Question

What is the vertex form of `7y = - 13x^2 -15x + 2`?

Answer 1

`y=-13/7(x+15/26)^2+329/364`

Explanation:

First, get the equation into its typical form by dividing both sides by `7`.

`y=-13/7x^2-15/7x+2/7`

Now, we want to get this into vertex form:

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

First, factor the `-13/7` from the first two terms. Note that factoring a `-13/7` from a term is the same as multiplying the term by `-7/13`.

`y=-13/7(x^2+15/13x)+2/7`

Now, we want the term in the parentheses to be a perfect square. Perfect squares come in the pattern `(x+a)^2=x^2+2ax+a^2`.

Here, the middle term `15/13x` is the middle term of the perfect square trinomial, `2ax`. If we want to determine what `a` is, divide `15/13x` by `2x` to see that `a=15/26`.

This means that we want to add the missing term in the parentheses to make the group equal to `(x+15/26)^2`.

`y=-13/7overbrace((x^2+15/13x+?))^((x+15/26)^2)+2/7`

The missing term at the end of the perfect square trinomial is `a^2`, and we know that `a=15/26`, so `a^2=225/676`.

Now we add `225/676` to the terms in the parentheses. However, we can't go adding numbers to equations willy-nilly. We must balance what we just added on the same side of the equation. (For example, if we had added `2`, we would need to add `-2` to the same side of the equation for a net change of `0`).

`y=color(blue)(-13/7)(x^2+15/13x+color(blue)(225/676))+2/7+color(blue)?`

Notice that we haven't actually added `225/676`. Since it's inside of the parentheses, the term on the outside is being multiplied in. Thus, the `225/676` actually has a value of

`225/676xx-13/7=225/52xx-1/7=-225/364`

Since we have actually added `-225/364`, we must add a positive `225/364` to the same side.

`y=-13/7(x+15/26)^2+2/7+225/364`

Note that `2/7=104/364`, so

`color(red)(y=-13/7(x+15/26)^2+329/364`

This is in vertex form, where the parabola's vertex is at `(h,k)->(-15/26,329/364)`.

We can check our work by graphing the parabola:

graph{7y = - 13x^2 -15x + 2 [-4.93, 4.934, -2.466, 2.466]}

Note that `-15/26=-0.577` and `329/364=0.904`, which are the values obtained by clicking on the vertex.