Question

use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to the give equations of the asymptotes. use the graphs to determine each functions domain and range. if applicable, use a graphing utility to confirm the hand - drawn graphs.
( g(x)=e^{x}-5 )
which transformation is needed to graph the function ( g(x)=e^{x}-5 )? choose the correct answer below.
a. the graph of ( f(x)=e^{x} ) should be shifted 5 units to the right.
b. the graph of ( f(x)=e^{x} ) should be shifted 5 units upward.
c. the graph of ( f(x)=e^{x} ) should be shifted 5 units to the left.
d. the graph of ( f(x)=e^{x} ) should be shifted 5 units downward.

Explanation:

Step1: Recall transformation rules

For a function \(y = f(x)+k\), if \(k>0\), the graph is shifted \(k\) units upward; if \(k < 0\), the graph is shifted \(|k|\) units downward. For a function \(y=f(x - h)\), if \(h>0\), the graph is shifted \(h\) units to the right; if \(h<0\), the graph is shifted \(|h|\) units to the left. The function \(g(x)=e^{x}-5\) can be written as \(g(x)=f(x)-5\) where \(f(x)=e^{x}\).

Step2: Apply the transformation rule

Since \(g(x)=f(x)-5\) (in the form \(y = f(x)+k\) with \(k=- 5\)), according to the rule for vertical shifts of the form \(y = f(x)+k\), the graph of \(y = f(x)\) is shifted \(|k| = 5\) units downward.

Answer:

D. The graph of \(f(x)=e^{x}\) should be shifted 5 units downward.