# What is the vertex form of y=(3x – 4) (2x – 1) ?

May 12, 2018

$y = 6 {\left(x - \frac{11}{12}\right)}^{2} - \frac{25}{24}$

#### Explanation:

In vertex form, a is stretch factor, h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

$y = a {\left(x - h\right)}^{2} + k$

So, we must find the vertex.

The zero product property says that, if $a \cdot b = 0$, then $a = 0$ or $b = 0$, or $a , b = 0$.

Apply the zero product property to find the roots of the equation.

$\textcolor{red}{\left(3 x - 4\right) = 0}$

$\textcolor{red}{3 x = 4}$

$\textcolor{red}{{x}_{1} = \frac{4}{3}}$

$\textcolor{b l u e}{\left(2 x - 1\right) = 0}$

$\textcolor{b l u e}{2 x = 1}$

$\textcolor{b l u e}{{x}_{2} = \frac{1}{2}}$

Then, find the midpoint of the roots to find the x-value of the vertex. Where $M = \text{midpoint}$:

$M = \frac{{x}_{1} + {x}_{2}}{2}$

$\text{ } = \frac{\frac{4}{3} + \frac{1}{2}}{2}$

$\text{ } = \frac{11}{12}$

$\therefore h = \frac{11}{12}$

We can input this value for x in the equation to solve for y.

$y = \left(3 x - 4\right) \left(2 x - 1\right)$

$y = \left[3 \left(\frac{11}{12}\right) - 4\right] \left[2 \left(\frac{11}{12}\right) - 1\right]$

$y = - \frac{25}{24}$

$\therefore k = - \frac{25}{24}$

Input these values respectively into a vertex-form equation.

$y = a {\left(x - \frac{11}{12}\right)}^{2} - \frac{25}{24}$

Solve for the a value by inputting a known value along the parabola, for this example, we'll use a root.

$0 = a {\left[\left(\frac{1}{2}\right) - \frac{11}{12}\right]}^{2} - \frac{25}{24}$

$\frac{25}{24} = a {\left(\frac{- 5}{12}\right)}^{2}$

$\frac{25}{24} = \frac{25}{144} a$

$a = 6$

$\therefore y = 6 {\left(x - \frac{11}{12}\right)}^{2} - \frac{25}{24}$