Question

1/14 - show solution/reason. (5 pts)
arwen studies the two functions shown below.
$f(x) = 4(2^x)$ $g(x) = 4(2^x) + 5$
what can she say about how the graph of $g(x)$ compares to the graph of $f(x)$?
the graph of $g(x)$ is the graph of $f(x)$ shifted 5 units to the right.
the graph of $g(x)$ is the graph of $f(x)$ shifted 5 units to the left.
the graph of $g(x)$ is the graph of $f(x)$ shifted 5 units up.
the graph of $g(x)$ is the graph of $f(x)$ shifted 5 units down.

Explanation:

Step1: Recall function transformation rules

For a function \( y = f(x) + k \), if \( k>0 \), the graph of \( y = f(x) \) is shifted up by \( k \) units; if \( k < 0 \), it is shifted down by \( |k| \) units. For horizontal shifts, the form is \( y=f(x - h) \), where \( h>0 \) shifts right and \( h < 0 \) shifts left.

Step2: Compare \( f(x) \) and \( g(x) \)

Given \( f(x)=4(2^{x}) \) and \( g(x)=4(2^{x})+5 \). We can see that \( g(x)=f(x)+5 \).

Step3: Determine the transformation

Using the vertical shift rule (since we are adding a constant to the entire function \( f(x) \)), adding 5 to \( f(x) \) means the graph of \( f(x) \) is shifted up by 5 units to get \( g(x) \). So the graph of \( g(x) \) is the graph of \( f(x) \) shifted 5 units up.

Answer:

The graph of \( g(x) \) is the graph of \( f(x) \) shifted 5 units up. (Corresponding to the option: "The graph of \( g(x) \) is the graph of \( f(x) \) shifted 5 units up.")