Question

find the vertex, axis of symmetry, x - intercept(s), y - intercept, focus, and directrix for the parabola. sketch the correct graph of the parabola.
( y = - \frac { 1 } { 4 } ( x + 4 ) ^ { 2 } - 3 )
the vertex is ( ( - 4, - 3 ) ).
(type an ordered pair.)
the axis of symmetry is ( x = - 4 ).
select the correct choice below.
a. there are no x - intercepts.
b. the x - intercepts are ( ( - 1,0 ) ) and ( ( - 7,0 ) ).
c. the x - intercept is ( ( - 1,0 ) ).

Explanation:

Step1: Find x - intercepts

Set \(y = 0\):

$$
LATEXBLOCK0
$$

Since the square of a real number \((x + 4)^2\geq0\) for all real \(x\), and \(-12<0\), there are no real solutions for \(x\).

Step2: Find y - intercept

Set \(x = 0\):

$$
LATEXBLOCK1
$$

The \(y\) - intercept is \((0,-7)\)

Step3: Find focus and directrix

The standard form of a parabola is \(y=a(x - h)^2+k\), where \((h,k)\) is the vertex. Here \(h=-4,k =-3,a=-\frac{1}{4}\).
For a parabola \(y=a(x - h)^2+k\), the focus is \((h,k+\frac{1}{4a})\) and the directrix is \(y=k-\frac{1}{4a}\)
Substitute \(a =-\frac{1}{4},h=-4,k=-3\)

$$
LATEXBLOCK2
$$

$$
LATEXBLOCK3
$$

The focus is \((-4,-4)\) and the directrix is \(y=-2\)

Answer:

A. There are no x - intercepts.
The \(y\) - intercept is \((0,-7)\)
The focus is \((-4,-4)\)
The directrix is \(y =-2\)