# What is the vertex of y=2(x-2)^2-3x ?

Jul 11, 2018

The vertex is $\left(\frac{11}{4} , - \frac{57}{8}\right)$ or $\left(2.75 , - 7.125\right)$.

#### Explanation:

Given:

Expand ${\left(x - 2\right)}^{2}$.

$y = 2 \left({x}^{2} - 4 x + 4\right) - 3 x$

Simplify.

$y = 2 {x}^{2} - 8 x + 8 - 3 x$

Simplify.

$y = 2 {x}^{2} - 11 x + 8$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$,

where:

$a = 2$, $b = - 11$, $c = 8$

Vertex: the maximum or minimum point of a parabola

The x-coordinate of the vertex can be calculated using the formula for the axis of symmetry:

$x = \frac{- b}{2 a}$

$x = \frac{- \left(- 11\right)}{2 \cdot 2}$

$x = \frac{11}{4}$ or $2.75$

The y-coordinate is determined by substituting $\frac{11}{4}$ for $x$ in the equation and solving for $y$.

$y = 2 {\left(\frac{11}{4}\right)}^{2} - 11 \left(\frac{11}{4}\right) + 8$

Simplify.

$y = 2 \left(\frac{121}{16}\right) - \frac{121}{4} + 8$

Simplify.

$y = \frac{242}{16} - \frac{121}{4} + 8$

Simplify $\frac{242}{16}$ to $\frac{121}{8}$.

$y = \frac{121}{8} - \frac{121}{4} + 8$

The least common denominator is $8$. Multiply $\frac{121}{4}$ by $\frac{2}{2}$ and multiply $8$ by $\frac{8}{8}$ in order to get equivalent fractions with $8$ as the denominator. Since $\frac{n}{n} = 1$, the numbers will change but the value will remain the same.

$y = \frac{121}{8} - \left(\frac{121}{4} \times \frac{2}{2}\right) + \left(8 \times \frac{8}{8}\right)$

Simplify.

$y = \frac{121}{8} - \frac{242}{8} + \frac{64}{8}$

Simplify.

$y = \frac{121 - 242 + 64}{8}$

$y = - \frac{57}{8}$ or $\left(- 7.125\right)$

The vertex is $\left(\frac{11}{4} , - \frac{57}{8}\right)$ or $\left(2.75 , - 7.125\right)$.

$g r a p h \left\{y = 2 \left({x}^{2} - 4 x + 4\right) - 3 x \left[- 9.58 , 10.42 , - 10.44 , - 0.44\right]\right\}$