Write an equation for a parabola with x-intercepts (-2,0) and (4,0) which passes through the point (1,-18)

Answer 1

To find the equation for a parabola, we can use the vertex form, which is given by:

y = a(x - h)^2 + k

where (h, k) represents the vertex of the parabola.

In this case, since the parabola passes through the points (-2,0) and (4,0), the vertex lies on the axis of symmetry, which is the line x = (−2 + 4)/2 = 1. Therefore, the vertex has x-coordinate 1.

Since the vertex (1, k) has a y-coordinate of -18, we can substitute these values into the vertex form equation:

-18 = a(1 - 1)^2 + k
-18 = a(0) + k
-18 = 0 + k
k = -18

Now that we know the values of h and k, we can substitute them into the vertex form equation:

y = a(x - 1)^2 - 18

To find the value of a, we can substitute the point (4,0) into the equation:

0 = a(4 - 1)^2 - 18
0 = a(3)^2 - 18
0 = 9a - 18
9a = 18
a = 18/9
a = 2

Therefore, the equation for the parabola is:

y = 2(x - 1)^2 - 18