Math Core

In Algebra 1 you modeled quantities that change by a fixed percent. Now you'll treat those models as full-fledged functions: you'll graph them, shift them, read their asymptotes, and build an equation from any two points on the curve. This is the foundation for everything else in the unit, including logarithms.

Exponential functions

An exponential function has the variable in the exponent:

f(x) = a \cdot b^x, \qquad a \ne 0,\; b > 0,\; b \ne 1.

The number $a$ is the initial value (because $f(0) = a \cdot b^0 = a$ ), and $b$ is the base, or growth factor. Each time $x$ increases by $1$ , the output is multiplied by $b$ . That constant multiplier is what separates exponential functions from linear ones, which add the same amount each step.

Why the restrictions on $b$ ? If $b = 1$ , then $b^x = 1$ for every $x$ and the function is just the constant $a$ . If $b$ were negative, expressions like $(-4)^{1/2}$ would not be real numbers, so the graph would have gaps everywhere.

Definition

Exponential growth and decay

For $f(x) = a \cdot b^x$ with $a > 0$ :

If $b > 1$ , $f$ shows exponential growth: the outputs increase faster and faster.
If $0 < b < 1$ , $f$ shows exponential decay: the outputs decrease toward $0$ but never reach it.

When the base is written as $b = 1 + r$ (growth) or $b = 1 - r$ (decay), the decimal $r$ is the percent rate of change per unit of $x$ .

The shape of the graph

Here are $y = 2^x$ (growth) and $y = \left(\tfrac{1}{2}\right)^x$ (decay) on the same axes.

y = 2^x rises to the right; y = (1/2)^x falls to the right. Both cross the y-axis at 1.Open in grapher →

Both graphs share some key features:

Domain: all real numbers. You can raise a positive base to any power.
Range: $y > 0$ . A positive base raised to any power is positive.
y-intercept: $(0, 1)$ , since $b^0 = 1$ .
Horizontal asymptote: the line $y = 0$ . On one side the graph gets closer and closer to the $x$ -axis without touching it.

Notice that $\left(\tfrac{1}{2}\right)^x = 2^{-x}$ , so the decay graph is the reflection of the growth graph across the $y$ -axis.

Transformations

The transformations you learned for parent functions work here too. In

y = a \cdot b^{\,x - h} + k,

$h$ shifts the graph right, $k$ shifts it up, and $a$ stretches it vertically (and reflects it across the $x$ -axis if $a < 0$ ). The most important consequence: the horizontal asymptote moves to $y = k$ , and the range becomes $y > k$ (or $y < k$ if $a < 0$ ).

Worked example: Graph a transformed exponential

Describe the graph of $g(x) = 2^{x - 1} + 3$ . Give its asymptote, domain, range, and $y$ -intercept.

Solution. Start from $y = 2^x$ . Subtracting $1$ from $x$ shifts it right $1$ ; adding $3$ shifts it up $3$ .

Asymptote: $y = 3$ .
Domain: all real numbers. Range: $y > 3$ .
$y$ -intercept: $g(0) = 2^{-1} + 3 = 0.5 + 3 = 3.5$ , so $(0, 3.5)$ .

g(x) = 2^(x−1) + 3 with its asymptote y = 3 (dashed).Open in grapher →

Recognizing exponential data

In a table with equally spaced $x$ -values, a linear function has a constant difference between outputs. An exponential function has a constant ratio.

Worked example: Write the function from a table

$x$	$0$	$1$	$2$	$3$
$y$	$5$	$15$	$45$	$135$

Solution. The ratios are $\dfrac{15}{5} = \dfrac{45}{15} = \dfrac{135}{45} = 3$ , a constant, so the data are exponential with $b = 3$ . At $x = 0$ the value is $5$ , so $a = 5$ :

y = 5 \cdot 3^x.

Writing an equation from two points

Often you don't know the $y$ -intercept, just two points on the curve. Substitute both into $y = a \cdot b^x$ and divide the equations. The $a$ 's cancel, leaving an equation in $b$ alone.

Worked example: Two points

Find the exponential function through $(1, 40)$ and $(4, 5)$ .

Solution. The points give

40 = a b^1 \qquad\text{and}\qquad 5 = a b^4.

Divide the second equation by the first:

\frac{5}{40} = \frac{a b^4}{a b^1} \quad\Longrightarrow\quad b^3 = \frac{1}{8} \quad\Longrightarrow\quad b = \frac{1}{2}.

Substitute back: $40 = a \cdot \tfrac{1}{2}$ , so $a = 80$ . The function is $y = 80\left(\tfrac{1}{2}\right)^x$ , a decay function that halves every time $x$ goes up by $1$ .

Check: $80\left(\tfrac{1}{2}\right)^4 = 80 \cdot \tfrac{1}{16} = 5$ . ✓

Common mistake

When you divide the two equations, the exponents subtract: $\dfrac{b^4}{b^1} = b^3$ , not $b^4$ and not $b^{4/1}$ . The exponent tells you which root to take. Here you needed a cube root because the $x$ -values were $3$ apart.

Changing the time period

A growth factor always belongs to a particular unit of time. Suppose an investment is modeled by $A = 500(1.08)^t$ , with $t$ in years. What's the monthly growth factor? A month is $\tfrac{1}{12}$ of a year, so use the power rule for exponents:

A = 500(1.08)^t = 500\left(1.08^{1/12}\right)^{12t} \approx 500(1.006434)^{12t}.

Each month the value is multiplied by about $1.006434$ , a monthly rate of about $0.6434\%$ . Notice that is not $\tfrac{8\%}{12} \approx 0.667\%$ : twelve monthly increases of $0.6434\%$ , compounded, produce exactly $8\%$ for the year.

The same trick handles doubling models. If $y = 200 \cdot 2^{t/5}$ , the yearly factor is $2^{1/5} \approx 1.1487$ , so the quantity grows about $14.87\%$ per year.

Tip

To convert a growth factor from one period to another, raise it to the power $\dfrac{\text{new period}}{\text{old period}}$ . From yearly to monthly: power $\tfrac{1}{12}$ . From yearly to per decade: power $10$ .

Practice

Practice 1

Does $f(x) = 7(0.6)^x$ represent exponential growth or exponential decay, and at what rate?

Practice 2

Find the $y$ -intercept of $f(x) = 3 \cdot 4^x - 5$ . Give the $y$ -coordinate.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

The graph of $y = 5(0.5)^{x + 2} - 4$ has a horizontal asymptote $y = c$ . What is $c$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Find the range of $g(x) = -2 \cdot 3^x + 1$ . Write it as an inequality in $y$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Practice 5

The table shows an exponential function. Find $f(5)$ .

$x$	$0$	$1$	$2$	$3$
$f(x)$	$2$	$6$	$18$	$54$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Write the exponential function $y = a \cdot b^x$ whose graph passes through $(2, 36)$ and $(4, 324)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

An exponential function passes through $(0, 800)$ and $(3, 409.6)$ . By what percent does it decrease each time $x$ increases by $1$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

An account's value is $A = 1000(1.12)^t$ , where $t$ is in years. What is the equivalent monthly percent growth rate? Round to the nearest hundredth of a percent.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.