# How do you graph the parabola y= - x^2 - 6x - 8 using vertex, intercepts and additional points?

Feb 25, 2018

See below

#### Explanation:

Firstly, complete the square to put the equation in vertex form,

$y = - {\left(x + 3\right)}^{2} + 1$

This implies that the vertex, or local maximum (since this is a negative quadratic) is $\left(- 3 , 1\right)$. This can be plotted.

The quadratic can also be factorised,

$y = - \left(x + 2\right) \left(x + 4\right)$

which tells us that the quadratic has roots of -2 and -4, and crosses the $x a \xi s$ at these points.

Finally, we observe that if we plug $x = 0$ into the original equation, $y = - 8$, so this is the $y$ intercept.

All of this gives us enough information to sketch the curve:

graph{-x^2-6x-8 [-10, 10, -5, 5]}

Feb 25, 2018

First, turn this equation to vertex form:

$y = a \left(x - h\right) + k$ with $\left(h , k\right)$ as the $\text{vertex}$. You can find this by completing the square:

$y = - \left({x}^{2} + 6 x + {\left(3\right)}^{2} - {\left(3\right)}^{2}\right) - 8$

$y = - {\left(x + 3\right)}^{2} + 1$

So the $\text{vertex}$ is at $\left(- 3 , 1\right)$

To find the $\text{zeroes}$ also known as $\text{x-intercept(s)}$, set $y = 0$ and factor (if it is factorable):

$0 = - \left({x}^{2} + 6 x + 8\right)$

$0 = - \left(x + 4\right) \left(x + 2\right)$

$x = - 4 , - 2$

The $\text{x-intercepts}$ are at $\left(- 4 , 0\right)$ and $\left(- 2 , 0\right)$.

You can also use the quadratic formula to solve if it is not factorable (A discriminant that is a perfect square indicates that the equation is factorable):

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

$x = \frac{- \left(- 6\right) \pm \sqrt{{\left(- 6\right)}^{2} - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

$x = \frac{6 \pm \sqrt{4}}{-} 2$

$x = \frac{6 \pm 2}{-} 2$

$x = - 4 , - 2$

The $\text{y-intercept}$ is $c$ in $a {x}^{2} + b x + c$:

The y-intercept here is $\left(0 , - 8\right)$.

To find additional points, plug in values for $x$:

$- {\left(1\right)}^{2} - 6 \cdot 1 - 8 \implies - 15 \implies \left(1 , - 15\right)$

$- {\left(2\right)}^{2} - 6 \cdot 2 - 8 \implies - 24 \implies \left(2 , - 24\right)$

etc.

A graph below is for reference:

graph{-x^2-6x-8 [-12.295, 7.705, -7.76, 2.24]}