What are the values of y when x=1,2,3 given the points (0,5) and (4,25) of an exponential function

To find the values of \( y \) when \( x = 1, 2, 3 \) for an exponential function that passes through the points \( (0, 5) \) and \( (4, 25) \), we need to first determine the equation of the exponential function.

The general form of an exponential function is:

\[
y = ab^x
\]

where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)) and \( b \) is the base (the growth or decay factor).

### Step 1: Determine \( a \)
From the point \( (0, 5) \), we know that:

\[
y = ab^0 = a = 5
\]

So, the equation becomes:

\[
y = 5b^x
\]

### Step 2: Determine \( b \)
We now use the second point, \( (4, 25) \), to find \( b \). Substituting \( x = 4 \) and \( y = 25 \) into the equation:

\[
25 = 5b^4
\]

Dividing both sides by 5:

\[
b^4 = 5
\]

Taking the fourth root of both sides:

\[
b = \sqrt[4]{5}
\]

### Step 3: Find \( y \) for \( x = 1, 2, 3 \)
Now that we have \( a = 5 \) and \( b = \sqrt[4]{5} \), the equation is:

\[
y = 5 \left( \sqrt[4]{5} \right)^x
\]

We can now compute the values of \( y \) for \( x = 1, 2, 3 \).

1. **For \( x = 1 \):**

\[
y = 5 \times \left( \sqrt[4]{5} \right)^1 = 5 \times \sqrt[4]{5}
\]

2. **For \( x = 2 \):**

\[
y = 5 \times \left( \sqrt[4]{5} \right)^2 = 5 \times \sqrt{5}
\]

3. **For \( x = 3 \):**

\[
y = 5 \times \left( \sqrt[4]{5} \right)^3 = 5 \times 5^{3/4}
\]

If you'd like, I can provide the exact numerical values for these calculations. Let me know!

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Would you like more detailed steps, or do you have any questions about this? Here are 5 questions to explore further:

1. How is the base \( b \) of an exponential function related to the growth rate?
2. How would the equation change if the function were decaying rather than growing?
3. Can you use a logarithmic approach to solve for \( b \) in an exponential function?
4. How would this method differ if the points represented a different type of curve, such as a linear function?
5. What happens to the function as \( x \) increases indefinitely?

**Tip**: The initial value \( a \) in an exponential function always represents the \( y \)-intercept (the value when \( x = 0 \)).

What are the values of y when x=1,2,3 given the points (0,5) and (4,25) of an exponential function?

Discover how to find the values of y for x = 1, 2, 3 in an exponential function passing through points (0,5) and (4,25). This problem involves finding the growth factor using the general exponential function y = ab^x and computing intermediate values.

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