Lesson 6.1 : Exponential Functions

New Concept

An exponential function models rapid growth or decay through a constant percent change. We'll master evaluating functions like $f(x) = ab^x$, finding their equations, and applying them to compound interest and continuous growth problems involving the number $e$.

What’s next

This card is just the start. Next, you'll work through interactive examples of evaluating functions and tackle real-world challenge problems involving compound interest.

Property

For any real number $x$, an exponential function is a function with the form

where

• $a$ is a non-zero real number called the initial value and
• $b$ is any positive real number such that $b \neq 1$.
• The domain of $f$ is all real numbers.
• The range of $f$ is all positive real numbers if $a > 0$.
• The range of $f$ is all negative real numbers if $a < 0$.
• The $y$-intercept is $(0, a)$, and the horizontal asymptote is $y = 0$.

Property

To evaluate an exponential function with the form $f(x) = b^x$, we simply substitute $x$ with the given value, and calculate the resulting power. For functions in the form $f(x) = a(b)^x$, it is important to follow the order of operations: simplify the power $b^x$ first, then multiply by $a$.

Examples

• Let $f(x) = 4^x$. To find $f(3)$, substitute $x=3$: $f(3) = 4^3 = 64$.
• Let $g(x) = 10(2)^{x+1}$. To evaluate $g(2)$, simplify the exponent first: $g(2) = 10(2)^{2+1} = 10(2^3) = 10(8) = 80$.
• Let $h(x) = 9(\frac{1}{3})^x$. To find $h(2)$, calculate the power first: $h(2) = 9(\frac{1}{3})^2 = 9(\frac{1}{9}) = 1$.

Explanation

When evaluating, always handle the exponent first before any other operations like multiplication or addition. This ensures you're correctly calculating the exponential growth or decay before applying the initial value or other transformations.

Property

A function that models exponential growth grows by a rate proportional to the amount present. For any real number $x$ and any positive real numbers $a$ and $b$ such that $b \neq 1$, an exponential growth function has the form

where

• $a$ is the initial or starting value of the function.
• $b$ is the growth factor or growth multiplier per unit $x$.

Property

To write an exponential model $f(x)=ab^x$ given two data points:

1. If one of the data points has the form $(0, a)$, then $a$ is the initial value. Using $a$, substitute the second point into the equation $f(x) = ab^x$, and solve for $b$.
2. If neither of the data points have the form $(0, a)$, substitute both points into two equations with the form $f(x) = ab^x$. Solve the resulting system of two equations in two unknowns to find $a$ and $b$.

Examples

• Given points $(0, 5)$ and $(2, 45)$: we know $a=5$. Substitute the second point into $y=5b^x$: $45 = 5b^2$, so $b^2=9$, and $b=3$. The equation is $f(x)=5(3)^x$.
• Given points $(2, 12)$ and $(4, 108)$: create a system of equations: $12=ab^2$ and $108=ab^4$. Divide the second by the first: $\frac{108}{12} = \frac{ab^4}{ab^2}$, which gives $9=b^2$, so $b=3$. Substitute $b=3$ into the first equation: $12 = a(3)^2$, so $a=\frac{12}{9}=\frac{4}{3}$. The equation is $f(x)=\frac{4}{3}(3)^x$.
• A wolf population was 50 in 2010 and 80 in 2012. Letting $t=0$ for 2010, the points are $(0, 50)$ and $(2, 80)$. Here, $a=50$. Then $80=50b^2$, so $b = \sqrt{1.6} \approx 1.265$. The function is $N(t)=50(1.265)^t$.

Explanation

Two distinct points on a graph provide enough information to define a unique exponential curve. By substituting the coordinates into the general formula $f(x)=ab^x$, you can solve for the initial value $a$ and the base $b$.

Property

Compound interest can be calculated using the formula

where

• $A(t)$ is the account value,
• $t$ is measured in years,
• $P$ is the starting amount of the account, often called the principal, or more generally present value,
• $r$ is the annual percentage rate (APR) expressed as a decimal, and
• $n$ is the number of compounding periods in one year.

Property

The letter $e$ represents the irrational number

Its approximation is $e \approx 2.718282$. For all real numbers $t$, and all positive numbers $a$ and $r$, continuous growth or decay is represented by the formula