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Part 1
Use the graph of the function to write the equation in factored form.
-8
-4
4
8
-8
-4
4
8
x
y
left parenthesis 0 comma 6 right parenthesisleft parenthesis negative 1 comma 0 right parenthesisleft parenthesis negative 6 comma 0 right parenthesis

x y graph
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Part 1
Write the equation.

enter your response here ​(Type your answer in factored​ form.)
Write the equation.(Type your answer in factored form.)
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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.

Learn how to write the equation of a polynomial function in factored form using its graph. This problem involves identifying roots from the graph and using them to construct the factored equation. Suitable for high school students studying polynomial functions and graph analysis.

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