Exponential Functions: Fresh Take

  • Evaluate exponential functions.
  • Find the equation of an exponential function.
  • Use compound interest formulas.

Defining Exponential Functions

The Main Idea

  • Exponential Growth and Decay:
    • Exponential growth: Increase based on a constant multiplicative rate over equal time increments
    • Exponential decay: Decrease based on a constant multiplicative rate over equal time increments
  • Contrast with Linear Growth:
    • Exponential: Changes by the same percentage over equal increments
    • Linear: Changes by the same amount over equal increments
  • General Form of Exponential Function:
    • [latex]f(x) = ab^x[/latex]
    • [latex]a[/latex] is any nonzero number
    • [latex]b[/latex] is a positive real number, [latex]b \neq 1[/latex]
    • If [latex]b > 1[/latex]: function grows
    • If [latex]0 < b < 1[/latex]: function decays
  • Evaluating Exponential Functions:
    • Substitute the given value for [latex]x[/latex]
    • Follow the order of operations carefully

Which of the following equations represent exponential functions?

  • [latex]f(x) = 2x^2 - 3x + 1[/latex]
  • [latex]g(x) = 0.875^x[/latex]
  • [latex]h(x) = 1.75x + 2[/latex]
  • [latex]j(x) = 1095.6^{2x}[/latex]

Let [latex]f\left(x\right)=8{\left(1.2\right)}^{x - 5}[/latex]. Evaluate [latex]f\left(3\right)[/latex] using a calculator. Round to four decimal places.

You can view the transcript for “Introduction to Exponential Functions – Nerdstudy” here (opens in new window).

 

Exponential Growth and Decay

The Main Idea

  • Exponential Growth:
    • Output increases by a constant factor over equal intervals
    • General form: [latex]f(x) = ab^x[/latex], where [latex]b > 1[/latex]
    • Example: [latex]f(x) = 2^x[/latex]
  • Exponential Decay:
    • Output decreases by a constant factor over equal intervals
    • General form: [latex]f(x) = ab^x[/latex], where [latex]0 < b < 1[/latex]
    • Example: [latex]g(x) = (\frac{1}{2})^x[/latex]
  • Key Characteristics of Exponential Functions:
    • Domain: [latex](-\infty, \infty)[/latex]
    • Range: [latex](0, \infty)[/latex]
    • Horizontal asymptote: [latex]y = 0[/latex]
    • [latex]y[/latex]-intercept: [latex](0, 1)[/latex] for [latex]f(x) = b^x[/latex]
    • No [latex]x[/latex]-intercept
The population of China was about [latex]1.39[/latex] billion in the year 2013 with an annual growth rate of about [latex]0.6 \%[/latex]. This situation is represented by the growth function [latex]P\left(t\right)=1.39{\left(1.006\right)}^{t}[/latex] where [latex]t[/latex] is the number of years since 2013. To the nearest thousandth, what will the population of China be in the year 2031? How does this compare to the population prediction we made for India in the previous example?

You can view the transcript for “Exponential growth and decay word problems | Algebra II | Khan Academy” here (opens in new window).

 

Finding Equations of Exponential Functions

The Main Idea

  • General Form of Exponential Functions:
    • [latex]f(x) = ab^x[/latex]
    • [latex]a[/latex]: initial value
    • [latex]b[/latex]: growth factor (if [latex]b > 1[/latex]) or decay factor (if [latex]0 < b < 1[/latex])
  • Methods for Finding Equations:
    • Using two points
    • Using a graph
  • Key Principle:
    • Every point on the graph satisfies the equation of the function

Problem-Solving Strategies

  1. When Given Two Points:
    • If one point is [latex](0, a)[/latex], use it as the initial value
    • If no [latex](0, a)[/latex] point, set up a system of equations
  2. When Given a Graph:
    • Choose the [latex]y[/latex]-intercept as one point if possible
    • Select a second point with integer coordinates
    • Use points far apart to minimize rounding errors
  3. General Steps:
    1. Identify or calculate the initial value [latex]a[/latex]
    2. Use the second point to solve for [latex]b[/latex]
    3. Write the equation in the form [latex]f(x) = ab^x[/latex]
A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population N of wolves over time t.

Given the two points [latex]\left(1,3\right)[/latex] and [latex]\left(2,4.5\right)[/latex], find the equation of the exponential function that passes through these two points.

Find an equation for the exponential function graphed below.

Graph of an increasing function with a labeled point at (0, sqrt(2)).
Graph of f(x) with y-intercept labeled

You can view the transcript for “Ex: Find an Exponential Function Given Two Points – Initial Value Not Given” here (opens in new window).

You can view the transcript for “Writing Exponential Functions from a Graph” here (opens in new window).