Question

the graph of an exponential function is given. select the function for the graph from the functions listed. f(x)=5^{x+2} f(x)=5^{x} f(x)=5^{x}+2 f(x)=5^{x}-2

Explanation:

Step1: Analyze the parent function $f(x)=5^x$

The parent exponential function $f(x) = 5^x$ has a y - intercept at $(0,1)$ because when $x = 0$, $f(0)=5^0 = 1$.

Step2: Analyze the y - intercept of the given graph

From the graph, when $x = 0$, we can see that the y - value is $1 + 2=3$? Wait, no, looking at the graph, the y - intercept seems to be at $(0,3)$? Wait, no, let's re - evaluate. Wait, the standard form of a vertical shift of an exponential function is $f(x)=a^x + k$, where $k$ is the vertical shift. If we consider the parent function $y = 5^x$ (which passes through $(0,1)$), and we want to find the vertical shift. Let's check the options:

• For $f(x)=5^{x + 2}$, when $x=0$, $f(0)=5^{2}=25$, which is not matching the graph's y - intercept.
• For $f(x)=5^{x}$, when $x = 0$, $f(0)=1$, but the graph's y - intercept is higher.
• For $f(x)=5^{x}+2$, when $x = 0$, $f(0)=5^{0}+2=1 + 2=3$. Looking at the graph, the y - intercept is around $(0,3)$ (since it's above $y = 1$ and below $y = 5$).
• For $f(x)=5^{x}-2$, when $x = 0$, $f(0)=5^{0}-2=1 - 2=- 1$, which is below the x - axis, not matching the graph.

So the function that matches the graph is $f(x)=5^{x}+2$.

Answer:

$f(x) = 5^x + 2$