Sure, I'd be happy to help you with this!
First, let's start by understanding a few basics about quadratic functions and parabolas.
1. Quadratic Function in Factored Form: A quadratic function can be written in factored form as:
[tex]\[ y = a(x - x_1)(x - x_2) \][/tex]
where [tex]\( (x_1, 0) \)[/tex] and [tex]\( (x_2, 0) \)[/tex] are the x-intercepts (zeros) of the function and [tex]\( a \)[/tex] is a coefficient that affects the direction and width of the parabola.
2. Given Zeros: In our case, the parabola has zeros at [tex]\( (-6, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex].
### Step-by-Step Solution:
1. Identify the Zeros:
- Zero 1: [tex]\( x_1 = -6 \)[/tex]
- Zero 2: [tex]\( x_2 = -2 \)[/tex]
2. Parabola Opening Downward: Since the parabola opens downward, the coefficient [tex]\( a \)[/tex] must be negative.
Let's choose [tex]\( a = -1 \)[/tex] for simplicity. You can choose other negative values for [tex]\( a \)[/tex], but [tex]\( -1 \)[/tex] provides an easy representation.
3. Substitute the Values:
[tex]\[ y = a(x - x_1)(x - x_2) \][/tex]
Substitute [tex]\( a = -1 \)[/tex], [tex]\( x_1 = -6 \)[/tex], and [tex]\( x_2 = -2 \)[/tex]:
[tex]\[ y = -1(x - (-6))(x - (-2)) \][/tex]
4. Simplify the Expression:
[tex]\[ y = -1(x + 6)(x + 2) \][/tex]
Thus, the quadratic function in factored form, representing a parabola that opens downward and has zeros at [tex]\( (-6, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex], is:
[tex]\[ y = -(x + 6)(x + 2) \][/tex]
This can also be written as:
[tex]\[ y = -(x + 6)(x + 2) \][/tex]
And that’s the quadratic function in factored form you were looking for!
