Question

the table represents an exponential function.

x	y
2	\\(\frac{2}{5}\\)
3	\\(\frac{2}{25}\\)
4	\\(\frac{2}{125}\\)

what is the multiplicative rate of change of the function?
\\(\circ\\) \\(\frac{1}{5}\\)
\\(\circ\\) \\(\frac{2}{5}\\)
\\(\circ\\) 2
\\(\circ\\) 5

Explanation:

Step1: Recall the formula for multiplicative rate of change

The multiplicative rate of change of an exponential function is found by dividing a consecutive \( y \)-value by the previous \( y \)-value. So, if we have \( y_1, y_2, y_3, \dots \), the rate \( r=\frac{y_{n + 1}}{y_n} \).

Step2: Choose two consecutive \( y \)-values from the table

Let's take \( y_1 = 2\) (when \( x = 1\)) and \( y_2=\frac{2}{5}\) (when \( x = 2\)).

Step3: Calculate the ratio

Now, calculate the ratio \( r=\frac{y_2}{y_1}=\frac{\frac{2}{5}}{2}\).
Simplify the expression: \(\frac{\frac{2}{5}}{2}=\frac{2}{5}\times\frac{1}{2}=\frac{1}{5}\)? Wait, no, wait. Wait, maybe I picked the wrong values. Wait, when \( x = 2\), \( y=\frac{2}{5}\); when \( x = 3\), \( y=\frac{2}{25}\). Let's use \( y_2=\frac{2}{5}\) ( \( x = 2\)) and \( y_3=\frac{2}{25}\) ( \( x = 3\) ). Then \( r=\frac{y_3}{y_2}=\frac{\frac{2}{25}}{\frac{2}{5}}\).
Simplify: \(\frac{\frac{2}{25}}{\frac{2}{5}}=\frac{2}{25}\times\frac{5}{2}=\frac{5}{25}=\frac{1}{5}\)? Wait, no, that's not right. Wait, wait, maybe I made a mistake. Wait, the exponential function has the form \( y = ab^x \). Let's check the values. When \( x = 1\), \( y = 2\); \( x = 2\), \( y=\frac{2}{5}\); \( x = 3\), \( y=\frac{2}{25}\); \( x = 4\), \( y=\frac{2}{125}\). Let's see the ratio between \( x = 1\) and \( x = 2\): \(\frac{\frac{2}{5}}{2}=\frac{2}{5}\times\frac{1}{2}=\frac{1}{5}\). Between \( x = 2\) and \( x = 3\): \(\frac{\frac{2}{25}}{\frac{2}{5}}=\frac{2}{25}\times\frac{5}{2}=\frac{1}{5}\). Between \( x = 3\) and \( x = 4\): \(\frac{\frac{2}{125}}{\frac{2}{25}}=\frac{2}{125}\times\frac{25}{2}=\frac{1}{5}\). Wait, but the options have \(\frac{1}{5}\) as an option? Wait, no, the options are \(\frac{1}{5}\), \(\frac{2}{5}\), \(2\), \(5\). Wait, maybe I messed up the direction. Wait, maybe the multiplicative rate of change is the base \( b \) in \( y=ab^x \). Let's see, when \( x = 1\), \( y = 2=ab^1\); \( x = 2\), \( y=\frac{2}{5}=ab^2\). Divide the second equation by the first: \(\frac{ab^2}{ab}=\frac{\frac{2}{5}}{2}\), so \( b=\frac{1}{5}\). Wait, but let's check with \( x = 2\) and \( x = 3\): \(\frac{ab^3}{ab^2}=\frac{\frac{2}{25}}{\frac{2}{5}}\), so \( b=\frac{1}{5}\). So the multiplicative rate of change is \(\frac{1}{5}\)? But wait, the options have \(\frac{1}{5}\) as the first option. Wait, but let me check again. Wait, maybe I made a mistake. Wait, the problem says "multiplicative rate of change". Wait, in exponential functions, the multiplicative rate of change is the common ratio between consecutive terms. Let's check the ratio between \( x = 1\) ( \( y = 2\)) and \( x = 2\) ( \( y=\frac{2}{5}\)): \(\frac{2/5}{2}=\frac{1}{5}\). Between \( x = 2\) ( \( y = 2/5\)) and \( x = 3\) ( \( y = 2/25\)): \(\frac{2/25}{2/5}=\frac{1}{5}\). Between \( x = 3\) ( \( y = 2/25\)) and \( x = 4\) ( \( y = 2/125\)): \(\frac{2/125}{2/25}=\frac{1}{5}\). So the multiplicative rate of change is \(\frac{1}{5}\). Wait, but let me check the options again. The first option is \(\frac{1}{5}\), so that should be the answer.

Wait, but wait, maybe I misread the table. Let me check the table again. The table is:

\( x \): 1, \( y \): 2

\( x \): 2, \( y \): 2/5

\( x \): 3, \( y \): 2/25

\( x \): 4, \( y \): 2/125

Yes, so the ratio between each \( y \)-value and the previous one is \( (2/5)/2 = 1/5 \), \( (2/25)/(2/5)=1/5 \), \( (2/125)/(2/25)=1/5 \). So the multiplicative rate of change is \( \frac{1}{5} \).

Answer:

\(\boldsymbol{\frac{1}{5}}\) (corresponding to the first option: \(\frac{1}{5}\))