Question

question 7 of 16, step 1 of 1
indicate which of the following four graphs is the correct graph of this function.
q(x) = 2^{x - 3}

Explanation:

Step1: Analyze the function type

The function \( q(x) = 2^{x - 3} \) is an exponential function with base \( 2>1 \), so it should be an increasing function (as exponential functions with base \( a > 1 \) are increasing). This eliminates the first graph (which is decreasing).

Step2: Find the horizontal asymptote

For exponential functions of the form \( a^{x - h}+k \), the horizontal asymptote is \( y = k \). Here, \( q(x)=2^{x - 3}=2^{x - 3}+0 \), so the horizontal asymptote is \( y = 0 \) (the x - axis). Now, check the y - intercept by setting \( x = 0 \): \( q(0)=2^{-3}=\frac{1}{8}=0.125 \), which is a small positive number. Also, when \( x = 3 \), \( q(3)=2^{0}=1 \).

Looking at the remaining graphs:

• The second graph seems to pass through the origin (\( x = 0,y = 0 \)), but \( q(0)=\frac{1}{8}

eq0 \), so it's incorrect.

• The third graph also seems to pass through the origin, incorrect for the same reason.
• The fourth graph has a horizontal asymptote \( y = 0 \) (approaches the x - axis as \( x

ightarrow-\infty \)) and for \( x = 0 \), it has a small positive y - value, and it's an increasing function, which matches \( q(x)=2^{x - 3} \).

Answer:

The bottom - right graph (the fourth graph)