We are given an exponential function that passes through the points $(-2, \frac{4}{81}), (-1, \frac{2}{9}), (0, 1), (1, \frac{9}{2}), (2, \frac{81}{4})$ and has a horizontal asymptote of $y = 0$. To determine the multiplicative rate of change of this exponential function, we need to observe how the function values change as the $x$-values increase by 1 unit.

Step-by-Step Approach:

An exponential function is generally of the form:

$y = ab^x$

where:

• $a$ is the initial value (value of $y$ when $x = 0$),
• $b$ is the base, which represents the multiplicative rate of change.

Looking at the given point $(0, 1)$, we know that when $x = 0$, $y = 1$. So, $a = 1$. This simplifies the equation to:

$y = b^x$

Now, let's use another point to determine $b$. Take the point $(1, \frac{9}{2})$. Plugging this into the equation $y = b^x$:

$\frac{9}{2} = b^1$

This gives us $b = \frac{9}{2}$.

Thus, the multiplicative rate of change of the exponential function is:

$\boxed{\frac{9}{2}}$

Conclusion:

The multiplicative rate of change of the exponential function is $\frac{9}{2}$, which corresponds to the answer Nine-halves.

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. What is the general form of an exponential function?
2. How does the base $b$ in an exponential function affect its growth or decay?
3. Can an exponential function have a negative base? Why or why not?
4. How would you graph an exponential function using a table of values?
5. What is the significance of the horizontal asymptote in an exponential function?

Tip: In exponential functions, the base $b$ determines whether the function represents growth ($b > 1$) or decay ($0 < b < 1$).