Mastery Check Solutions

Show What You Know

Write the equation of an exponential equation that passes through $(2, \frac{9}{4})$ and $(- 1, \frac{1}{12}) .$

$\begin{matrix} y = a b^{x} \\ \frac{9}{4} = a b^{2} & \frac{1}{12} = (\frac{9}{4 b^{2}}) b^{- 1} & a = \frac{9}{4 {(3)}^{2}} \\ a = \frac{9}{4 b^{2}} & \frac{1}{12} = \frac{9}{4 b^{3}} & a = \frac{9}{4 (9)} \\ \frac{4 b^{3}}{4} = \frac{(12) (9)}{4} & a = \frac{1}{4} \\ b^{3} = 27 \\ b = 3 & y = \frac{1}{4} {(3)}^{x} \end{matrix}$

Note

Q: Does this equation represent growth or decay? Explain.

A: This is a growth function because $b > 1$ .

You can check that your equation is correct by using technology and confirming that the points are on the equation’s graph.

Write a new equation for the exponential function in part A, translated right two spaces and down three spaces.

$a = \frac{1}{4}, h = 2, k = - 3 y = \frac{1}{4} {(3)}^{x - 2} - 3$

Note

Remember to identify a, h, and k for translations.

Graph part A and part B on the same coordinate plane.

Note

Q: How does the end behavior change as the graphs approach negative infinity?

A: The graph from part A approaches zero, and the graph from part B approaches –3.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Graph an exponential function.
Describe transformations on exponential functions.
Transform an exponential function.
Write an exponential function given two points.