To find the formula for the exponential function passing through the given points, let’s analyze the problem.

Given points:

1. $(-2, \frac{5}{4})$
2. $(2, 20)$

We want to find an exponential function of the form: $y = ab^x$ where $a$ is the initial value and $b$ is the base of the exponential function.

Step 1: Set up the equations

Using the point $(-2, \frac{5}{4})$: $\frac{5}{4} = ab^{-2}$

Using the point $(2, 20)$: $20 = ab^2$

Step 2: Solve for $a$ and $b$

Divide the second equation by the first to eliminate $a$: $\frac{20}{\frac{5}{4}} = \frac{ab^2}{ab^{-2}}$ $16 = b^4$ $b = 2 \quad (\text{since } b > 0 \text{ for an exponential function})$

Step 3: Substitute $b$ back to find $a$

Using $b = 2$ in the first equation: $\frac{5}{4} = a \cdot 2^{-2}$ $\frac{5}{4} = a \cdot \frac{1}{4}$ $a = 5$

Final Answer

The exponential function passing through the points is: $y = 5 \cdot 2^x$

This matches the solution shown in the image.

Would you like further details, or do you have any questions about this process?

Here are five related questions to deepen understanding:

1. How do you determine whether an exponential function should have a positive base?
2. Can you find another pair of points that lie on the graph of $y = 5 \cdot 2^x$?
3. How would the equation change if the base were a fraction instead of an integer?
4. How does the initial value $a$ affect the position of the exponential curve?
5. What would the function look like if the points had indicated a decreasing exponential function?

Tip: When solving for exponential functions through points, dividing equations helps isolate the base, simplifying the process.