Given the equation (x + 4)^2 = -12y + 24 determine: a) The Vertex b) If the parabola opens up, down, left, right c) The Focus
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)` .