Question

A parabola passes through the points (-3,-16) (-2,0) (-1,8) (1,0) and (2, 16)

a) Determine an equation for the parabola in factored from.
b) Express your equation in standard form

Answer 1

The equation of the parabola in factored form is y = -4(x + 2)(x - 1), and in standard form, it is y = -4x² + 4x + 8.

Step-by-step explanation:

Finding the Equation of a Parabola in Factored and Standard Form

To find the equation of a parabola that passes through the given points (-3,-16), (-2,0), (-1,8), (1,0), and (2, 16), we start by recognizing that since these points are symmetric concerning the vertical axis of symmetry and the parabola opens upwards, the factored form of the quadratic equation will have factors corresponding to its x-intercepts and a leading coefficient that determines the vertical stretch or compression.

From the given points, we can see the parabola crosses the x-axis at x = -2 and x = 1. Thus, our equation in factored form may look like:

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

To find the value of a, we can use one of the other given points, for example (-1, 8).

8 = a(-1 + 2)(-1 - 1)

8 = a(1)(-2)

a = -4

So the equation of the parabola in factored form is:

y = -4(x + 2)(x - 1)

To express this equation in standard form, which is y = ax² + bx + c, we expand the factored form:

y = -4(x² - x - 2)

y = -4x² + 4x + 8

So, the equation of the parabola in standard form is:

y = -4x² + 4x + 8