Question

What is the equation of the parabola that has a vertex at `(9, -23)` and passes through point `(35,17)`?

Answer 1

We can solve this using the vertex formula, `y=a(x-h)^2+k`

Explanation:

The standard format for a parabola is

`y = ax^2 + bx + c`

But there is also the vertex formula,

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

Where `(h,k)` is the location of the vertex.

So from the question, the equation would be

`y=a(x-9)^2-23`

To find a, substitute the x and y values given: `(35,17)` and solve for `a`:

`17=a(35-9)^2-23`

`(17+23)/(35-9)^2=a`

`a=40/26^2 = 10/169`

so the formula, in vertex form, is

`y = 10/169(x-9)^2-23`

To find the standard form, expand the `(x-9)^2` term, and simplify to
`y = ax^2 + bx + c` form.

Answer 2

For problems of this type, use vertex form, y = a`(x - p)^2` + q.

Explanation:

In vertex form, mentioned above, the vertex's coordinates are (p, q) and a point (x , y) that is on the parabola.

When finding the equation of the parabola, we have to solve for a, which influences the width and the direction of opening of the parabola.

y = a`(x - p)^2` + q
17 = a`(35 - 9)^2` - 23
17 = 576a - 23
17 + 23 = 576a
`5/72` = a

So, the equation of the parabola is y = `5/72` `(x - 9)^2` - 23.

Hopefully you understand now!