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

Question

Ethan Adkins · Accepted Answer

#y#-intercept: #y=-7#
#x#intercepts: #x=-1# and #x=-7#
vertex at: #(-4,9)#

Given
#color(white)("XXX")y=(-x-1)(x+7)#

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:
#color(white)("XXX")y=(-0-1)(0+7)=-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)
#color(white)("XXX")0=(-x-1)(x+7)#

#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=((-1)+(-7))/2==-4#
Substituting #x=-4# into the given equation gives the #y# coordinate of the vertex
#color(white)("XXX")y=(-(-4)-1)((-4)+7)=3 * 3 =9#
So the vertex is at #(-4,9)#

Savannah Anderson · Answer

undefined To find the vertex of the quadratic equation $y = (-x - 1)(x + 7)$, first, expand the equation to obtain the standard form $y = ax^2 + bx + c$. Then, use the formula $x_v = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Next, substitute $x_v$ into the equation to find the corresponding y-coordinate. To find the x-intercepts, set $y = 0$ and solve for $x$. To find the y-intercept, set $x = 0$ and solve for $y$.