# How do you find the vertex and intercepts for x =-1/2(y-2)^2-4?

May 21, 2016

$\textcolor{b l u e}{\text{Vertex } \to \left(x , y\right) \to \left(- 4 , 2\right)}$
$\textcolor{b l u e}{{x}_{\text{intercept}} = - 6}$
color(blue)(y_("intercept")=2+-(2sqrt(2)) i color(red)(" "y_("intercept")!inRR

(No 'Real Number' solution for y intercept)

#### Explanation:

Instead of an equation in $x$ we have an equation in $y$. What that does is 'rotate' the graph ${90}^{o}$ to the right

So instead the general shape being $\cap$ it is $\supset$

Also the format of the given equation is in Vertex form.

$\textcolor{b l u e}{\text{Determine the vertex}}$
'..................................................................
$\textcolor{red}{\text{If this had been an equation in "x) color(magenta)(larr" For comparison only}}$

Then $\textcolor{red}{{x}_{\text{vertex}} = \left(- 1\right) \times \left(- 2\right) = + 2}$

color(red)(y_("vertex")=-4
'.......................................................................

But our equation is in $y$ so we need to reverse them in that

$\textcolor{g r e e n}{{y}_{\text{vertex}} = \left(- 1\right) \times \left(- 2\right) = + 2}$
$\textcolor{g r e e n}{{x}_{\text{vertex}} = - 4}$

$\textcolor{b l u e}{\text{Vertex } \to \left(x , y\right) \to \left(- 4 , 2\right)}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Determine x intercept}}$

Set $y = 0$ giving

color(brown)(x=-1/2(y-2)^2-4)color(green)(" "->" "x=-1/2(0-2)^2-4

$\textcolor{b l u e}{{x}_{\text{intercept}} = - 6}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine y intercept}}$

Set $x = 0$

color(brown)(x=-1/2(y-2)^2-4)color(green)(" "->" "0=-1/2(y-2)^2-4

Add 4 to both sides

$4 = - \frac{1}{2} {\left(y - 2\right)}^{2}$

Multiply both sides by (-2)

$- 8 = {\left(y - 2\right)}^{2}$

Square root both sides

$\sqrt{- 8} = y - 2$

$y = 2 \pm \sqrt{- 8}$

As you have square root of a negative number the graph does not cross the y axis.

So the only solution for the values of y are in the complex numbers set of values

$y = 2 \pm \left(2 \sqrt{2}\right) i$