# How do you graph and label the vertex and axis of symmetry of y=-3(x-2)^2+2?

May 21, 2017

vertex (2,2), axis of symmetry x=2

#### Explanation:

The coefficient of ${x}^{2}$ is negative, signifies the parabola opens down wards.

The given equation is in vertex form, it is easily identifiable (2,2). Axis of symmetry is x=2

y intercept would be $y = - 3 {\left(- 2\right)}^{2} + 2$ =-10

With these inputs the parabola can be easily graphed as shown below. May 21, 2017

$\text{see explanation}$

#### Explanation:

$\text{to graph the parabola, would require}$

• " coordinates of the vertex"

• " x and y intercepts"

• " shape of parabola max or min"

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

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

$\text{with } h = 2 , k = 2$

$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(2 , 2\right)$

$\text{the axis of symmetry passes through the vertex}$
$\text{and is a vertical line with equation } x = 2$

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

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

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

$x = 0 \to y = - 3 {\left(- 2\right)}^{2} + 2 = - 10 \leftarrow \textcolor{red}{\text{ y-intercept}}$

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

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

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

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

$\Rightarrow x = 2 \pm \sqrt{\frac{2}{3}} \leftarrow \text{ exact values}$

$x \approx 2.82 , x \approx 1.18 \text{ to 2 dec. places "larrcolor(red)" x-intercepts}$

$\textcolor{b l u e}{\text{Shape of parabola}}$

• " if " a>0" then minimum " uuu

• " if " a<0" then maximum " nnn

$\text{here " a=-3<0" hence maximum}$
graph{(y+3x^2-12x+10)(y-1000x+2000)=0 [-10, 10, -5, 5]}