To write the equation of the function in factored form using the graph, we need to analyze the key points provided:

1. The graph passes through the points $(-1, 0)$ and $(-6, 0)$. These points are the roots of the function.
2. The graph also passes through the point $(0, 6)$, which is not a root but can be used to find the leading coefficient.

Factored Form

The factored form of a polynomial function with roots $x_1$ and $x_2$ is:

$y = a(x - x_1)(x - x_2)$

Given the roots $-1$ and $-6$, the equation will be:

$y = a(x + 1)(x + 6)$

Finding the Leading Coefficient $a$

To determine the value of $a$, use the point $(0, 6)$:

$6 = a(0 + 1)(0 + 6)$

Simplifying:

$6 = a(1)(6)$

$6 = 6a$

$a = 1$

Final Equation

Thus, the equation of the function in factored form is:

$y = (x + 1)(x + 6)$

Would you like any more details or have any questions?

Related Questions:

1. What would the equation be if the point $(0, 6)$ were instead $(0, -6)$?
2. How would the equation change if there were an additional root at $x = 3$?
3. How can you determine the end behavior of the function based on the equation?
4. What is the importance of the leading coefficient $a$ in the factored form?
5. How would you graph the equation $y = (x + 1)(x + 6)$ from scratch?

Tip: Always verify your equation by substituting the roots and other known points to ensure accuracy.