Find the equation of the exponential function that goes through the points (0,3) and (3,375).

Do not include f(x)= in your answer. Do not include fractions in your answer

To find the equation of an exponential function that passes through the points \( (0, 3) \) and \( (3, 375) \), we can use the general form of an exponential function:

\[
y = ab^x
\]

### Step 1: Use the point \( (0, 3) \)

Substitute \( x = 0 \) and \( y = 3 \) into the equation:

\[
3 = ab^0
\]

Since \( b^0 = 1 \), this simplifies to:

\[
3 = a
\]

So, we now know that \( a = 3 \). The equation becomes:

\[
y = 3b^x
\]

### Step 2: Use the point \( (3, 375) \)

Substitute \( x = 3 \) and \( y = 375 \) into the equation:

\[
375 = 3b^3
\]

Solve for \( b \):

\[
375 = 3b^3 \quad \Rightarrow \quad b^3 = \frac{375}{3} = 125
\]

Now, take the cube root of both sides:

\[
b = \sqrt[3]{125} = 5
\]

### Final equation

Now that we have \( a = 3 \) and \( b = 5 \), the exponential function is:

\[
y = 3 \cdot 5^x
\]

This is the required equation.

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Would you like further details or have any questions about this solution?

### Here are 5 related questions:
1. How do you solve for the base \( b \) in an exponential function when given two points?
2. How does the exponential function change if the value of \( b \) is greater than 1 versus less than 1?
3. What happens if you are given negative exponents in exponential equations?
4. How can you graph an exponential function based on its equation?
5. How does the value of \( a \) affect the vertical shift of an exponential function?

**Tip:** To solve exponential equations systematically, isolate the exponential term, then use logarithms if necessary to find the unknown exponent or base.

Find the equation of the exponential function that goes through the points (0,3) and (3,375).

Learn how to find the equation of an exponential function that passes through the points (0,3) and (3,375). This detailed solution explains the process of determining the values of a and b for the equation y = 3 * 5^x, making it suitable for students in Grades 9-11.

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