(-2, 8/25) (2, 50/4) how would i find the exponential growth function without using a calculator, leaving it in the simplest form

To find the exponential growth function given two points, we follow these steps:

### Step 1: General form of the exponential function
The general form of an exponential growth function is:

\[
y = ab^x
\]

Where:
- \(a\) is the initial value (the value of \(y\) when \(x = 0\)),
- \(b\) is the growth factor.

We are given two points \((-2, \frac{8}{25})\) and \((2, \frac{50}{4})\). These points give us two equations.

### Step 2: Create two equations
Substitute the first point \((-2, \frac{8}{25})\) into the general form of the exponential function:

\[
\frac{8}{25} = ab^{-2}
\]

Now substitute the second point \((2, \frac{50}{4})\) into the same general form:

\[
\frac{50}{4} = ab^2
\]

### Step 3: Solve for \(a\) and \(b\)
We now have two equations:

1. \(\frac{8}{25} = ab^{-2}\)
2. \(\frac{50}{4} = ab^2\)

Divide equation 2 by equation 1 to eliminate \(a\):

\[
\frac{\frac{50}{4}}{\frac{8}{25}} = \frac{ab^2}{ab^{-2}}
\]

Simplify both sides:

\[
\frac{50}{4} \cdot \frac{25}{8} = b^{2 - (-2)} = b^4
\]

Simplify the left side:

\[
\frac{50 \times 25}{4 \times 8} = b^4
\]

\[
\frac{1250}{32} = b^4
\]

Now simplify \( \frac{1250}{32} \):

\[
b^4 = \frac{625}{16}
\]

### Step 4: Solve for \(b\)
To solve for \(b\), take the fourth root of both sides:

\[
b = \sqrt[4]{\frac{625}{16}} = \frac{\sqrt[4]{625}}{\sqrt[4]{16}} = \frac{5}{2}
\]

So, \(b = \frac{5}{2}\).

### Step 5: Solve for \(a\)
Now substitute \(b = \frac{5}{2}\) into one of the original equations to solve for \(a\). Let's use the first equation \(\frac{8}{25} = ab^{-2}\):

\[
\frac{8}{25} = a \left(\frac{5}{2}\right)^{-2}
\]

\[
\frac{8}{25} = a \cdot \left(\frac{2}{5}\right)^2
\]

\[
\frac{8}{25} = a \cdot \frac{4}{25}
\]

Now solve for \(a\):

\[
a = \frac{8}{25} \div \frac{4}{25} = \frac{8}{25} \times \frac{25}{4} = 2
\]

### Final Exponential Function
The exponential growth function is:

\[
y = 2 \left(\frac{5}{2}\right)^x
\]

Would you like more details, or have any other questions?

Here are 5 related questions:
1. How can I check if a function is exponential or linear?
2. How do I graph exponential functions by hand?
3. What happens if the base \(b\) in an exponential function is less than 1?
4. How do exponential functions model population growth?
5. How do transformations affect exponential functions?

**Tip**: When solving for unknowns in exponential equations, dividing equations is a powerful method to eliminate common factors like \(a\).

Learn how to find the exponential growth function given two points (-2, 8/25) and (2, 50/4) without using a calculator. This detailed solution involves creating and solving exponential equations, ideal for students in Grades 10-12.

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