Let’s solve the given problem step-by-step:

Part (a): Find the equation of the exponential function.

The general form of the exponential function is:

$y = a \cdot b^x$

We are given three points:

1. $(1, 4.8)$
2. $(2, 5.76)$
3. $(3, 6.912)$

We will use these points to determine the constants $a$ and $b$.

Step 1: Solve for $b$.

From the exponential function:

$y = a \cdot b^x$

Using the points $(1, 4.8)$ and $(2, 5.76)$, we set up the ratio of their $y$-values to eliminate $a$:

$\frac{y_2}{y_1} = \frac{a \cdot b^2}{a \cdot b^1} = b$

Substitute $y_1 = 4.8$ and $y_2 = 5.76$:

$b = \frac{5.76}{4.8} = 1.2$

So, $b = 1.2$.

Step 2: Solve for $a$.

Using the point $(1, 4.8)$, substitute $x = 1$, $y = 4.8$, and $b = 1.2$ into the equation:

$4.8 = a \cdot 1.2^1$

$a = \frac{4.8}{1.2} = 4$

Thus, $a = 4$.

Final Equation:

Substitute $a = 4$ and $b = 1.2$ into the general form:

$y = 4 \cdot (1.2)^x$

Part (b): Evaluate the function at $x = 5$.

Substitute $x = 5$ into the equation $y = 4 \cdot (1.2)^x$:

$y = 4 \cdot (1.2)^5$

First, calculate $1.2^5$:

$1.2^5 = 1.2 \cdot 1.2 \cdot 1.2 \cdot 1.2 \cdot 1.2 = 2.48832$

Now multiply by $4$:

$y = 4 \cdot 2.48832 = 9.95328$

Thus, at $x = 5$:

$y \approx 9.95$

Final Answers:

(a) The equation is: $y = 4 \cdot (1.2)^x$

(b) At $x = 5$: $y \approx 9.95$

Let me know if you want detailed steps for calculations or any clarifications!

Here are 5 related questions you might find helpful:

1. How do you calculate the base $b$ in an exponential function with two points?
2. What is the role of $a$ in an exponential equation?
3. How does changing $b$ affect the shape of an exponential graph?
4. How do you solve for $y$ when the equation involves fractional exponents?
5. Can exponential growth be modeled with a logarithmic equation?

Tip: When solving for constants in an exponential function, always use the ratio of $y$-values for consecutive points to simplify finding $b$.