Exponential Functions

In an exponential function, the variable is in the exponent, as in $f(x) = 2^x$. Instead of adding the same amount each step, an exponential function multiplies by the same amount each step. That’s how populations, savings accounts, and viral videos grow, and how medicines and radioactive materials fade.

Key ideas

The exponential function f(x) = aˣ

$f(x) = a^x$ is an exponential function when the base $a$ is a positive number other than $1$.

• If $a \lt 0$, many values aren’t real numbers: $(-4)^{\frac{1}{2}} = \sqrt{-4}$ isn’t defined.
• If $a = 1$, then $1^x = 1$ for every $x$, which is just a horizontal line.

Thanks to rational exponents, $a^x$ has exactly one value for every real $x$, so it is a function: every vertical line crosses its graph once.

Growth and decay

• $a \gt 1$: exponential growth. The graph rises from left to right, slowly at first and then very steeply.
• $0 \lt a \lt 1$: exponential decay. The graph falls from left to right and levels off.

Both graphs pass through $(0, 1)$ and approach the $x$-axis without ever touching it.

Key properties of y = aˣ

Property	Value
domain	$\{x \in \mathbb{R}\}$
range	$\{y \in \mathbb{R} \mid y \gt 0\}$
$y$-intercept	$1$, since $a^0 = 1$
$x$-intercept	none, since $a^x$ is never $0$
horizontal asymptote	$y = 0$ (the $x$-axis)
increasing or decreasing	increasing if $a \gt 1$, decreasing if $0 \lt a \lt 1$

Telling patterns apart from a table

When the $x$-values go up in equal steps:

• Linear: the first differences (subtract consecutive $y$-values) are constant.
• Quadratic: the second differences are constant.
• Exponential: the ratios of consecutive $y$-values are constant. Each $y$ is the previous one times the same number.

Worked examples

Example 1: A table and its properties

Make a table of values for $y = 3^x$ from $x = -2$ to $x = 2$, and state its key properties.

Solution.

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$\tfrac{1}{9}$	$\tfrac{1}{3}$	$1$	$3$	$9$

Each value is $3$ times the one before. Since $3 \gt 1$, the function is increasing. Domain $\{x \in \mathbb{R}\}$, range $\{y \in \mathbb{R} \mid y \gt 0\}$, $y$-intercept $1$, horizontal asymptote $y = 0$.

Example 2: A decay function

Describe the graph of $y = \left(\tfrac{1}{4}\right)^x$, and find $y$ when $x = -1$.

Solution. The base $\tfrac{1}{4}$ is between $0$ and $1$, so this is exponential decay: the graph falls from left to right. It passes through $(0, 1)$, approaches the asymptote $y = 0$ on the right, and has range $y \gt 0$.

$$\left(\tfrac{1}{4}\right)^{-1} = 4$$

So the point $(-1, 4)$ is on the graph.

Example 3: Which kind of function?

Decide whether each table is linear, quadratic, or exponential.

$x$	$0$	$1$	$2$	$3$	$4$
A	$3$	$6$	$12$	$24$	$48$
B	$3$	$5$	$9$	$15$	$23$
C	$2$	$5$	$8$	$11$	$14$

Solution.

• A: the ratios are all $\tfrac{6}{3} = \tfrac{12}{6} = \dots = 2$. Exponential.
• B: the first differences are $2, 4, 6, 8$, and the second differences are all $2$. Quadratic.
• C: the first differences are all $3$. Linear.

Example 4: Finding the base

Find $a$ if $y = a^x$ passes through $(3, 64)$. Then find $a$ if it passes through $(-2, 25)$.

Solution. $a^3 = 64$, so $a = \sqrt[3]{64} = 4$.

$a^{-2} = 25$ means $\dfrac{1}{a^2} = 25$, so $a^2 = \dfrac{1}{25}$ and $a = \dfrac{1}{5}$ (the base must be positive).

Common mistakes

Drawing the graph crossing the $x$-axis. $a^x$ is never $0$ or negative. The graph gets closer and closer to the axis but never reaches it.

Mixing up $2^x$ and $x^2$. In $2^x$ the variable is the exponent; in $x^2$ it’s the base. $2^x$ eventually grows much faster.

Saying the $y$-intercept is $0$. At $x = 0$, $a^0 = 1$, so every $y = a^x$ crosses the $y$-axis at $1$.

Checking differences instead of ratios. Exponential patterns have constant ratios. Their differences keep changing.

Calling a decreasing graph “negative”. $y = \left(\tfrac{1}{2}\right)^x$ is decreasing, but all its $y$-values are still positive.

Practice

1. (Warm-up) Which of these are exponential functions?

• (a) $y = 5^x$
• (b) $y = x^5$
• (c) $y = 0.3^x$
• (d) $y = 1^x$

Solution

(a) and (c). (b) has the variable in the base, not the exponent. (d) has base $1$, so it’s just the line $y = 1$.

2. (Warm-up) State the $y$-intercept and the horizontal asymptote of $y = 7^x$.

Solution

$y$-intercept $1$; horizontal asymptote $y = 0$.

3. (Warm-up) Is each function increasing or decreasing?

• (a) $y = 1.5^x$
• (b) $y = \left(\tfrac{2}{3}\right)^x$
• (c) $y = 0.9^x$

Solution

(a) Increasing, since $1.5 \gt 1$. (b) and (c) are decreasing, since their bases are between $0$ and $1$.

4. (Core) Is this table linear, quadratic, or exponential? Explain.

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$80$	$40$	$20$	$10$	$5$

Solution

Exponential. Each value is half the one before: the ratio is always $\tfrac{1}{2}$. (The first differences, $-40, -20, -10, -5$, are not constant, so it isn’t linear.)

5. (Core) Is this table linear, quadratic, or exponential? Explain.

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$1$	$4$	$9$	$16$	$25$

Solution

Quadratic. The first differences are $3, 5, 7, 9$, and the second differences are all $2$. (The ratios $4, 2.25, \dots$ change, so it isn’t exponential.)

6. (Core) Find the base $a$ of $y = a^x$ if the graph passes through each point.

• (a) $(2, 36)$
• (b) $\left(-1, \tfrac{1}{3}\right)$

Solution

(a) $a^2 = 36$, so $a = 6$.

(b) $a^{-1} = \tfrac{1}{3}$ means $\tfrac{1}{a} = \tfrac{1}{3}$, so $a = 3$.

7. (Core) Compare $2^x$ and $x^2$ for $x = 1, 2, 3, 4, 5, 6$. Where are they equal, and which is bigger for large $x$?

Solution

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$2^x$	$2$	$4$	$8$	$16$	$32$	$64$
$x^2$	$1$	$4$	$9$	$16$	$25$	$36$

They’re equal at $x = 2$ and $x = 4$. After $x = 4$, $2^x$ pulls ahead and stays ahead, because doubling eventually beats squaring.

8. (Challenge) Explain why $y = (-2)^x$ isn’t studied as an exponential function.

Solution

Many of its values aren’t real numbers: $(-2)^{\frac{1}{2}} = \sqrt{-2}$ is undefined. Even at whole numbers, the values jump between positive and negative ($-2, 4, -8, 16, \dots$), so there’s no smooth curve. That’s why the base must be positive.

9. (Challenge) Show that $y = \left(\tfrac{1}{3}\right)^x$ is the reflection of $y = 3^x$ in the $y$-axis.

Solution

$$\left(\frac{1}{3}\right)^x = \left(3^{-1}\right)^x = 3^{-x}$$

Replacing $x$ with $-x$ in $y = 3^x$ reflects its graph in the $y$-axis. For example, $(2, 9)$ on $y = 3^x$ matches $(-2, 9)$ on $y = \left(\tfrac{1}{3}\right)^x$, since $\left(\tfrac{1}{3}\right)^{-2} = 9$.