Question

Create a formula/equation to calculate the x-value for the vertex of any parabola?

Answer 1

The formula is:

`h=-b/(2a)`

where h is the x coordinate of the vertex and the other values are obtained from the standard form of the equation of a parabola:

`y = ax^2+bx+c`

Answer 2

For a parabola in standard form, `y=ax^2+bx+c`,

the x value of the vertex is given by `-b/(2a)`

Explanation:

Here are two forms of the parabola

Standard form

`y=ax^2+bx+c`

Vertex form

`y=a(x+b)^2+c`

The vertex is given by `(-b,c)` in the vertex form. Since the x-value of the vertex is simply `-b` in the vertex form, we are going to derive a formula for the vertex that can be read off the standard form.

*Note that a, b and c are not necessarily the same numbers in each form. You can't just put the a, b and c from one form into the other. You will see this later.

So, you need to be happy with the vertex form giving the vertex as `(-b,c)`. If you are satisfied with that then it's just a matter of converting the standard form into the vertex form by completing the square (I won't explain how to do that but here are the steps)

`y=ax^2+bx+c`

`y=a(x^2+(bx)/a+c/a)`

`y=a(color(green)(x^2+(bx)/a+(b/(2a))^2)+c/a-(b/(2a))^2)`

`y=a(color(green)(x^2+(bx)/a+(b^2/(4a^2)))+c/a-b^2/(4a^2))`

`y=a(color(green)((x+b/(2a))^2)+c/a-b^2/(4a^2))`

`y=a(x+b/(2a))^2+c-b^2/(4a)`

Now that we are in vertex form, equate the new b and c values to b and c in the general vertex form (this is what I meant by saying that a, b and c aren't the same in both forms)

`y=a(x+color(blue)(b))^2+color(red)(c)`

`y=a(x+color(blue)(b/(2a)))^2+color(red)(c-b^2/(4a))`

The x-value of the vertex is given by `-b` so it is now `-b/(2a)`

The y-value of the vertex is given by `c` and becomes `c-b^2/(4a)`

So to summarise, when you have a parabola in standard form, the x-value of the vertex is given by `-b/(2a)`

E.g.

`y=x^2+6x+8`

`"x-value of the vertex"=-6/(2*1)=-3`

`"y-value of the vertex"=c-b^2/(4a)=8-(6^2)/(4*1)=-1`