In Exercises 5–16, find the focus and directrix of the parabola w...

Question

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.

y^2 = - 8x

Accepted Answer

Hey, everyone here we are asked to solve for the focus and the dia tris as well as graph. The given equation of a parabola Y squared is equal to negative 24 X. Here we have four answer choice options. Each of which display a graph of our parabola and the di tris and each solution states the focus and the dia So to begin solving this problem first, we need to recall the given equation for our parabola here which is again, Y squared is equal to negative 24 X. And we see that this given equation is in the form of the equation, the quantity of Y minus K squared is equal to four A times the quantity of X minus H. So now we just want to pair our formula with our equation to determine our value for A. So here we see the constant or the coefficient of X in our given equation is negative 24. So we can just equate negative 24 to 4 A from our formula. And now dividing both sides by four, we see that A is going to be negative six. And so now we want to bind our vertex. And if we recall the vertex is at the point H K. And so from our given equation, we see H K is just going to be the origin here zero and zero. So our vertex in this case is at the origin, now we can identify the focus as well. And if we recall the focus is going to be at the point H plus A for the X coordinate and K for the Y coordinate. And now using our given equation, we see that this point is going to be at the 0.0.0 plus negative six for the X coordinate and at zero for the Y coordinate. And now just simplifying, we see the focus is at negative and zero. So now we need to note that the directs and the focus lie at equal distances from the vertex of the parabola. And the equation of the dia tris can be calculated based on the equation of the parabola. So since the vertex is at the origin zero and zero, and the focus is at a, we know the equation of the dia is now going to be at X is equal to negative A. So at X is equal to negative A, we have X is equal to negative quantity of negative six. So we see that our dix is going to be at X is equal to positive six. So now with our dix identified as well as our vertex and our focus, we can now grab our parabola. So graphing, we see we have a vertical line for a vertical dash line for our di tris at X is equal to six. And then we have our parabola which is opening towards the left side of our graph here. And we see the vertex is still at the origin at 00. And we see the focus is at negative six and zero. So with this graph and the components properly identified, we can see here that our answer is going to be answer choice B where again we have our graph of our para parabola. And we see that the focus is at negative six and zero and the dire is also graphed at X is equal to six. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. y^2 = - 8x

Key Concepts

Parabola Definition

Focus and Directrix

Graphing Parabolas

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