# How do you find the vertex and intercepts for y=(-x-1)(x+7)?

Oct 27, 2017

$y$-intercept: $y = - 7$
$x$intercepts: $x = - 1$ and $x = - 7$
vertex at: $\left(- 4 , 9\right)$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} y = \left(- x - 1\right) \left(x + 7\right)$

The y-intercept is the value of $y$ when $x = 0$ (that is, on the Y-axis, since for all values on the Y-axis, $x = 0$)
Substituting $0$ for $x$ in the given equation:
$\textcolor{w h i t e}{\text{XXX}} y = \left(- 0 - 1\right) \left(0 + 7\right) = - 7$
The y-intercept is at $y = - 7$

The x-intercepts are the values of $x$ for which $y = 0$ (that is on the X-axis)
$\textcolor{w h i t e}{\text{XXX}} 0 = \left(- x - 1\right) \left(x + 7\right)$

color(white)("XXX"){:((-x-1)=0," or ",(x+7)=0), (rarr x=-1,,rarrx=-7):}
The x-intercepts are at $x = - 1$ and $x = - 7$

The vertex can be determined in a couple ways:
1. by converting the given equation into vertex form (ask if you need to see this version)
or
2. by noting that the axis of symmetry will cross the X-axis a t he mid point between the 2 x-intercepts;
namely at $x = \frac{\left(- 1\right) + \left(- 7\right)}{2} = = - 4$
Substituting $x = - 4$ into the given equation gives the $y$ coordinate of the vertex
$\textcolor{w h i t e}{\text{XXX}} y = \left(- \left(- 4\right) - 1\right) \left(\left(- 4\right) + 7\right) = 3 \cdot 3 = 9$
So the vertex is at $\left(- 4 , 9\right)$ 