Question

What is the graph of `f(x) = 2x^2 - 3x + 7`?

Answer 1

To graph a quadratic equation we first need to factorise it into a different form.

First we check what the discriminant is equal to
Where `f(x)=ax^2+bx^2+c`
`Delta`(Discriminant)`=b^2-4ac`

In this case `Delta`=`3^2-4*2*7`
`Delta=-47`

Because it is less than zero it can't be factored normally
Therefore we must use the The Quadratic Formula or Completing the Square

Here I have completed the square

`f(x)=2x^2-3x+7`

Remove factor from `x^2` term

`f(x)=2*(x^2-3/2x+7/2)`

Take `x` term, half it and then square it

`f(x)=-3/2->-3/4->9/16`

Add and then subtract this number inside the equation

`f(x)=2*(x^2-3/2x+9/16-9/16+7/2)`

Combine the first three terms in a perfect square

`f(x)=2*((x-3/4)^2-9/16+7/2)`

Equate left over terms

`f(x)=2*((x-3/4)^2+47/16)`

Multiply coefficient back in

`f(x)=2(x-3/4)^2+47/8`

This gives a turning point of `(3/4,47/8)=(0.75,5.875)`
and a `y` intercept of `2*(3/4)^2+47/8`
`=(0,7)`