Question

which graph represents the function $f(x) = \frac{1}{2}(2)^x$? (four graphs are shown with orange curves and points on a coordinate plane, each with x-axis from -3 to 3 and y-axis from 0 to 10, with options to select one of the four graphs)

Explanation:

Step1: Find y-intercept (x=0)

Substitute $x=0$ into $f(x)=\frac{3}{2}(2)^x$:
$f(0)=\frac{3}{2}(2)^0=\frac{3}{2}(1)=1.5$

Step2: Calculate f(1)

Substitute $x=1$ into the function:
$f(1)=\frac{3}{2}(2)^1=\frac{3}{2}(2)=3$

Step3: Calculate f(2)

Substitute $x=2$ into the function:
$f(2)=\frac{3}{2}(2)^2=\frac{3}{2}(4)=6$

Step4: Calculate f(-1)

Substitute $x=-1$ into the function:
$f(-1)=\frac{3}{2}(2)^{-1}=\frac{3}{2}\times\frac{1}{2}=0.75$

Step5: Match points to graph

The points $(0,1.5)$, $(1,3)$, $(2,6)$, $(-1,0.75)$ correspond to the third graph (left to right).

Answer:

The third graph (leftmost to rightmost: 3rd option) with points matching $(x=-1, y=0.75)$, $(x=0, y=1.5)$, $(x=1, y=3)$, $(x=2, y=6)$