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 = x 2 -4y=x^2 − 4 y = x 2

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 0 negative 1, and the directrix line y equals 1, 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 4p times y minus k is equal to x minus h squared. And what we want to 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 on 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 0 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 parabola's vertex is going to be somewhere in between these or somewhere in the center because the parabola's vertex is between the directrix and focus. So this would be the parabolas center which would be at 0. 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 1 unit to get to the directrix line, and 1 unit to get down to the focus point. So p is going to be 1, 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 0. So that means our equation is going to be 4 times negative one, times y is equal to x squared, and 4 times negative one is negative 4. So we'll have negative 4 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 = 1 y=1 y = 1 , find the standard equation for the parabola.

− 4 y = x 2 -4y=x^2 − 4 y = x 2

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

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

19. Conic Sections

Parabolas

Precalculus

19. Conic Sections

Parabolas

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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.

$4y=x^2$

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

$-4y=x^2$

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

Verified step by step guidance

Identify the given elements of the parabola: the focus at (0, -1) and the directrix y = 1.

Recall that the standard form of a parabola with a vertical axis of symmetry is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.

Determine the vertex of the parabola, which is the midpoint between the focus and the directrix. Calculate the midpoint between (0, -1) and y = 1.

Calculate the value of p, which is the distance from the vertex to the focus. Since the focus is below the directrix, p will be negative.

Substitute the values of h, k, and p into the standard form equation (x - h)^2 = 4p(y - k) to find the equation of the parabola.

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