Determine the equation of the function with 
zeros at +1 -1  and -2 and a y-intercept of -6

To determine the equation of a function with given zeros and a y-intercept, we can follow these steps:

### Step 1: Identify the factors from the given zeros.
The function has zeros at \( x = 1 \), \( x = -1 \), and \( x = -2 \). This means the function will have the factors:
\[
(x - 1), (x + 1), (x + 2)
\]

So, the general form of the function can be written as:
\[
f(x) = a(x - 1)(x + 1)(x + 2)
\]
where \( a \) is a constant that we will determine using the y-intercept.

### Step 2: Use the y-intercept to find \( a \).
The y-intercept is given as -6, meaning \( f(0) = -6 \). Substituting \( x = 0 \) into the function:
\[
f(0) = a(0 - 1)(0 + 1)(0 + 2) = a(-1)(1)(2) = -2a
\]
Now, set this equal to the y-intercept:
\[
-2a = -6
\]
Solving for \( a \):
\[
a = \frac{-6}{-2} = 3
\]

### Step 3: Write the final equation.
Substitute \( a = 3 \) into the function:
\[
f(x) = 3(x - 1)(x + 1)(x + 2)
\]

This is the equation of the function.

Would you like to expand this or simplify further?

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Here are some follow-up questions:
1. What is the significance of the leading coefficient \( a \) in the function?
2. How does the degree of the polynomial relate to the number of zeros?
3. How would the function change if one of the zeros were repeated?
4. Can you identify the end behavior of this function based on its degree and leading coefficient?
5. What is the general form of a cubic function?

**Tip:** Always check for y-intercepts by substituting \( x = 0 \) in any function to verify or solve for constants.

Determine the equation of the function with zeros at +1, -1, and -2 and a y-intercept of -6.

Learn how to determine the equation of a cubic function with zeros at +1, -1, and -2, and a y-intercept of -6. This solution includes key algebraic concepts, such as polynomial equations, factorization, and intercepts. Ideal for students in Grades 9-12.

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