Question

What is the vertex form of `y=-13x^2-x-19`?

Answer 1

`y = -13(x+1/26)^2-987/52` is vertex form

This gives the vertex as `(-1/26, -18 51/52)`

Explanation:

The equation of a parabola can be written as:

`y = ax^2 +bx+c" "` or in vertex form: `" "y = a(x+b)^2 +c`

We have `" "y = -13x^2-x-19`

To change an equation into vertex form:

Step 1. Make `1x^2`. Divide out `a`, the coefficient of `x^2`

`y = -13(x^2+x/13+19/13)`

Step 2. Add and subtract `color(blue)((b/2)^2)` (same as `+0)`

`y = -13(x^2+x/13 color(blue)(+(1/26)^2 -(1/26)^2)+19/13)`

Step 3. Write 3 terms as a perfect square and simplify the others

`y = -13((color(red)(x^2+x/13 +(1/26)^2)) -color(green)((1/676)+19/13))`

`y = -13(color(red)((x+1/26)^2) color(green)(+987/676))`

Step 4: Multiply by `a` outside the bracket

`y = -13color(red)((x+1/26)^2) color(green)(-987/52)" "larr` vertex form

This gives the vertex as `(-1/26, -18 51/52)`

graph{y = -13x^2-x-19 [-0.2135, 0.4115, -19.2013, -18.8888]}