# How do you find the vertex and intercepts for y= -4x^2 - 16x -11?

Aug 25, 2017

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

X-Intercepts: $\left(- \frac{4 + \sqrt{5}}{2} , 0\right)$ and $\left(- \frac{4 - \sqrt{5}}{2} , 0\right)$

Refer to the explanation for the process.

#### Explanation:

Given:

$y = - 4 {x}^{2} - 16 x - 11$ is a quadratic equation in standard form:

$a {x}^{2} + b x + c$,

where:

$a = - 4$, $b = - 16$, and $c = - 11$.

The axis of symmetry, $x$, is $\frac{- b}{2 a}$.

$x = \frac{- \left(- 16\right)}{2 \cdot - 4}$

Simplify.

$x = \frac{16}{- 8}$

Simplify.

$x = - 2$

This is also the $x$ value of the vertex.

Vertex: maximum or minimum point of the parabola.

$x = - 2$

To determine $y$, substitute $- 2$ for $x$ in the equation and solve.

$y = - 4 {\left(- 2\right)}^{2} - 16 \left(- 2\right) - 11$

Simplify.

$y = - 4 \left(4\right) + 32 - 11$

$y = - 16 + 32 - 11$

$y = 5$

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

Since $a < 0$, the vertex is the maximum point and the parabola opens downward.

X-Intercepts: values of $x$ when $y = 0$.

To determine the x-intercepts, substitute $0$ for $y$ and solve for $x$.

$0 = - 4 {x}^{2} - 16 x - 11$

Use the quadratic formula to solve for $x$.

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Plug in the known values from the quadratic equation.

$x = \frac{- \left(- 16\right) \pm \sqrt{{\left(- 16\right)}^{2} - 4 \cdot - 4 \cdot - 11}}{2 \cdot - 4}$

Simplify.

$x = \frac{16 \pm \sqrt{256 - 176}}{- 8}$

Simplify.

$x = \frac{16 \pm \sqrt{80}}{- 8}$

Prime factorize $80$.

$x = \frac{16 \pm \sqrt{\left(2 \times 2\right) \times \left(2 \times 2\right) \times 5}}{- 8} 8$

Simplify.

$x = \frac{16 \pm 4 \sqrt{5}}{- 8}$

Simplify.

$x = \frac{4 \pm \sqrt{5}}{- 2}$

Solutions for $x$.

$x = - \frac{4 + \sqrt{5}}{2} ,$$- \frac{4 - \sqrt{5}}{2}$

x-intercepts: $\left(- \frac{4 + \sqrt{5}}{2} , 0\right)$ and $\left(- \frac{4 - \sqrt{5}}{2} , 0\right)$

Approximate values of x-intercepts:

$\left(- 0.882 , 0\right)$ and $\left(- 3.12 , 0\right)$

Summary

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

X-Intercepts: $\left(- \frac{4 + \sqrt{5}}{2} , 0\right)$ and $\left(- \frac{4 - \sqrt{5}}{2} , 0\right)$

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

graph{y=-4x^2-16x-11 [-11.25, 11.25, -5.625, 5.625]}