Question

Use the given information to write the equation of the parabola.

Answer 1

Answer:

x² = -2y

Step-by-step explanation:

The focus is p away from the vertex, and so is the directrix.

To find the equation of the parabola, we must first determine if the parabola is horizontal or vertical.

• Horizontal parabola [Standard form]: (y – k)² = 4p(x – h)
• Vertical parabola [Standard form]: (x – h)² = 4p(y – k)

If the parabola is vertical, the directrix, and focus will have the same x value but different y value compared to the vertex (h, k). You can also tell if the directrix in in the form y = k – p, and if the focus is in the form (h, k + p).

Likewise, if the parabola is horizontal, the directrix, and focus will have the same y value but different x value compared to the vertex (h,k) . You can also tell if the directrix is in the form x = h – p, and if the focus is in the form (h + p, k).

For this problem, given that the vertex is at the origin (0,0), and that the focus is at the point (0, -½).

You can tell that the x value is the same for the vertex, and focus so this must be a vertical parabola. Because this is a vertical parabola, we can use the form mentioned as earlier (x – h)² = 4p(y – k).

If h = 0, and k = 0, the p value must be the difference between the k of the vertex, and the k of the focus: -½ - 0 → -½.

Now we can just plug in our known information to derive the equation!

h = 0, k = 0, p = -½ → (x - h)² = 4p(y - k) →

(x - 0)² = 4(-½)(y - 0) → x² = -2(y - 0) →

x² = -2y.

Also p = 1/4a, if you are wondering.

So because this is a vertical parabola, x² = -2y is generally the same as y = -1/2x² in standard quadratic form. I just like to think of the horizontal parabola as an inverse quadratic because it is like reflecting over the line y = x.

Answer 2

Answer:

$\boxed {\boxed {\sf x^2=-2y}}$

Step-by-step explanation:

First, plot the two points. We can see the focus is below the vertex. The result will be a vertical parabola. Since the parabola must contain the focus, it will have to open downward.

From this information, we know that the following equation must be used:

$(x-h)^2=-4p(y-k)$

where (h,k) is the vertex and p is the distance from the focus to the vertex.

Next, find p, the distance between the focus and vertex. We can use the distance formula.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The vertex is (0,0) and the focus is (0, -1/2). Therefore:

$x_1=0\\y_1=0\\x_2=0\\y_2= -1/2$

$d=\sqrt{(0-0)^2+(0--1/2)^2}$

$d=\sqrt{(0)^2+(0+1/2)^2}$

$d=\sqrt{0+(1/2)^2}$

$d=\sqrt{0+1/4}$

$d=\sqrt{1/4}$

$d=1/2$

Now we know the distance and vertex. We can plug the values into the equation.

d=1/2 and (h,k)= (0,0)

$(x-0)^2=-4(1/2)(y-0)$

$(x)^2=-2(y)$

$x^2=-2y$

The equation of the parabola is x²= -2y