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

Vertex: [tex](0,0)[/tex]
Focus: [tex](0,-\dfrac{1}{2})[/tex]

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.

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:

[tex]\boxed {\boxed {\sf x^2=-2y}}[/tex]

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:

[tex](x-h)^2=-4p(y-k)[/tex]

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.

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

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

[tex]x_1=0\\y_1=0\\x_2=0\\y_2= -1/2[/tex]

[tex]d=\sqrt{(0-0)^2+(0--1/2)^2}[/tex]

[tex]d=\sqrt{(0)^2+(0+1/2)^2}[/tex]

[tex]d=\sqrt{0+(1/2)^2}[/tex]

[tex]d=\sqrt{0+1/4}[/tex]

[tex]d=\sqrt{1/4}[/tex]

[tex]d=1/2[/tex]

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

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

[tex](x-0)^2=-4(1/2)(y-0)[/tex]

[tex](x)^2=-2(y)[/tex]

[tex]x^2=-2y[/tex]

The equation of the parabola is x²= -2y

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