Question

Write an equation for a parabola with​ x-intercepts (-1,0) and (5,0) which passes through the point (3,-16)

Write the equation : _________

Answer 1

y = -1/4sqrt(2)(x - 2)^2 - 17

Explanation:

To write the equation of a parabola, we need to know the vertex and either the focus or the directrix. However, we can use the x-intercepts and a point on the parabola to find the equation.

The x-intercepts of the parabola are (-1,0) and (5,0). Therefore, the axis of symmetry is the line x = 2. The vertex is at the midpoint of the line segment connecting the x-intercepts, which is (2,0).

The point (3,-16) lies on the parabola. Since it is not on the axis of symmetry, it must be equidistant from the vertex and the directrix. Let d be the distance from (3,-16) to the directrix. Then, by definition of a parabola, d is also the distance from (3,-16) to the vertex.

The distance between (3,-16) and (2,0) is sqrt((3-2)^2 + (-16-0)^2) = sqrt(1^2 + 16^2) = 17. Therefore, d = 17.

Since the directrix is a horizontal line, it has an equation of y = k, where k is a constant. The distance from any point on the parabola to the directrix is equal to d = 17. Therefore, we have:

|k - 0| = 17

Solving for k gives k = 17 or k = -17.

Since the vertex is at (2,0), we know that the equation of the parabola has a form of y = a(x - 2)^2 + 0.

If k = 17, then the directrix is y = 17. The focus is then at (2,-17). Using this point and (3,-16), we can find a:

sqrt((3-2)^2 + (-16+17)^2) = sqrt(1^2 + 1^2) = sqrt(2)

a = -1/4sqrt(2)

Therefore, the equation of the parabola with x-intercepts (-1,0) and (5,0) which passes through (3,-16) is:

y = -1/4sqrt(2)(x - 2)^2 - 17

Answer 2

So, the equation of the parabola is:

y = -2(x + 1)(x - 5)

or, if you want to multiply it out:

y = -2x^2 + 8x + 10

Explanation:

To write the equation of a parabola given its x-intercepts and a point it passes through, we can use the factored form:

[y = a(x - r)(x - s)]

where (r) and (s) are the x-intercepts, and (a) is a constant.

Given that the x-intercepts are (-1,0) and (5,0), we have:

[r = -1 text{ and } s = 5]

We also know that the parabola passes through the point (3,-16), which gives us another point:

[x = 3, quad y = -16]

Substituting these values into the equation, we get:

[-16 = a(3 - (-1))(3 - 5)]

Solving for (a), we get:

[-16 = 8a]

[a = -2]

So, the equation of the parabola is:

[y = -2(x + 1)(x - 5)]

or, if you want to multiply it out:

[y = -2x^2 + 8x + 10]