# How do you find the vertex and the intercepts for e(x) = -x² - 8x - 16?

Jul 19, 2018

The vertex is $\left(- 4 , 0\right)$.

There are no x-intercepts.

#### Explanation:

Given:

$e \left(x\right) = - {x}^{2} - 8 x - 16$,

where:

$a = - 1$, $b = - 8$, $c = - 16$

Vertex: minimum or maximum point of the parabola

The x-coordinate of the vertex is found using the formula:

$x = \frac{- b}{2 a}$

$x = \frac{- \left(- 8\right)}{2 \cdot - 1}$

$x = \frac{8}{- 2}$

$x = - \frac{8}{2}$

$x = - 4$

Substitute $y$ for $e \left(x\right)$. The y-coordinate of the vertex is found by substituting $- 4$ for $x$ and solving for $y$.

$y = - {\left(- 4\right)}^{2} - 8 \left(- 4\right) - 16$

$y = - 16 + 32 - 16$

$y = 0$

The vertex is $\left(- 4 , 0\right)$.

Use the discriminant (D) from the quadratic formula to determine whether there are any x-intercepts.

$\text{D} = \sqrt{{b}^{2} - 4 a c}$

Plug in the known values.

$\text{D} = \sqrt{{\left(- 8\right)}^{2} - 4 \cdot - 1 \cdot - 16}$

$\text{D} = \sqrt{- 128}$

Since the discriminant is the square root of a negative number, there are no real solutions (x-intercepts).

You could determine the y-intercept and the point opposite the y-intercept to help graph the parabola.

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

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

$y = - {\left(0\right)}^{2} - 8 \left(0\right) - 16$

$y = - 16$

The y-intercept is $\left(0 , - 16\right)$

Opposite point

$x = - 8$

Substitute $- 8$ for $x$ and solve for $y$.

$y = - \left({8}^{2}\right) - 8 \left(- 8\right) - 16$

$y = - 16$

The opposite point is $\left(- 8 , - 16\right)$

Plot the vertex, y-intercept, and opposite point. Sketch a parabola through the points. Do not connect the dots.

graph{-x^2-8x-16 [-10, 10, -5, 5]}