Given the equation (x + 4)^2 = -12y + 24 determine:
a) The Vertex
b) If the parabola opens up, down, left, right
c) The Focus
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To determine the vertex, express the given equation of the parabola to its vertex form which is:
where (h,k) is the vertex.
To convert the equation in this form, subtract both sides by 24.
And divide both sides by -12.
So the vertex form of the given equation is:
Hence, the vertex of the parabola is `(-4,2)` .
(b) Direction of the parabola.
Note that when it is the variable x that is squared, the parabola either opens up or down.
And if it is the variable y that is squared, the parabola either opens to the left or right.
In the given equation, it is the variable x that is squared. So, it is either upward or downward. To determine which of these two is the direction of the parabola, consider the value of a.
If a is positive, the parabola opens up. And if it is negative, it open down.
Base on the vertex form, `a=-1/12` .
Since a is negative, therefore, the parabola opens down.
Since the parabola is downward, then its axis of symmetry is vertical. And a parabola with vertical axis, its focus has a coordinates (h, k+p).
To solve for p, use the formula:
Since a=-1/12, the value of p will be:
Then, substitute h=-4, k=2 and p=-3 to the coordinate form of the focus.
Hence, the focus of the parabola is `(-4 , -1)` .
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