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: Recall Transformations of Exponential Functions

For a function \( f(x) = a^{x} \), the transformation \( f(x + h) \) represents a horizontal shift. If \( h>0 \), it is a shift to the left by \( h \) units; if \( h < 0 \), it is a shift to the right by \( |h| \) units. For vertical shifts, it is of the form \( f(x)+k \), where \( k>0 \) is up and \( k < 0 \) is down.

Here, \( g(x)=3^{x + 2}=f(x + 2) \) where \( f(x)=3^{x} \). So, comparing with \( f(x+h) \), we have \( h = 2>0 \), so it is a shift to the left by 2 units.

Step2: Find the Asymptote of \( g(x)=3^{x+2} \)

The parent function \( f(x)=3^{x} \) has a horizontal asymptote \( y = 0 \) (since as \( x
ightarrow-\infty \), \( 3^{x}
ightarrow0 \)). For a horizontal shift (left or right), the horizontal asymptote remains the same in terms of its \( y \)-value (only the graph shifts horizontally). So, the asymptote of \( g(x)=3^{x + 2} \) is also \( y=0 \)? Wait, no, wait. Wait, actually, let's re - check. Wait, the general form of an exponential function \( a^{x - h}+k \) has horizontal asymptote \( y = k \). Wait, in our case, \( g(x)=3^{x+2}=3^{x-(-2)}+0 \). So the horizontal asymptote is \( y = 0 \)? Wait, no, that's a mistake. Wait, no, the parent function \( f(x)=3^{x} \) has horizontal asymptote \( y = 0 \). When we do a horizontal shift (left or right), the horizontal asymptote does not change its \( y \)-coordinate. So the horizontal asymptote of \( g(x)=3^{x + 2} \) is \( y=0 \)? Wait, no, wait, let's think again. Let's take the limit as \( x
ightarrow-\infty \). \( \lim_{x
ightarrow-\infty}3^{x + 2}=\lim_{x
ightarrow-\infty}3^{x}\times3^{2}=0\times9 = 0 \). So the horizontal asymptote is \( y = 0 \).

Step3: Find the Domain of \( g(x)=3^{x+2} \)

The domain of an exponential function \( a^{x} \) (where \( a>0,a
eq1 \)) is all real numbers, because we can plug in any real number for \( x \) and \( a^{x} \) is defined. For \( g(x)=3^{x + 2} \), we can substitute any real number for \( x \), so the domain is \( (-\infty,\infty) \) or in interval notation \( (-\infty,+\infty) \).

For the transformation question:

The function \( g(x)=3^{x + 2} \) is of the form \( f(x+2) \) where \( f(x)=3^{x} \). Using the rule for horizontal shifts of functions: if we have a function \( y = f(x) \), then \( y=f(x + h) \) is a horizontal shift. If \( h>0 \), the graph shifts \( h \) units to the left; if \( h<0 \), it shifts \( |h| \) units to the right. Here, \( h = 2>0 \), so the graph of \( f(x)=3^{x} \) is shifted 2 units to the left. So the correct option is D.

For the asymptote:

The parent function \( f(x)=3^{x} \) has a horizontal asymptote \( y = 0 \). Horizontal shifts (left or right) do not change the horizontal asymptote (since the horizontal asymptote is determined by the end - behavior as \( x
ightarrow\pm\infty \), and a horizontal shift does not affect the limit as \( x
ightarrow-\infty \) or \( x
ightarrow\infty \) in terms of the \( y \)-value of the asymptote). So the equation of the asymptote for \( g(x)=3^{x+2} \) is \( y = 0 \).

For the domain:

The function \( g(x)=3^{x+2} \) is an exponential function. Exponential functions of the form \( a^{x} \) (where \( a>0,a
eq1 \)) are defined for all real numbers \( x \). So the domain of \( g(x) \) is all real numbers, which in interval notation is \( (-\infty,\infty) \).

Answer:

Transformation: D. The graph of \( f(x) = 3^{x} \) should be shifted 2 units to the left.

Asymptote: \( y = 0 \)

Domain: \( (-\infty,\infty) \)