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We are given two points, \( A = (1, 3) \) and \( B = (4, 81) \), and need to find a possible formula of the form \( y = C a^x \) for the function graphed.

### Step 1: Use the form \( y = C a^x \)
For the two points, we can substitute the coordinates into the equation \( y = C a^x \) to create a system of equations.

1. From \( A(1, 3) \):
\[
3 = C a^1 \implies 3 = C a
\]

2. From \( B(4, 81) \):
\[
81 = C a^4
\]

### Step 2: Solve the system of equations
We now have two equations:
1. \( 3 = C a \)
2. \( 81 = C a^4 \)

We can solve for \( C \) in terms of \( a \) from the first equation:
\[
C = \frac{3}{a}
\]

Substitute this into the second equation:
\[
81 = \frac{3}{a} a^4
\]
Simplify:
\[
81 = 3 a^3
\]
Divide by 3:
\[
27 = a^3
\]
Solve for \( a \):
\[
a = 3
\]

### Step 3: Find \( C \)
Now substitute \( a = 3 \) into the first equation:
\[
3 = C \times 3
\]
Solve for \( C \):
\[
C = 1
\]

### Final Formula:
The possible formula for the function is:
\[
y = 1 \times 3^x \quad \text{or simply} \quad y = 3^x
\]

Would you like more details or have any questions?

### Relative questions:
1. How do we determine the values of \( C \) and \( a \) for exponential functions in other cases?
2. What if the points given didn't satisfy an exponential function—how would we approach it?
3. How can we verify that the function \( y = 3^x \) passes through the points \( A \) and \( B \)?
4. What changes if the function was in a different form, such as \( y = C e^{kx} \)?
5. How can we graph \( y = 3^x \) and interpret its behavior?

### Tip:
In exponential functions, the base \( a \) determines the growth rate, while \( C \) is a scaling factor that adjusts the vertical stretch of the graph.

Consider the function y = f(x) graphed below. The two points marked on the graph are A = (1,3) and B = (4,81). Give a possible formula of the form C a^x for the function graphed.

Solve for the formula of the exponential function y = C a^x using the points A(1, 3) and B(4, 81). This step-by-step guide walks through solving for constants C and a.

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