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

Vertex: $\left(0.5 , 4.5\right)$
Axis of Symmetry: $x = 0.5$

#### Explanation:

First, we have to convert $y = 2 {x}^{2} - 2 x + 5$ into vertex form, because it is currently in standard form $\left(a {x}^{2} + b x + c\right)$. 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: $2 {x}^{2} \mathmr{and} {x}^{2}$.

This becomes $2 \left({x}^{2} - x\right) + 5$.

Now, use ${x}^{2} - x$ to complete the square, adding and subtracting ${\left(\frac{b}{2}\right)}^{2}$.

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

${\left(\frac{- 1}{2}\right)}^{2}$ = $0.25$

$2 \left({x}^{2} - x + 0.25 - 0.25\right) + 5$

Now, we can write this as a binomial squared.

$2 \left[{\left(x - 0.5\right)}^{2} - 0.25\right] + 5$

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

This becomes $2 {\left(x - 0.5\right)}^{2} - 0.5 + 5$

Which simplifies to $2 {\left(x - 0.5\right)}^{2} + 4.5$

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

Vertex: $\left(0.5 , 4.5\right)$
Axis of Symmetry: $x = 0.5$

Hope this helps!

Best wishes,
A fellow highschool student