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!

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!