Answer:
Option C.
Step-by-step explanation:
Use the biquadratic formula.
Positive sign:
Negative sign:
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You can rewrite the equation in vertex form to find some of the parameters of interest.
x² +8x +4 = -4y
x² +8x +16 -12 = -4y
(x+4)² -12 = -4y
y = (-1/4)(x +4)² +3
From this, we see the vertex is (-4, 3).
The scale factor, (-1/4) is 1/(4p), where p is the distance from the vertex to the focus. Solving for p, we have
-1/4 = 1/(4p)
p = -1 . . . . . multiply by -4p
So, the focus is 1 unit below the vertex. The focus is (-4, 2).
The directrix is the same distance from the vertex, but in the opposite direction, hence the directrix is y = 4.
_____
On a graph, if all you know is the vertex, you can draw a line with slope 1/2 through the vertex. It intersects the parabola at the y-value of the focus. Then the distance from that point of intersection to the axis of symmetry (where the focus lies) is the same as the distance from that point to the directrix.
Every point on the parabola, including the vertex and the point of intersection just described, is the same distance from the focus and the directrix. This point of intersection is on a horizontal line through the focus, so finding the distance is made easy.
x² +8x +4 = -4y
x² +8x +16 -12 = -4y
(x+4)² -12 = -4y
y = (-1/4)(x +4)² +3
From this, we see the vertex is (-4, 3).
The scale factor, (-1/4) is 1/(4p), where p is the distance from the vertex to the focus. Solving for p, we have
-1/4 = 1/(4p)
p = -1 . . . . . multiply by -4p
So, the focus is 1 unit below the vertex. The focus is (-4, 2).
The directrix is the same distance from the vertex, but in the opposite direction, hence the directrix is y = 4.
_____
On a graph, if all you know is the vertex, you can draw a line with slope 1/2 through the vertex. It intersects the parabola at the y-value of the focus. Then the distance from that point of intersection to the axis of symmetry (where the focus lies) is the same as the distance from that point to the directrix.
Every point on the parabola, including the vertex and the point of intersection just described, is the same distance from the focus and the directrix. This point of intersection is on a horizontal line through the focus, so finding the distance is made easy.
Given the piecewise function:
So, for the given function, we will graph it as follows:
1) graph the line (-0.5x-4) as x less than or equal to -2
To graph the line, we need two points
We will substitute with two values less than -2 and calculate the corresponding values of f(x)
When x = -4, f(x) = -0.5 * -4 - 4 = -2
When x = -6, f(x) = -0.5 * -6 - 4 = -1
So, the line will pass through the points ( -4, -2) and ( -6, -1)
The line will start from the point ( -2, -3)
The dot at x = -2 will be closed.
2) Graph the line (y = 5) as x between -2 and 4
The line will be horizontal at this interval
the dot at x = -2 will be open
The dot at x = 4 will be closed
3) Graph the line ( -x + 5 ) as x greater than 4
As in (1), substitute with 2 values of x greater than 4
When x = 5, f(x) = -5 + 5 = 0
When x = 6, f(x) = -6 + 5 = -1
So, the line will pass through the points ( 5, 0 ) and ( 6, -1 )
The graph of the function will be as shown in the following picture:
