Question

determining rate of change from a graph
what is the multiplicative rate of change for the exponential function graphed to the left?
graph with points (-2, 2/9), (-1, 2/3), (0, 2), (1, 6), (2, 18), (3, 54)

Explanation:

Step1: Recall exponential function form

An exponential function is $y = ab^x$, where $b$ is the multiplicative rate of change, $a$ is the initial value.

Step2: Identify initial value $a$

When $x=0$, $y=2$, so $a=2$.

Step3: Solve for $b$ using a point

Use $(1,6)$: Substitute $x=1$, $y=6$, $a=2$ into $y=ab^x$
$6 = 2b^1$
$\frac{6}{2} = b$
$b=3$

Step4: Verify with another point

Use $(2,18)$: $y=2(3)^2=2\times9=18$, which matches.

Answer:

3