Question

given the equation: $f(x) = 4^{x+2}+2$, answer the following:
a. growth or decay?
b. h =
c. k =
d. horizontal asymptote:
e. find f(-1). write as a coordinate pair

Explanation:

Step1: Identify function type

The function is $f(x)=4^{x+2}+2$, an exponential function $b^x$ with $b=4>1$, so it's growth.

Step2: Rewrite to match form $b^{x-h}+k$

Rewrite as $f(x)=4^{x-(-2)}+2$, so $h=-2$, $k=2$.

Step3: Find horizontal asymptote

For $f(x)=b^{x-h}+k$, asymptote is $y=k$, so $y=2$.

Step4: Calculate $f(-1)$

Substitute $x=-1$:

$$\begin{align*} f(-1)&=4^{(-1)+2}+2\\ &=4^{1}+2\\ &=4+2\\ &=6 \end{align*}$$

The coordinate pair is $(-1,6)$.

Answer:

A. growth
B. $-2$
C. $2$
D. $y=2$
E. $(-1, 6)$