Question

Given the exponential function y=
3
4
​
(
7
2
​
)
x
, what is the growth rate?

4
3
​


7
2
​


3
4
​


2
7
​

Answer 1

The exponential function given is

\[
y = \frac{3}{4} \left( \frac{7}{2} \right)^x.
\]

In an exponential function of the form

\[
y = a \cdot b^x,
\]

where \(a\) is the initial value and \(b\) is the base of the exponential, the growth rate is determined by \(b\).

In this case, the base \(b\) is \(\frac{7}{2}\). To determine the growth rate, we can interpret this as

\[
b - 1.
\]

Calculating the growth rate:

\[
\text{Growth rate} = \frac{7}{2} - 1 = \frac{7}{2} - \frac{2}{2} = \frac{5}{2}.
\]

So, the growth rate in terms of a fractional increase is \( \frac{5}{2} \) or 2.5.

If you were looking for answers from the options you provided:

- \( \frac{4}{3} \)
- \( \frac{7}{2} \)
- \( \frac{3}{4} \)
- \( \frac{2}{7} \)

None of these represent the growth rate correctly. The growth rate is \( \frac{5}{2} \), which is equivalent to 2.5.