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The function \( y = f(x) \) appears to be an exponential function, given that the points \( A = (1, 5) \) and \( B = (3, 125) \) are on the graph. The general form of an exponential function is:

\[
y = C \cdot a^x
\]

We will determine the values of \( C \) and \( a \) by solving for them using the given points.

### Step 1: Use the point \( A(1, 5) \)

Substitute \( x = 1 \) and \( y = 5 \) into the equation:

\[
5 = C \cdot a^1 \quad \Rightarrow \quad 5 = C \cdot a
\]
This gives us the equation:
\[
C \cdot a = 5
\]

### Step 2: Use the point \( B(3, 125) \)

Substitute \( x = 3 \) and \( y = 125 \) into the equation:

\[
125 = C \cdot a^3
\]
This gives us the second equation:
\[
C \cdot a^3 = 125
\]

### Step 3: Solve the system of equations

We now have the system:
1. \( C \cdot a = 5 \)
2. \( C \cdot a^3 = 125 \)

We can solve this by dividing the second equation by the first:

\[
\frac{C \cdot a^3}{C \cdot a} = \frac{125}{5}
\]

Simplifying:

\[
a^2 = 25 \quad \Rightarrow \quad a = 5
\]

### Step 4: Find \( C \)

Substitute \( a = 5 \) into the first equation:

\[
C \cdot 5 = 5 \quad \Rightarrow \quad C = 1
\]

### Final Answer:

The formula for the function is:

\[
y = 1 \cdot 5^x \quad \Rightarrow \quad y = 5^x
\]

Let me know if you'd like more details or have any questions!

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Here are 5 related questions:
1. How would the graph change if \( C \) were a value other than 1?
2. Can you derive a logarithmic function from this exponential graph?
3. How would you interpret the value of \( a \) in terms of the graph's growth rate?
4. What would happen if you swapped the points \( A \) and \( B \)? 
5. How can you verify this solution by plugging in the points into the equation?

**Tip:** Always check for the growth factor in exponential functions by examining how quickly the \( y \)-values increase or decrease for given \( x \)-values!

Consider the function y = f(x) graphed below. The two points marked on the graph are A = (1, 5) and B = (3, 125). Give a possible formula, of the form Ca^x, for the function graphed.

Solve for the exponential function y = C * a^x passing through points A(1,5) and B(3,125). Learn how to derive the values of C and a using algebraic techniques in this high school-level math problem.

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