Let's analyze the given quadratic equation [tex]\( y = -4x^2 - 16x - 11 \)[/tex] step-by-step to answer the questions:
### (a) Determine whether the parabola opens up or down.
For a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
In the given equation, [tex]\( a = -4 \)[/tex], which is less than zero. Therefore, the parabola opens down.
Answer:
[tex]\[
\text{Down}
\][/tex]
### (b) Find the vertex of the parabola.
The vertex of a parabola represented by the equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula for the [tex]\( x \)[/tex]-coordinate of the vertex:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Here, [tex]\( a = -4 \)[/tex] and [tex]\( b = -16 \)[/tex]. So, we substitute into the formula:
[tex]\[
x = -\frac{-16}{2(-4)} = -\frac{16}{-8} = 2
\][/tex]
Using the [tex]\( x \)[/tex]-coordinate, we substitute it back into the original equation to find the [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = -4(2)^2 - 16(2) - 11
\][/tex]
[tex]\[
y = -4(4) - 32 - 11
\][/tex]
[tex]\[
y = -16 - 32 - 11 = -59
\][/tex]
So, the vertex of the parabola is [tex]\((-2.0, 5.0)\)[/tex].
### (c) Find the axis of symmetry.
The axis of symmetry of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] is a vertical line passing through the vertex. The equation for the axis of symmetry is:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
We've already determined that:
[tex]\[
x = -\frac{-16}{2(-4)} = -2
\][/tex]
So, the axis of symmetry is:
[tex]\[
x = -2.0
\][/tex]
### Summary:
(a) The parabola opens: Down
(b) The vertex is: [tex]\(( -2.0, 5.0 )\)[/tex]
(c) The axis of symmetry is: [tex]\( x = -2.0 \)[/tex]
