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Exercises Exponentials and logarithms

Example 1 from Algebra and Trigonometry, 6.2 Graphs of Exponential Functions

Sketching the Graph of an Exponential Function of the Form f(x) = bx

Sketch a graph of $f (x) = {0.25}^{x} .$ State the domain, range, and asymptote.

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
Before graphing, identify the behavior and create a table of points for the graph.
- Since $b = 0.25$ is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $y = 0.$
- Create a table of points as in Table 3.
  
  $x$ $- 3$ $- 2$ $- 1$ $0.015625$ $1$ $2$ $(0, 1),$
  
  $f (x) = {0.25}^{x}$ $64$ $16$ $4$ $1$ $0.25$ $0.0625$ $0.015625$
  
  Table 3
- Plot the y-intercept, $(0, 1),$ along with two other points. We can use $(- 1, 4)$ and $(1, 0.25) .$
Draw a smooth curve connecting the points as in Figure 4.

Figure 4

The domain is $(- \infty, \infty);$ the range is $(0, \infty);$ the horizontal asymptote is $y = 0.$

Your turn

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

Sketch the graph of $f (x) = 4^{x} .$ State the domain, range, and asymptote.

How did it go?

PreviousCalculating Continuous Decay Next Graphing a Shift of an Exponential Function

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.