To write the quadratic equation in factored form based on the given table of values, I will analyze the data provided:

From the table, the function $f(x)$ appears to depend quadratically on $x$. A quadratic equation in factored form is typically written as:

$f(x) = a(x - p)(x - q)$

Key Observations

• The values of $f(x)$ are zero when $x = 6$ and $x = 8$. These are the roots of the quadratic equation, so the equation can be written as:

$f(x) = a(x - 6)(x - 8)$

Determine the Coefficient $a$

To find $a$, substitute another point from the table into the equation. For example, use the point $(x, f(x)) = (2, 48)$:

$48 = a(2 - 6)(2 - 8)$

Simplify:

$48 = a(-4)(-6)$

$48 = 24a$

$a = 2$

Final Equation

The quadratic equation in factored form is:

$f(x) = 2(x - 6)(x - 8)$

Would you like a graph of the function or further steps explained?

Here are 5 related questions for deeper exploration:

1. How would the equation look in standard form after expanding the factors?
2. Can you find the vertex of this quadratic equation?
3. How do changes to the coefficient $a$ affect the shape of the parabola?
4. What are the domain and range of $f(x)$ based on the table values?
5. Can you identify the symmetry in the function from its equation?

Tip: Always verify your derived equation with additional points from the table to ensure accuracy.