Question

let $f(x) = 5^x$.
let $g(x) = 5^x - 7$.
which statement describes the graph of $g(x)$ with respect to the graph of $f(x)$?
\bigcirc $g(x)$ is translated 7 units right from $f(x)$.
\bigcirc $g(x)$ is translated 7 units up from $f(x)$.
\bigcirc $g(x)$ is translated 7 units down from $f(x)$.
\bigcirc $g(x)$ is translated 7 units left from $f(x)$.

Explanation:

Step1: Recall Vertical Shift Rule

For a function \( y = f(x) + k \), if \( k>0 \), it's a shift up by \( k \) units; if \( k<0 \), it's a shift down by \( |k| \) units.
Here, \( g(x)=5^x - 7=f(x)-7 \), so \( k = - 7 \).

Step2: Determine Translation Direction

Since \( k=-7 \) (negative), the graph of \( g(x) \) is the graph of \( f(x) \) shifted down by 7 units.

Answer:

\( g(x) \) is translated 7 units down from \( f(x) \) (the third option: \( g(x) \) is translated 7 units down from \( f(x) \))