Question

begin by graphing f(x) = 3^x. then use transformations of this graph to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the function’s domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs.

g(x) = 3^(x + 2)

which transformation is needed to graph the function g(x) = 3^(x + 2)? choose the correct answer below.

a. the graph of f(x) = 3^x should be shifted 2 units to the right.

b. the graph of f(x) = 3^x should be shifted 2 units upward.

c. the graph of f(x) = 3^x should be shifted 2 units downward.

d. the graph of f(x) = 3^x should be shifted 2 units to the left.

graph g(x) = 3^(x + 2) and its asymptote. use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

click to enlarge graph

the equation of the asymptote for g(x) = 3^(x + 2) is . (type an equation.)

the domain of g(x) = 3^(x + 2) is .
(type your answer in interval notation.)

Explanation:

Step1: Analyze Transformation

For exponential functions, the transformation \( f(x + h) \) shifts \( f(x) \) left by \( h \) units, and \( f(x) + k \) shifts up by \( k \) units. Here, \( g(x)=3^{x + 2}=f(x + 2) \), so it's a horizontal shift. A positive \( h \) (here \( h = 2 \)) in \( f(x+h) \) means shift left by 2 units. So option D is correct.

Step2: Find Asymptote

The parent function \( f(x)=3^x \) has a horizontal asymptote \( y = 0 \). For \( g(x)=3^{x+2} \), horizontal shifts don't change the horizontal asymptote. So the asymptote is \( y = 0 \).

Step3: Determine Domain

Exponential functions of the form \( a^{x + h} \) (where \( a>0,a
eq1 \)) have domain all real numbers. So domain of \( g(x)=3^{x+2} \) is \( (-\infty,\infty) \).

Answer:

1. Transformation: D. The graph of \( f(x) = 3^x \) should be shifted 2 units to the left.
2. Asymptote: \( y = 0 \)
3. Domain: \( (-\infty, \infty) \)