Exponential Function

A Variable in the Exponent

An exponential function has its independent variable in the exponent. The growth and decay models below use the general form $f(x) = a \cdot b^x$ with $a > 0$ , $b > 0$ , and $b \neq 1$ .

Components of exponential functions:

In the function $f(x) = a \cdot b^x$ :

$a$ is the multiplier constant that determines the initial value of the function
$b$ is the exponential base that determines the rate of growth or decay
$x$ is the independent variable (exponent)

Characteristics of Exponential Functions

Exponential functions have several special properties that distinguish them from other functions. The base decides whether the graph grows or falls, and the value at $x = 0$ is always the same.

How the Graph Behaves

Two identities fix the shape of every exponential graph: it passes through the point where the input is zero, and adding two inputs multiplies the two function values.

f(0) = a \cdot b^0 = a \cdot 1 = a

Types of Exponential Functions

Exponential Growth Function ( $b > 1$ ):

Function values increase as $x$ increases
Graph rises from left to right
Example: $f(x) = 2^x$

Exponential Decay Function ( $0 < b < 1$ ):

Function values decrease as $x$ increases
Graph falls from left to right
Example: $f(x) = (0.5)^x$

Graphs of Exponential Functions

The following is a visualization of various exponential functions:

Comparison of Exponential Functions

The graph shows the growth function

f(x) = 2^x

and decay function

g(x) = (0.5)^x

Comparison of function values $f(x) = 2^x$ and $g(x) = (0.5)^x$ :

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x) = 2^x$	$0.25$	$0.5$	$1$	$2$	$4$	$8$
$g(x) = (0.5)^x$	$4$	$2$	$1$	$0.5$	$0.25$	$0.125$

Transformations of Exponential Functions

A vertical shift moves the asymptote, while a horizontal shift moves a reference point on the graph. Each constant added to the function shifts the graph in one fixed direction, and the transformed equation shows which direction that is.

Vertical Translation

The function $f(x) = a \cdot b^x + c$ shifts the graph upward (if $c > 0$ ) or downward (if $c < 0$ ).

Vertical Translation

Comparison of

f(x) = 2^x

with

g(x) = 2^x + 2

Horizontal Translation

The function $f(x) = a \cdot b^{x-h}$ shifts the graph to the right (if $h > 0$ ) or to the left (if $h < 0$ ). The shift moves every point by the same amount, so the asymptote stays where it is and the shape of the curve does not change.

Applications of Exponential Functions

The quantity in each example below changes by a constant factor over equal intervals. That factor represents population growth, decay, or an increase in an account balance due to interest.

Population Growth

When a population has a constant proportional growth rate and its conditions do not change, an exponential model can describe its growth. If the initial population is $P_0$ and the growth rate is $r$ per time period, then:

P(t) = P_0 \cdot (1 + r)^t

Example: Bacterial population that reproduces every hour at a rate of $20\%$ :

Initial population: $1000 \text{ bacteria}$
Growth rate: $r = 0.2$
Function: $P(t) = 1000 \cdot (1.2)^t$

Bacterial Growth Table

Time (hours)	$0$	$1$	$2$	$3$	$4$	$5$
Population	$1000$	$1200$	$1440$	$1728$	$2074$	$2488$

The model gives $2073.6$ and $2488.32$ at hours $4$ and $5$ . The table rounds those populations to the nearest whole bacterium.

Radioactive Decay

Radioactive substances decay following exponential functions. If the initial mass is $M_0$ and the half-life is $t_{1/2}$ , then:

M(t) = M_0 \cdot \left(\frac{1}{2}\right)^{t/t_{1/2}}

Compound Interest

Investments with compound interest grow exponentially. If the initial capital is $P$ , interest rate is $r$ per year, and time is $t \text{ years}$ :

A = P \cdot \left(1 + \frac{r}{n}\right)^{nt}

In this formula, $n$ is the number of times interest is compounded per year.

Exponential Equations

An exponential equation is an equation that contains a variable in the exponent. General form:

b^x = c

Method 1: Equalizing Bases

If $b^{f(x)} = b^{g(x)}$ , then $f(x) = g(x)$

Example: Solve $2^{x+1} = 8$

2^{x+1} = 8

Method 2: Using Logarithms

For $b>0$ , $b\neq1$ , and $c>0$ , solve $b^x = c$ with logarithms:

x = \log_b c = \frac{\log c}{\log b}

Exercises

Each problem gives information about an exponential function and asks for the function itself or for one of its values, so write the general form first and substitute the given point.

Determine the value of $f(3)$ if $f(x) = 5 \cdot 2^x$
Solve the equation $3^{2x-1} = 27$
A city's population is $50{,}000 \text{ people}$ and grows $3\%$ per year. What will the population be after $10 \text{ years}$ ?
A radioactive substance has a half-life of $5 \text{ years}$ . If the initial mass is $100 \text{ grams}$ , how much mass remains after $15 \text{ years}$ ?

Worked Solutions

Cover a solution, work the problem yourself, then compare your steps and your final value. Work through the steps in order, because one wrong step carries into the final result.

Evaluate the function

Substitute $x = 3$ into $f(x) = 5 \cdot 2^x$ . Evaluate the exponent before multiplying by the initial factor.

f(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40

$f(3) = 40$ .

Match equal bases

Rewrite $27$ as $3^3$ . Equal positive bases have equal exponents.

3^{2x-1} = 3^3

Checking gives $3^{2(2)-1} = 3^3 = 27$ , so the solution is correct.

Model population growth

The initial population is $P_0 = 50{,}000$ , the annual growth rate is $r = 0.03$ , and the time is $t = 10$ years. Use the annual growth factor $1 + r = 1.03$ .

P(10) = 50{,}000 \cdot (1.03)^{10}

The population is rounded to the nearest whole person.

Apply the half life

Fifteen years contain $15/5 = 3$ half-life periods. The mass is therefore multiplied by $1/2$ three times.

M(15) = 100 \cdot \left(\frac{1}{2}\right)^3

After $15 \text{ years}$ , $12.5 \text{ grams}$ remains.

A Variable in the Exponent

An exponential function has its independent variable in the exponent. The growth and decay models below use the general form $f(x) = a \cdot b^x$ with $a > 0$ , $b > 0$ , and $b \neq 1$ .

Components of exponential functions:

In the function $f(x) = a \cdot b^x$ :

$a$ is the multiplier constant that determines the initial value of the function
$b$ is the exponential base that determines the rate of growth or decay
$x$ is the independent variable (exponent)

Characteristics of Exponential Functions

Exponential functions have several special properties that distinguish them from other functions. The base decides whether the graph grows or falls, and the value at $x = 0$ is always the same.

How the Graph Behaves

Two identities fix the shape of every exponential graph: it passes through the point where the input is zero, and adding two inputs multiplies the two function values.

f(0) = a \cdot b^0 = a \cdot 1 = a

Types of Exponential Functions

Exponential Growth Function ( $b > 1$ ):

Function values increase as $x$ increases
Graph rises from left to right
Example: $f(x) = 2^x$

Exponential Decay Function ( $0 < b < 1$ ):

Function values decrease as $x$ increases
Graph falls from left to right
Example: $f(x) = (0.5)^x$

Graphs of Exponential Functions

The following is a visualization of various exponential functions:

Comparison of Exponential Functions

The graph shows the growth function

f(x) = 2^x

and decay function

g(x) = (0.5)^x

Comparison of function values $f(x) = 2^x$ and $g(x) = (0.5)^x$ :

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x) = 2^x$	$0.25$	$0.5$	$1$	$2$	$4$	$8$
$g(x) = (0.5)^x$	$4$	$2$	$1$	$0.5$	$0.25$	$0.125$

Transformations of Exponential Functions

Vertical Translation

The function $f(x) = a \cdot b^x + c$ shifts the graph upward (if $c > 0$ ) or downward (if $c < 0$ ).

Vertical Translation

Comparison of

f(x) = 2^x

with

g(x) = 2^x + 2

Horizontal Translation

Applications of Exponential Functions

The quantity in each example below changes by a constant factor over equal intervals. That factor represents population growth, decay, or an increase in an account balance due to interest.

Population Growth

P(t) = P_0 \cdot (1 + r)^t

Example: Bacterial population that reproduces every hour at a rate of $20\%$ :

Initial population: $1000 \text{ bacteria}$
Growth rate: $r = 0.2$
Function: $P(t) = 1000 \cdot (1.2)^t$

Bacterial Growth Table

Time (hours)	$0$	$1$	$2$	$3$	$4$	$5$
Population	$1000$	$1200$	$1440$	$1728$	$2074$	$2488$

The model gives $2073.6$ and $2488.32$ at hours $4$ and $5$ . The table rounds those populations to the nearest whole bacterium.

Radioactive Decay

Radioactive substances decay following exponential functions. If the initial mass is $M_0$ and the half-life is $t_{1/2}$ , then:

M(t) = M_0 \cdot \left(\frac{1}{2}\right)^{t/t_{1/2}}

Compound Interest

Investments with compound interest grow exponentially. If the initial capital is $P$ , interest rate is $r$ per year, and time is $t \text{ years}$ :

A = P \cdot \left(1 + \frac{r}{n}\right)^{nt}

In this formula, $n$ is the number of times interest is compounded per year.

Exponential Equations

An exponential equation is an equation that contains a variable in the exponent. General form:

b^x = c

Method 1: Equalizing Bases

If $b^{f(x)} = b^{g(x)}$ , then $f(x) = g(x)$

Example: Solve $2^{x+1} = 8$

2^{x+1} = 8

Method 2: Using Logarithms

For $b>0$ , $b\neq1$ , and $c>0$ , solve $b^x = c$ with logarithms:

x = \log_b c = \frac{\log c}{\log b}

Exercises

Each problem gives information about an exponential function and asks for the function itself or for one of its values, so write the general form first and substitute the given point.

Determine the value of $f(3)$ if $f(x) = 5 \cdot 2^x$
Solve the equation $3^{2x-1} = 27$
A city's population is $50{,}000 \text{ people}$ and grows $3\%$ per year. What will the population be after $10 \text{ years}$ ?
A radioactive substance has a half-life of $5 \text{ years}$ . If the initial mass is $100 \text{ grams}$ , how much mass remains after $15 \text{ years}$ ?

Worked Solutions

Cover a solution, work the problem yourself, then compare your steps and your final value. Work through the steps in order, because one wrong step carries into the final result.

Evaluate the function

Substitute $x = 3$ into $f(x) = 5 \cdot 2^x$ . Evaluate the exponent before multiplying by the initial factor.

f(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40

$f(3) = 40$ .

Match equal bases

Rewrite $27$ as $3^3$ . Equal positive bases have equal exponents.

3^{2x-1} = 3^3

Checking gives $3^{2(2)-1} = 3^3 = 27$ , so the solution is correct.

Model population growth

The initial population is $P_0 = 50{,}000$ , the annual growth rate is $r = 0.03$ , and the time is $t = 10$ years. Use the annual growth factor $1 + r = 1.03$ .

P(10) = 50{,}000 \cdot (1.03)^{10}

The population is rounded to the nearest whole person.

Apply the half life

Fifteen years contain $15/5 = 3$ half-life periods. The mass is therefore multiplied by $1/2$ three times.

M(15) = 100 \cdot \left(\frac{1}{2}\right)^3

After $15 \text{ years}$ , $12.5 \text{ grams}$ remains.