Question

What is the vertex of `y= 2x^2 - 4x + 3`?

Answer 1

`color(purple)((x_("vertex"), y_("vertex")) = (-1,9))`

Explanation:

Given: `y=2x^2-4x+3.............................(1)`

Let the coordinates of the vertex be `(x_("vertex"), y_("vertex"))`
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`color(white)(.........................)color(green)(" Preamble")`

There are two ways of doing this.

It looks as though the fashionable way at the moment is 'completing the square' alternatively known as a 'vertex equation'.

The other, which I am going to show you, is the basis upon which completing the square is built on.
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`color(Blue)(underline(Stepcolor(white)(.)1))`

Write the given equation in the form of:
`y=2(x^2-2x+3/2)`

I have made it such that I have a part that starts with `x^2`

We now look at the part inside the brackets of `-2x`

We then do this: `-1/2xx-2=-1` ignoring the variable `x`

We have now found `color(green)(x_("vertex")=-1)....................(2)`

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`color(Blue)(underline(Stepcolor(white)(.)2))`

Substitute (2) into (1) giving:

`y_("vertex")=2(-1)^2-4(-1)+3`

`y_("vertex")=2+4+3`

`color(green)(y_("vertex")=9)`
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`color(purple)((x_("vertex"), y_("vertex")) = (-1,9))`