	$729$
$7$	$294$	$2, 187$

Alternatively, here is a graph showing the plotted values from the table and another showing $y = p (x)$ and $y = q (x)$ .

A graph showing the plotted values from the table. The x-axis has a scale of 2 that extends from 0 to 8. The y-axis has a scale of 200 that extends from 0 to 800. The blue points represent data for function q. The black points represent data for function p.

Graphs of the curves approximating the data for functions q and p on a coordinate grid. The x-axis has a scale of 2 that extends from 0 to 8. The y-axis has a scale of 200 that extends from 0 to 800. The blue points represent data for function q. The black points represent data for function p.

Are you ready for more?

Extending Your Thinking

Janna says that some exponential functions grow more slowly than the quadratic function as $x$ increases. Do you agree with Janna? Be prepared to show your reasoning.

Compare your answer:

Yes. This is true for the functions $g (x) = (1 \frac{1}{2}) x$ and $f (x) = x^{2}$ as $x$ increases for $x$ -values less than $10$ .

Could you have an exponential function $g (x) = b^{x}$ so that $g (x) < f (x)$ for all values of $x$ ?

Compare your answer:

No, $g (0) > f (0)$ for all values of $b$ (except $0$ ), because $b^{0} = 1$ and $0^{2} = 0$ . (If $b = 0$ , there are problems if $x \leq 0$ , so this is not usually considered an exponential function.)

Video: Comparing Exponential and Quadratic Functions

Watch the following video to learn more about functions.

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Self Check

Which of the following has the greatest value as

x

increases?

$y=(\frac {1}{4})^x$
$y=4x$
$y=4^x$
$y=4 \cdot x^2$

Additional Resources

Exponential versus Quadratic Functions

For each of the two functions, do the following:

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

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

$x$	$- 1$	$0$	$1$	$2$	$3$	$5$
$f (x)$	$\frac{3}{2}$ or equivalent	$3$	$6$	$12$	$24$	$96$

Values of the exponential function f(x) = 3 * 2^x for specific input values of x.

A graph of an exponential curve with plotted points. The x-axis ranges from -1 to 6 and the y-axis from 0 to 110. The curve rises slowly at first, then increases rapidly after x equals 3.

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

$g (x) = 3 \cdot x^{2}$

$x$	$- 1$	$0$	$1$	$2$	$3$	$5$
$g (x)$	$3$	$0$	$3$	$12$	$27$	$75$

Table showing corresponding values of x and g(x)=3x^2 across several data points.

A graph of an quadratic curve with plotted points. The x-axis ranges from -1 to 6 and the y-axis from 0 to 110. The curve rises slowly at first, then increases after x equals 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 $g (x) = 3 x^{2}$ , 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 $g (x)$ ?

$f (x) = 3 x^{2}$

$g (x) = 3^{x}$

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:

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

Table showing the evaluation of the function f(x) = 3x^2 for integer values of x from -2 to 4.

Graph of a parabola opening upward, labeled f(x) in orange. The vertex is at the origin (0,0), and the axes range from negative 10 to 10 on the x-axis and from 0 to 15 on the y-axis.

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

Values for the exponential function g(x) = 3^x for integer inputs x from -2 to 4.

An exponential curve rising steeply from near zero at x equals negative 2, passing through the point (0, 1) and reaching the point (2, 9), on a grid with labeled axes.

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.