# How to graph a parabola y=-3(x-2)^2 + 5?

Jun 29, 2018

Refer to the explanation.

#### Explanation:

Graph:

$y = - 3 {\left(x - 2\right)}^{2} + 5$ is a quadratic equation in vertex form:

$y = a {\left(x - h\right)}^{2} + k$,

where:

$h$ is the axis of symmetry and $\left(h , k\right)$ is the vertex.

In order to graph a parabola, you need the vertex, the y-intercept, x-intercepts, and one or more additional points.

Vertex: maximum or minimum point of the parabola. Since $a < 0$, the vertex is the maximum point and the parabola opens downward.

The vertex is $\left(2 , 5\right)$

Y-intercept: value of $y$ when $x = 0$

Substitute $0$ for $x$.

$y = - 3 {\left(0 - 2\right)}^{2} + 5$

$y = - 3 {\left(- 2\right)}^{2} + 5$

$y = - 3 \left(4\right) + 5$

$y = - 7$

The y-intercept is $\left(0 , - 7\right)$.

X-intercepts (Roots): values for $x$ when $y = 0$

Substitute $0$ for $y$ and solve for $x$.

$0 = - 3 {\left(x - 2\right)}^{2} + 5$

Subtract $5$ from both sides.

$- 5 = - 3 {\left(x - 2\right)}^{2}$

Divide both sides by $- 3$.

$\frac{5}{3} = {\left(x - 2\right)}^{2}$ $\leftarrow$ Two negatives make a positive.

Take the square root of both sides.

$\pm \sqrt{- \frac{5}{3}} = x - 2$

Use the rule $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

$\pm \frac{\sqrt{5}}{\sqrt{3}} = x - 2$

Add $2$ to both sides.

$2 \pm \frac{\sqrt{5}}{\sqrt{3}} = x$

Switch sides.

$x = 2 \pm \frac{\sqrt{5}}{\sqrt{3}}$

$x = 2 + \frac{\sqrt{5}}{\sqrt{3}}$, $2 - \frac{\sqrt{5}}{\sqrt{3}}$

$x = \approx 3.291 , 0.709$

The x-intercepts are: $\left(2 + \frac{\sqrt{5}}{\sqrt{3}} , 0\right)$, $\left(2 - \frac{\sqrt{5}}{\sqrt{3}}\right)$

Approximate x-intercepts: $\left(\approx 3.291 , 0\right)$, $\left(0.709 , 0\right)$

Substitute $4$ for $x$ and solve for $y$.

$y = - 3 {\left(4 - 2\right)}^{2} + 5$

$y = - 3 {\left(2\right)}^{2} + 5$

$y = - 3 \left(4\right) + 5$

$y = - 12 + 5$

$y = - 7$

Additional point: $\left(4 , - 7\right)$

Summary:

Vertex: $\left(2 , 5\right)$

Y-intercept: $\left(0 , - 7\right)$

X-intercepts: $\left(2 + \frac{\sqrt{5}}{\sqrt{3}} , 0\right)$, $\left(2 - \frac{\sqrt{5}}{\sqrt{3}}\right)$

Approximate x-intercepts: $\left(\approx 3.291 , 0\right)$, $\left(0.709 , 0\right)$

Additional point: $\left(4 , - 7\right)$

Plot the points and sketch a parabola through the points. Do not connect the dots.

graph{y=-3(x-2)^2+5 [-10, 10, -5, 5]}