use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to give the equation…

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.\n( h(x)=e^{x + 2}-3 )\nwhich transformations are needed to graph the function ( h(x)=e^{x + 2}-3 )? choose the correct answer below.\na. the graph of ( f(x)=e^{x} ) should be shifted to the right by 2 units and shift ( f(x) ) upward by 3 units.\nb. the graph of ( f(x)=e^{x} ) should be shifted to the left by 2 units and shift ( f(x) ) upward by 3 units.\nc. the graph of ( f(x)=e^{x} ) should be shifted to the left by 2 units and shift ( f(x) ) downward by 3 units.\nd. the graph of ( f(x)=e^{x} ) should be shifted to the right by 2 units and shift ( f(x) ) downward by 3 units.\ngraph ( h(x)=e^{x + 2}-3 ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.\nfind the equation of the asymptote for ( h(x)=e^{x + 2}-3 ) using the graph.\n(type an equation.)

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.\n( h(x)=e^{x + 2}-3 )\nwhich transformations are needed to graph the function ( h(x)=e^{x + 2}-3 )? choose the correct answer below.\na. the graph of ( f(x)=e^{x} ) should be shifted to the right by 2 units and shift ( f(x) ) upward by 3 units.\nb. the graph of ( f(x)=e^{x} ) should be shifted to the left by 2 units and shift ( f(x) ) upward by 3 units.\nc. the graph of ( f(x)=e^{x} ) should be shifted to the left by 2 units and shift ( f(x) ) downward by 3 units.\nd. the graph of ( f(x)=e^{x} ) should be shifted to the right by 2 units and shift ( f(x) ) downward by 3 units.\ngraph ( h(x)=e^{x + 2}-3 ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.\nfind the equation of the asymptote for ( h(x)=e^{x + 2}-3 ) using the graph.\n(type an equation.)

Answer

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).