Question

which exponential function is represented by the graph?
options:
$f(x) = 2(\frac{1}{2})^x$
$f(x) = \frac{1}{2}(2)^x$
$f(x) = 2(2)^x$
$f(x) = 3(\frac{1}{2})^x$

Explanation:

Step1: Recall exponential form

The general exponential function is $f(x)=ab^x$, where $a$ is the y-intercept.

Step2: Find $a$ from y-intercept

From the graph, the y-intercept is $(0,3)$. Substitute $x=0$, $f(x)=3$:
$3 = ab^0 = a(1) \implies a=3$

Step3: Test point to find $b$

Use the point $(1,6)$. Substitute $a=3$, $x=1$, $f(x)=6$:
$6 = 3b^1 \implies b=\frac{6}{3}=2$

Step4: Verify with third point

Use $(-1,1.5)$. Substitute $a=3$, $b=2$, $x=-1$:
$f(-1)=3(2^{-1})=3\times\frac{1}{2}=1.5$, which matches the point.

Answer:

$f(x)=3(2^x)$