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Exercises Exponentials and logarithms

Example 6 from Algebra and Trigonometry, 6.2 Graphs of Exponential Functions

Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

$f (x) = e^{x}$ is vertically stretched by a factor of $2$ , reflected across the y-axis, and then shifted up $4$ units.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
We want to find an equation of the general form $f (x) = a b^{x + c} + d .$ We use the description provided to find $a,$ $b,$ $c,$ and $d .$
- We are given the parent function $f (x) = e^{x},$ so $b = e .$
- The function is stretched by a factor of $2$ , so $a = 2.$
- The function is reflected about the y-axis. We replace $x$ with $- x$ to get: $e^{- x} .$
- The graph is shifted vertically 4 units, so $d = 4.$
Substituting in the general form we get,

$\begin{array}{l} f (x) & = a b^{x + c} + d \\ = 2 e^{- x + 0} + 4 \\ = 2 e^{- x} + 4 \end{array}$
The domain is $(- \infty, \infty);$ the range is $(4, \infty);$ the horizontal asymptote is $y = 4.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

$f (x) = e^{x}$ is compressed vertically by a factor of $\frac{1}{3},$ reflected across the x-axis and then shifted down $2$ units.

How did it go?

PreviousWriting and Graphing the Reflection of an Exponential Function Next Converting from Logarithmic Form to Exponential Form

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 6.2, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.