What is the equation of this parabola?

the points (-8, 0) and (8, 0) show the roots. The line of symmetry is halfway between these 2 points, at x = 0 The general equation for a parabola is ax^2 + bx + c, and the line of symmetry is at x = -b/2a. If this line is at x = 0, then the coefficient b = 0. This reduces the equation to ax^2 + c.

Since the problem states that the point (0, 2) lies on the graph, we know that c = 2.

The roots, in general, lie at +/-Sqrt((b^2)-(4ac)) to either side of the line of symmetry. Since we know that b = 0, and c = 2, we can calculate the roots: +/- sqrt(-4a(2)) = +/- sqrt(-8a) Since the roots are real, we know that the quantity under the radical sign must be positive, so the coefficient "a" must b.e negative. Knowing the roots are +/- 8, we know that sqrt(-8a) = sqrt(64), so

-8a = 64, and a = -8. The equation is: -8x^2 + 2 = y = f(x). This graphs as an inverted (opening downward) parabola, centered on the y-axis.

Since two of the points have their y values equaling zero, these points are the roots of the quadratic and, since the x-value of the other point is zero, the vertex is (0, 2), so

Vertex (0, 2):

h = 0

k = 2

One Root (8, 0):

x = 8

y = 0

Vertex Form of Equation:

y = a(x - h)² + k

Subbing known values,

0 = a(8 - 0)² + 2

0 = a(8)² + 2

0 = a(64) + 2

0 = 64a + 2

64a = - 2

a = - 2 / 64

a = - 1/32

Equation:

y = - 1/32 x² + 2

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