Question

algebra: concepts and connections - plc
exponential growth functions
what is the rate of change of the function shown on the graph? round to the nearest hundredth.
(options: 2.00, 2.50, 5.25, 10.50)

Explanation:

Step1: Recall exponential growth form

The general exponential growth function is $y = ab^x$, where $b=1+r$ ($r$ = growth rate), $a$ = initial value.

Step2: Substitute first point $(1,5)$

$5 = ab^1 \implies a = \frac{5}{b}$

Step3: Substitute second point $(2,12.5)$

Substitute $a=\frac{5}{b}$ into $12.5 = ab^2$:
$12.5 = \frac{5}{b} \times b^2 = 5b$

Step4: Solve for $b$

$b = \frac{12.5}{5} = 2.50$

Step5: Find growth rate $r$

Since $b=1+r$, $r = b-1 = 2.50 - 1 = 1.50$? No, wait: the question asks for the rate of change (growth factor multiplier)? Wait no, recheck: the options include 2.50. Wait, the problem says "rate of change" for exponential function, here the growth factor (the base $b$) is 2.50. Wait, let's confirm:
From $(1,5)$ to $(2,12.5)$, $\frac{12.5}{5}=2.50$, which is the multiplicative rate of change.

Answer:

2.50