How does the horizontal asymptote of $f(x)=2^{x+1}+4$ compare to the horizontal asymptote of $g(x)=2^{x+1}-4$ ? $\newline$ The horizontal asymptote of $f(x)$ is $4$ units less than the horizontal asymptote of $g(x)$ . $\newline$ The horizontal asymptote of $f(x)$ is $4$ units greater than the horizontal asymptote of $g(x)$ . $\newline$ The horizontal asymptote of $f(x)$ is $8$ units less than the horizontal asymptote of $g(x)$ . $\newline$ The horizontal asymptote of $f(x)$ is $8$ units greater than the horizontal asymptote of $g(x)$

View step-by-step help

Home

Math Problems

Algebra 2

Function transformation rules

Full solution

Q. How does the horizontal asymptote of

f(x)=2^{x+1}+4

compare to the horizontal asymptote of

g(x)=2^{x+1}-4

\newline

The horizontal asymptote of

f(x)

4

units less than the horizontal asymptote of

g(x)

\newline

The horizontal asymptote of

f(x)

4

units greater than the horizontal asymptote of

g(x)

\newline

The horizontal asymptote of

f(x)

8

units less than the horizontal asymptote of

g(x)

\newline

The horizontal asymptote of

f(x)

8

units greater than the horizontal asymptote of

g(x)

Identify Horizontal Asymptote: Identify the horizontal asymptote of the function $g(x) = 2^{(x+1)} - 4$ . $\newline$ The base function here is $2^{(x+1)}$ , which is an exponential function. The horizontal asymptote of an exponential function of the form $a^x$ , where $a > 0$ and $a \neq 1$ , is $y = 0$ . The transformation $-4$ shifts the graph vertically down by $4$ units.
Determine Asymptote for $g(x)$ : Determine the horizontal asymptote of $g(x)$ . Since the base function $2^{(x+1)}$ has a horizontal asymptote at $y = 0$ , shifting it down by $4$ units will result in a new horizontal asymptote at $y = -4$ for $g(x)$ .
Identify Horizontal Asymptote: Identify the horizontal asymptote of the function $f(x) = 2^{(x+1)} + 4$ . $\newline$ Using the same reasoning as in Step $1$ , the base function $2^{(x+1)}$ has a horizontal asymptote at $y = 0$ . The transformation $+4$ shifts the graph vertically up by $4$ units.
Determine Asymptote for $f(x)$ : Determine the horizontal asymptote of $f(x)$ . Since the base function $2^{(x+1)}$ has a horizontal asymptote at $y = 0$ , shifting it up by $4$ units will result in a new horizontal asymptote at $y = 4$ for $f(x)$ .
Compare Asymptotes: Compare the horizontal asymptotes of $f(x)$ and $g(x)$ . The horizontal asymptote of $f(x)$ is at $y = 4$ , and the horizontal asymptote of $g(x)$ is at $y = -4$ . To compare them, we calculate the difference: $4 - (-4) = 8$ .
Determine Correct Statement: Determine the correct statement based on the comparison. $\newline$ Since the horizontal asymptote of $f(x)$ is $8$ units above that of $g(x)$ , the correct statement is that the horizontal asymptote of $f(x)$ is $8$ units greater than the horizontal asymptote of $g(x)$ .