Question

Find the equation in factored form zeros=-2 and 2 with vertex (0,-4)

Answer 1

The equation in factored form given the zeros -2 and 2 with the vertex (0,-4) is f(x) = -(x + 2)(x - 2).

Step-by-step explanation:

To find the equation in factored form given the zeros = -2 and 2 with the vertex (0,-4), we start with the standard form of a quadratic function, which is f(x) = ax^2 + bx + c. Since the vertex is at the origin and the parabola opens upward (the coefficient of x^2 is positive), and the factor form of the quadratic equation is f(x) = a(x - p)(x - q), where p and q are the zeros of the function.

In this case, using the given zeros of -2 and 2, the factored form starts as f(x) = a(x + 2)(x - 2). To find the value of a, we use the vertex (0,-4). The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where the vertex is at the point (h,k). Comparing the vertex form with the vertex given, we get f(x) = a(x - 0)^2 - 4. Since when x = 0, f(x) = -4, we can see that a must be -1. Therefore, the factored form of the quadratic function is f(x) = -1(x + 2)(x - 2) or f(x) = -(x + 2)(x - 2).

Answer 2

`\large\boxed{y=(x+2)(x-2)}`

Step-by-step explanation:

Factored form of quadratic equation:

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

x₁, x₂ - zeros

We have zeros: x₁ = -2 and x₂ = 2. Substitute:

`y=a(x-(-2))(x-2)=a(x+2)(x-2)`

We have the vertex (0, -4). Put the coordinates of the vertex to the equation:

`-4=a(0+2)(0-2)`

`-4=a(2)(-2)`

`-4=-4a` divdie both sides by (-4)

`1=a\to a=1`

Finally:

`y=1(x+2)(x-2)=(x+2)(x-2)`