# How do you find the intercept and vertex of  y-4 = -(x-1)^2?

May 15, 2017

$\text{see explanation}$

#### Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form }$is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where ( h , k ) are the coordinates of the vertex and a is a constant.

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

$\text{add 4 to both sides}$

$y \cancel{- 4} \cancel{+ 4} = - {\left(x - 1\right)}^{2} + 4$

$\Rightarrow y = - {\left(x - 1\right)}^{2} + 4 \leftarrow \textcolor{red}{\text{ in vertex form}}$

$\text{with " h=1" and } k = 4$

$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(1 , 4\right)$

$\textcolor{b l u e}{\text{Intercepts}}$

• " let x = 0, in equation for y-intercept"

• " let y = 0, in equation for x-intercepts"

$x = 0 \to y = - {\left(0 - 1\right)}^{2} + 4 = 3 \leftarrow \textcolor{red}{\text{ y- intercept}}$

$y = 0 \to - {\left(x - 1\right)}^{2} + 4 = 0$

$\Rightarrow - {\left(x - 1\right)}^{2} = - 4$

$\text{multiply both sides by - 1}$

$\Rightarrow {\left(x - 1\right)}^{2} = 4$

$\textcolor{b l u e}{\text{take the square root of both sides}}$

$\sqrt{{\left(x - 1\right)}^{2}} = \pm \sqrt{4} \leftarrow \textcolor{red}{\text{ note plus or minus}}$

$\Rightarrow x - 1 = \pm 2 \leftarrow \text{ add 1 to both sides}$

$\Rightarrow x = 1 \pm 2$

$\Rightarrow x = 1 - 2 = - 1 , x = 1 + 2 = 3 \leftarrow \textcolor{red}{\text{ x- intercepts}}$
graph{-(x-1)^2+4 [-10, 10, -5, 5]}