Question

Given the vertex of a parabola is (-3,1) and also contains the point (-4,-1). State the exact equation of the parabola in vertex form, show all your work.

Answer 1

The equation of the parabola in vertex form is y = (-2/49)(x + 3)² + 1.

Step-by-step explanation:

To find the equation of a parabola in vertex form, we need to use the vertex and a point on the parabola. The vertex form of a parabola equation is y = a(x - h)² + k, where (h, k) is the vertex.

Using the given vertex (-3, 1), we have h = -3 and k = 1.

Since the point (-4, -1) also lies on the parabola, we can substitute these values into the vertex form equation and solve for a to find the equation of the parabola.

Substituting (-4, -1) into the equation:

-1 = a(-4 - (-3))² + 1

-1 = a(-7)² + 1

-1 = 49a + 1

49a = -2

a = -2/49

Therefore, the exact equation of the parabola in vertex form is y = (-2/49)(x + 3)² + 1.

Answer 2

y = -2(x + 3)^2 - 1

Step-by-step explanation:

The vertex equation of a parabola is y = a(x - h)^2 + k. If the vertex here is (-3, 1), then this equation becomes y = a(x + 3)^2 + 1.

Knowing that the graph passes through (-4, -1), we let y = -1 and x = -4, obtaining:

-1 = a(-4 + 3)^2 - 1, which must be solved for a:

-1 = a - 1. Therefore, a = -2, and the exact equation of this parabola is:

y = -2(x + 3)^2 - 1