Question

use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to give the equation of the asymptote. use the graphs to determine each functions domain and range. if applicable, use a graphing utility to confirm the hand - drawn graphs.
( h(x)=e^{x + 2}-3 )
which transformations are needed to graph the function ( h(x)=e^{x + 2}-3 )? choose the correct answer below.
a. the graph of ( f(x)=e^{x} ) should be shifted to the right by 2 units and shift ( f(x) ) upward by 3 units.
b. the graph of ( f(x)=e^{x} ) should be shifted to the left by 2 units and shift ( f(x) ) upward by 3 units.
c. the graph of ( f(x)=e^{x} ) should be shifted to the left by 2 units and shift ( f(x) ) downward by 3 units.
d. the graph of ( f(x)=e^{x} ) should be shifted to the right by 2 units and shift ( f(x) ) downward by 3 units.
graph ( h(x)=e^{x + 2}-3 ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.
find the equation of the asymptote for ( h(x)=e^{x + 2}-3 ) using the graph.
(type an equation.)

Explanation:

Step1: Recall transformation rules

For a function \(y = f(x + a)+b\), if \(a>0\), the graph of \(y = f(x)\) is shifted to the left by \(a\) units; if \(a < 0\), the graph is shifted to the right by \(|a|\) units. If \(b>0\), the graph is shifted upward by \(b\) units; if \(b<0\), the graph is shifted downward by \(|b|\) units.
For \(h(x)=e^{x + 2}-3\), compared with \(f(x)=e^{x}\), we have \(a = 2\) and \(b=-3\).
Since \(a = 2>0\), the graph of \(y = e^{x}\) is shifted to the left by \(2\) units. Since \(b=-3<0\), the graph of \(y = e^{x+2}\) is shifted downward by \(3\) units.

Step2: Find the asymptote

The asymptote of \(y = e^{x}\) is \(y = 0\).
For \(y=e^{x+2}-3\), using the transformation rule for the horizontal asymptote. If \(y = f(x)\) has an asymptote \(y = c\), then \(y=f(x + a)+b\) has an asymptote \(y=c + b\).
Since \(c = 0\) for \(y = e^{x}\) and \(b=-3\), the asymptote of \(y=e^{x+2}-3\) is \(y=-3\).

Answer:

C. The graph of \(f(x)=e^{x}\) should be shifted to the left by 2 units and shift \(f(x)\) downward by 3 units.
The equation of the asymptote is \(y=-3\).