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.