# What is the vertex form of 6y = 18x^2+18x+42?

Feb 28, 2016

Answered the wrong question: Typo must have double tap of the 2 key . One with shift and one without inserting a spurious 2: Error not spotted and carried through!!!
color(blue)("vertex equation"->y=9/13(x+(color(red)(1))/2)^(color(green)(2))+ 337/156

color(brown)(y_("vertex")=337/156~=2.1603" to 4 decimal places")

$\textcolor{b r o w n}{{x}_{\text{vertex}} = \left(- 1\right) \times \frac{1}{2} = - \frac{1}{2} = - 0.5}$

#### Explanation:

Given:$\text{ } 26 y = 18 {x}^{2} + 18 x + 42$

Divide both sides by 26

$y = \frac{18}{26} {x}^{2} + \frac{18}{26} x + \frac{42}{18}$

$y = \frac{9}{13} {x}^{2} + \frac{9}{13} x + \frac{7}{3}$..................(1)

Write as:$\text{ } y = \frac{9}{13} \left({x}^{\textcolor{g r e e n}{2}} + x\right) + \frac{7}{3}$.....(2)

$x \to \textcolor{red}{1} \times x$

Change equation (2) to be

$y = \frac{9}{13} {\left(x + \frac{\textcolor{red}{1}}{2}\right)}^{\textcolor{g r e e n}{2}} + \frac{7}{3} + k$ ......(3)

The correction constant $k$ is needed because we have changed the value of the whole RHS by changing the bracketed part as we did.

To find the value of k equate equation (1) to equation (3) through y

$\frac{9}{13} {x}^{2} + \frac{9}{13} x + \frac{7}{3} = y = \frac{9}{13} {\left(x + \frac{\textcolor{red}{1}}{2}\right)}^{\textcolor{g r e e n}{2}} + \frac{7}{3} + k$

$\frac{9}{13} {x}^{2} + \frac{9}{13} x + \frac{7}{3} = \frac{9}{13} \left({x}^{2} + x + \frac{1}{4}\right) + \frac{7}{3} + k$

$\cancel{\frac{9}{13} {x}^{2}} + \cancel{\frac{9}{13} x} + \cancel{\frac{7}{3}} = \cancel{\frac{9}{13} {x}^{2}} + \cancel{\frac{9}{13} x} + \frac{9}{52} + \cancel{\frac{7}{3}} + k$

$k = - \frac{9}{52}$

So equation (3) becomes

color(blue)("vertex equation"->y=9/13(x+(color(red)(1))/2)^(color(green)(2))+ 337/156

$\textcolor{red}{\text{As in the graph}}$

${y}_{\text{vertex}} = \frac{337}{156} \cong 2.1603$ to 4 decimal places

${x}_{\text{vertex}} = \left(- 1\right) \times \frac{1}{2} = - \frac{1}{2} = - 0.5$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Mar 22, 2016

Correct answer this time. Other solution left in place as an extended example of method.

$\textcolor{b l u e}{\text{ } y = 3 \left(x + 1\right) + 4}$

#### Explanation:

I have built this in the way I would do it for myself. The previous solution (incorrect question) shows the method in detail.

Given:$\text{ } 6 y = 18 {x}^{2} + 18 x + 42$

Divide both sides by 6

$\text{ } y = 3 {x}^{2} + 3 x + \frac{42}{6}$

$\text{ } y = 3 {\left(x + 1\right)}^{2} + k + \frac{42}{6}$

$\text{ "k= -3" and } \frac{42}{6} = 7$

$\textcolor{b l u e}{\text{ } y = 3 \left(x + 1\right) + 4}$