If a parabola has the focus at ( 0 , − 1 ) \left(0,-1\right) ( 0 , − 1 ) and a directrix line y = 1 y=1 y = 1 , find the standard equation for the parabola.

Hey everyone, let's give this problem a try. So we're told if a parabola has the focus point zero negative one, and the directrix line y equals one, find the standard equation for the parabola. Okay, so to solve this problem, what you need to do is remember what the standard form for any parabola is, And that is we have four p times y minus k is equal to x minus h squared. And what we wanna do is piece this equation together so there's only a y and a k remaining. The or excuse me, only a y and an x remaining because those are the only variables that we really have at the end. Now what I can do is work backwards given the focus and the directrix. And if I think about what this means as far as a graph goes and by the way, we don't actually need to graph this. That's not in the instructions. I'm just, doing this so we can have this for reference. What we can do is recognize that the focus point is at zero negative one, which is right down there. And the directrix line is at y equals one, which would be somewhere up here. Now if the focus point and directrix line are both this distance apart, the parabolas vertex is going to be somewhere in between these or somewhere in the center because the parabolas vertex is between the directrix and focus. So this would be the parabolas center which would be at zero zero. So we're at the origin in this problem. Now, what we need to do from here is find out what the p value is. And we can find the p value by recognizing that it takes one unit to get to the directrix line and one unit to get down to the focus point. So p is going to be one, But since we see that the focus point is down, it's going to be negative one because the directrix line is gonna be opposite the direction it opens. So this is going to curve down, meaning we would have a negative p value. And because of this, this allows us to assemble our equation because we have the vertex and we have the p value. So I recognize that p is negative one, and the h and the k are both going to be zero. So that means our equation is going to be four times negative one times y is equal to x squared, and four times negative one is negative four. So we'll have negative four y equals x squared, and that is going to be the equation for our parabola. So hope you found this helpful. Thanks for watching.

If a parabola has the focus at (0,−1)\left(0,-1\right)(0,−1) and a directrix line y=1y=1y=1, find the standard equation for the parabola.

4(y−1)=x24\left(y-1\right)=$$x^24(y$$−1)=x2

−4(y+1)=x2-4\left(y+1\right)=$$x^2$$−4(y+1)=x2

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Multiple Choice

If a parabola has the focus at $$\left$(0,-1$\right$)$ and a directrix line $y=1$ , find the standard equation for the parabola.

$4y=x^2$

$4$\left$(y-1$\right$)=x^2$

$-4y=x^2$

$-4$\left$(y+1$\right$)=x^2$

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Verified step by step guidance

Step 1: Recall the definition of a parabola. A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Step 2: Identify the given focus and directrix. The focus is at (0, -1), and the directrix is the horizontal line y = 1.

Step 3: Use the formula for a parabola with a vertical axis of symmetry. The distance from any point (x, y) on the parabola to the focus must equal its distance to the directrix. This gives the equation: |y - (-1)| = |y - 1|.

Step 4: Simplify the equation to find the relationship between x and y. The vertex of the parabola is midway between the focus and the directrix, at (0, 0). The equation becomes: (y + 1) = (y - 1).

Step 5: Rearrange the equation into standard form. For a parabola opening downward, the equation takes the form -4(y + 1) = x^2, where the negative sign indicates the direction of opening.