Graph the parabola − 4 ( y + 1 ) = ( x + 1 ) 2 -4\left(y+1\right)=\left(x+1\right)^2 − 4 ( y + 1 ) = ( x + 1 ) 2 , and find the focus point and directrix line.

Welcome back everyone. So let's give this problem a try. We are asked here to graph the parabola, negative 4 times y plus 1 equals x plus 1 squared, and to find the focus point and directrix line. Okay. So to solve this problem, I'm going to write the standard equation for any parabola, which is 4p times y minus k is equal to x minus h squared. Now what I'm first going to do is find the vertex of our parabola, which is at the position hk, And find our vertex, we can look at the h and k that correspond to the equation. I can see that h corresponds with 1, but notice that there's a different sign. There's a plus sign here and a minus sign there. So that means that the h that we're looking for is actually negative one, since there's a different sign there. Now likewise, we can look at this k, and see that it corresponds with this one. But again, there's a plus sign instead of a minus sign, so that means k is also going to be negative one. So our vertex is going to be at the position negative one negative one, which on our graph is going to be right about there. So this is the vertex. Now the next thing I'm going to do is calculate the p value, I can do that by setting everything in front of the y minus k equal to 4p, which in this case is negative 4. So we have that negative 4 is equal to 4p, and to solve for p, I can just divide 4 on both sides of this equation. This will get the force to cancel giving me that p is equal to negative 4 over 4, which is negative one. Now notice that the p value comes out negative. When the p value comes out negative that means that we have a downward opening parabola, rather than an upward opening parabola. And because it opens downward that means the focus point is going to be downward as well, because recall that the focus is gonna be whatever direction the parabola opens. Now what I can do is take the absolute value of p, which gets rid of the negative sign, giving us that p or the absolute value of p is equal to 1. And so if I go over here to our vertex, we need to go down 1 unit to get to the focus point. And to find the directrix, we need to go up 1 unit, because the directrix is opposite the direction the parabola opens. So since the parabola opens down, the directrix is going to be p units up which is right here at this value. So it's right along the x axis which means our directrix is at y equals 0, and that our focus is at the point negative one negative two. So this is how you can find the focus and the directrix. Now our last step is going to be to graph the parabola. And in order to draw the graph, well, what we can do is recognize that our parabola, starting from the focus point, is going to be 2 p units to the left and to the right. That's gonna allow us to find 2 points on the parabola. So it's 2 times the absolute value of P, which is the same thing as 2 times 1, which is equal to 2. So if we start here at this focus point we can go 1 to 2 units to the left and 1 to 2 units to the right and our parabola is going to look something like that. So this is our graph along with the focus point and the directrix line. So that's how you can solve this problem. Hope you found this video helpful. Thanks for watching.

8. Conic Sections

Parabolas

College Algebra

8. Conic Sections

Parabolas

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Multiple Choice

Graph the parabola $-4\left(y+1\right)=\left(x+1\right)^2$ , and find the focus point and directrix line.

Graph of a parabola with focus at (-1,0) and directrix line y = -2.

Graph of a parabola with focus at (0,1) and directrix line y = -1.

Graph of a parabola with focus at (0,-1) and directrix line y = 1.

Graph of a parabola with focus at (-1,-2) and directrix line y = 0.

Verified step by step guidance

Start by rewriting the given equation in the standard form of a parabola. The given equation is -4(y+1) = (x+1)^2. Divide both sides by -4 to isolate (y+1): y + 1 = -1/4(x+1)^2.

Identify the vertex of the parabola. The equation y + 1 = -1/4(x+1)^2 is in the form y = a(x-h)^2 + k, where (h, k) is the vertex. Here, h = -1 and k = -1, so the vertex is (-1, -1).

Determine the direction in which the parabola opens. Since the coefficient of (x+1)^2 is negative (-1/4), the parabola opens downwards.

Find the focus of the parabola. The distance from the vertex to the focus is 1/(4|a|), where a = -1/4. Calculate this distance and add it to the y-coordinate of the vertex to find the focus.

Determine the equation of the directrix. The directrix is a horizontal line located the same distance from the vertex as the focus, but in the opposite direction. Subtract the distance calculated in the previous step from the y-coordinate of the vertex to find the equation of the directrix.