Question

Find the equation of a parabola with a vertex of (2,8) and passes through the point (10,-4)?

Answer 1

`\displaystyle y = -(3)/(16)\, (x - 2)^(2) + 8`.

Explanation:

In general, if the vertex of a parabola is
`(h,\, k)` (where
`h` and
`k` are constants,) the equation of that parabola would be
`y = a\, (x - h)^(2) + k` for some constant
`a` (
`a \\e 0`.) This equation is the vertex form equation of this parabola.

In this question, it is given that the vertex of this parabola is
`(2,\, 8)`. Thus,
`h = 2` and
`k = 8`. The equation of this parabola would be
`y = a\, (x - 2)^(2) + 8` for some constant
`a`.

Finding the value of this constant
`a` requires the coordinates of a point on this parabola other than the vertex
`(2,\, 8)`.

Since
`(10,\, -4)` is a point on this parabola,
`x = 10` and
`y = (-4)` should satisfy the equation of this parabola
`y = a\, (x - 2)^(2) + 8`.

Thus:

`(-4) = a\, (10 - 2)^(2) + 8`.

Solve this equation for
`a`:

`\displaystyle a = -(3)/(16)`.

Thus, the equation of this parabola in vertex form would be:

`\displaystyle y = -(3)/(16)\, (x - 2)^(2) + 8`.