Question

what is the multiplicative rate of change of the exponential function shown on the graph? options: \\(\frac{2}{9}\\), 1, 4, \\(\frac{9}{2}\\) (graph has points (-2, \\(\frac{4}{81}\\)), (-1, \\(\frac{2}{9}\\)), (0, 1), (1, \\(\frac{9}{2}\\)), (2, \\(\frac{81}{4}\\)) and the exponential curve)

Explanation:

Step1: Recall exponential function form

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

Step2: Identify initial value $a$

From the point $(0,1)$, substitute $x=0, y=1$ into $y=ab^x$:
$1 = ab^0 \implies a=1$ (since $b^0=1$ for any $b
eq0$)

Step3: Use another point to find $b$

Take the point $(1, \frac{9}{2})$, substitute $a=1, x=1, y=\frac{9}{2}$ into $y=b^x$:
$\frac{9}{2} = b^1 \implies b=\frac{9}{2}$
Verify with another point, e.g., $(2, \frac{81}{4})$: $y=(\frac{9}{2})^2=\frac{81}{4}$, which matches.

Answer:

D. $\frac{9}{2}$