EXPLANATION
Since we have the given quadratic function:
[tex]y=-4x^2-8x-8[/tex]
a)
Since the value of the squared term coefficient is negative, thus the graph opens up.
b)
[tex]\mathrm{The\: vertex\: of\: an\: up-down\: facing\: parabola\: of\: the\: form}\: y=ax^2+bx+c\: \mathrm{is}\: x_v=-\frac{b}{2a}[/tex][tex]\mathrm{The\: parabola\: params\: are\colon}[/tex][tex]a=-4,\: b=-8,\: c=-8[/tex][tex]x_v=-\frac{b}{2a}[/tex][tex]x_v=-\frac{\left(-8\right)}{2\left(-4\right)}[/tex][tex]\mathrm{Simplify}[/tex][tex]x_v=-1[/tex]
Plugging in the x_v value into the equation:
[tex]y_v=-4\mleft(-1\mright)^2-8\mleft(-1\mright)-8[/tex]
Computing the powers:
[tex]y_v=-4\cdot1-8(-1)-8[/tex]
Multiplying numbers:
[tex]y_v=-4+8-8[/tex]
Adding numbers:
[tex]y_v=-4[/tex]
[tex]\mathrm{Therefore\: the\: parabola\: vertex\: is}[/tex][tex]\mleft(-1,\: -4\mright)[/tex]
c) For a quadratic equation of the form ax^2 + bx + c = 0 the discriminant is b^2 -4ac
[tex]\mathrm{For\: }\quad a=-4,\: b=-8,\: c=-8\colon\quad \mleft(-8\mright)^2-4\mleft(-4\mright)\mleft(-8\mright)[/tex]
Computing the powers and multiplying numbers:
[tex]=-64[/tex]
Since the discriminant cannot be negative, there are not x-intercepts.
d) Identifying y-intercepts:
[tex]y\mathrm{-intercept\: is\: the\: point\: on\: the\: graph\: where\: }x=0[/tex][tex]y=-4\cdot\: 0^2-8\cdot\: 0-8[/tex][tex]\mathrm{Apply\: rule}\: 0^a=0[/tex][tex]y=-4\cdot\: 0-8\cdot\: 0-8[/tex][tex]\mathrm{Subtract\: the\: numbers\colon}\: -0-0-8=-8[/tex][tex]y=-8[/tex]
The y-intercept is at (0,-8)
