# How do you find the vertex and the intercepts for y = -3(x - 2)^2 + 4?

Sep 17, 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 multiplier.

$y = - 3 {\left(x - 2\right)}^{2} + 4 \text{ is in vertex form}$

$\text{with "(h,k)=(2,4)larrcolor(red)" vertex}$

$\text{to find the intercepts}$

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

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

$x = 0 \to y = - 3 {\left(- 2\right)}^{2} + 4 = - 8 \leftarrow \textcolor{red}{\text{ y-intercept}}$

$y = 0 \to - 3 {\left(x - 2\right)}^{2} + 4 = 0$

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

$\Rightarrow {\left(x - 2\right)}^{2} = \frac{4}{3}$

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

$\Rightarrow x - 2 = \pm \sqrt{\frac{4}{3}} \leftarrow \text{ note plus or minus}$

$\Rightarrow x = 2 \pm \frac{2}{\sqrt{3}} = 2 + \frac{2 \sqrt{3}}{3}$

$\Rightarrow x \approx 0.85 \text{ or "x~~ 3.15larrcolor(red)" x-intercepts}$
graph{-3(x-2)^2+4 [-10, 10, -5, 5]}