Question

a student wants to graph the exponential function $f(x)=3(2)^x$. which characteristics will help the student graph the function?
○ the function has an $x$-intercept at $(3,0)$ and contains the points $(1,6)$, $(2,12)$, and $(3,24)$.
○ the function has an $x$-intercept at $(3,0)$ and contains the points $(6,1)$, $(12,2)$, and $(24,3)$.
○ the function has a $y$-intercept at $(0,3)$ and contains the points $(1,6)$, $(2,12)$, and $(3,24)$.
○ the function has a $y$-intercept at $(0,3)$ and contains the points $(6,1)$, $(12,2)$, and $(24,3)$.

Explanation:

Step1: Find y-intercept (x=0)

Substitute $x=0$ into $f(x)=3(2)^x$:
$f(0)=3(2)^0=3(1)=3$
So y-intercept is $(0,3)$. Exponential functions $a(b)^x$ (with $a>0, b>1$) never cross the x-axis, so no x-intercept.

Step2: Verify given points

Test $x=1$: $f(1)=3(2)^1=3(2)=6$ → point $(1,6)$
Test $x=2$: $f(2)=3(2)^2=3(4)=12$ → point $(2,12)$
Test $x=3$: $f(3)=3(2)^3=3(8)=24$ → point $(3,24)$

Answer:

The function has a y-intercept at (0, 3) and contains the points (1, 6), (2, 12), and (3, 24).