Question

What is the axis of symmetry and vertex for the graph `y = 2x^2 - 2x + 5`?

Answer 1

Vertex: `(0.5,4.5)`
Axis of Symmetry: `x = 0.5`

Explanation:

First, we have to convert `y=2x^2 - 2x + 5` into vertex form, because it is currently in standard form `(ax^2 + bx + c)`. To do this, we must complete the square and find the perfect square trinomial that corresponds with the equation.

First, factor the 2 out of our first two terms: `2x^2 and x^2`.

This becomes `2(x^2 - x) + 5`.

Now, use `x^2-x` to complete the square, adding and subtracting `(b/2)^2`.

Since there is no coefficient in front of x, we can assume that it is -1 because of the sign.

`([-1]/2)^2` = `0.25`

`2(x^2-x+0.25-0.25)+5`

Now, we can write this as a binomial squared.

`2[(x - 0.5)^2-0.25] + 5`

We must multiply the -0.25 by 2 to get rid of its brackets.

This becomes `2(x-0.5)^2-0.5+5`

Which simplifies to `2(x-0.5)^2+4.5`

It's finally in vertex form! We can easily see that the vertex is `(0.5,4.5)`, and the axis of symmetry is simply the x coordinate of the vertex.

Vertex: `(0.5,4.5)`
Axis of Symmetry: `x = 0.5`

Hope this helps!

Best wishes,
A fellow highschool student