5.13.3 Comparing Exponential Functions

Algebra 15.13.3 Comparing Exponential Functions

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Activity

Here are graphs representing two functions and descriptions of two functions.

A graph with area (square inches) on the y-axis and time (months) on the x-axis; two upward-curving lines are shown, with a labeled point (p, q) where the lines intersect. The solid line has an y-intercept closer to 0 then the dashed line.

Function $f$ : The area of a wall that is covered by Mold A, in square inches, doubling every month.
Function $g$ : The area of a wall that is covered by Mold B, in square inches, tripling every month.

1. Which graph represents each function? Explain your answer.

Compare your answer:

Since Mold B is growing more quickly, the graph that shows a greater growth factor (the one starting lower and ending higher) is function $g$ (solid).

2. When the mold was first spotted and measured, was there more of Mold A or Mold B? Explain how you know.

Compare your answer:

There was more of Mold A. The graph for $f$ (dashed) shows a greater vertical intercept.

3. What does the point $(p, q)$ tell us in this situation? Explain your answer.

Compare your answer:

After $p$ months, there was the same amount of Mold A and Mold B on the wall ( $q$ square inches), after which Mold B started to have larger area than Mold A.

Self Check

In the formula

f(x)=a \cdot b^x

, what changes on the graph when

a

changes?

The growth factor
The slope
The $y$ -intercept
The $x$ -intercept

Additional Resources

Reading Exponential Graphs

The graphs of $f$ and $g$ are located below and are functions of the form $y = a \cdot b^{x}$ .

Graph of exponential growth functions, f and g. Function g has a smaller y-intercept and climbs more quickly than function f.

Which curve has the larger $a$ value?

$f (x)$ because its $y$ -intercept is higher.

Which curve has the larger $b$ value?

$g (x)$ because it is growing more quickly.

Try it

Try It: Reading Exponential Graphs

The graphs of $f$ and $g$ are located below and are functions of the form $y = a b^{x}$ .

Graph of exponential growth functions, f and g. Function f has a smaller y-intercept and climbs more quickly than function g.

Which curve has the larger $b$ value?

Compare your answer:

$f (x)$ has the larger $b$ value because it is growing faster than $g (x)$ .

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-1/pages/about-this-course
- Section URL: https://openstax.org/books/algebra-1/pages/5-13-3-comparing-exponential-functions

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