Question

1. describe the transformation of $h(x) = 2^{x + 3}$. list the following...

a. y - intercept:
b. domain:
c. range:
d. asymptote:

Explanation:

Step1: Find y-intercept (set x=0)

Substitute $x=0$ into $h(x)$:
$h(0)=2^{0+3}=2^3=8$

Step2: Identify domain of exponential function

Exponential functions accept all real numbers as inputs, so domain is all real $x$.

Step3: Identify range of exponential function

The base $2>1$, so $2^{x+3}$ is always positive and approaches 0 but never reaches it.

Step4: Find horizontal asymptote

As $x\to-\infty$, $2^{x+3}\to0$, so asymptote is $y=0$.

Answer:

a. y-intercept: $8$
b. Domain: All real numbers ($(-\infty, \infty)$)
c. Range: All positive real numbers ($(0, \infty)$)
d. Asymptote: $y=0$