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Question

A parabola has a focus located at     and a directrix of   .

What are the coordinates for the vertex?     

The equation for this parabola in vertex form is as follows:

         

Answer 1

In your first post of this type, I provided this general procedure (and I also solved the particular problem). If you have questions on how to apply this to each different problem, you can post a comment.

The general form of a parabola:

Opening left or right: y² = 4ax (Opening left: -4ax)

Opening up or down: x² = 4ay (Opening down: -4ay)

To remember which is which: If it opens up or down, then there are 2 x values for every y value--similar logic for right and left.

If the vertex is not at the origin, then we apply offsets: If the origin is at (j, k), then replace x with x - k, and replace y with y = k

a is the distance from the vertex to either the focus or directrix. (2a is the distance from focus to directrix)

So...the steps:

1. Put the equation into the form described above
2. Find a and use that to determine the relative location of focus, vertex, and directrix
3. PLOT the function to verify your results

Answer 2

vertex is half way between the directrix and focus.

The parabola opens downward. vertex has the same x value as the focus, 6

Its y value is the average of 11/2 and 9/2 or 10/2=5 vertex is (6,5)

Find the zeroes, Let y = 0 and solve the function for x. a=-1/2, it has a negative sign when the parabola opens downward.

0=a(x-6)²+5

x= approximately 2.84 and 9.16, the two x-intercepts, where the parabola intersects the x-axis

The general form of the parabola equation is y-k=a(x-h)² which could be written as y=a(x-h)²+k. Since

the vertex is (h,k) = (6,5), plug those in and you have y=a(x-6)²+5. As a=-1/2, that leaves

y=(-1/2)(x-6)²+5

Find a few more points to sketch the graph, such as the y-intercept. That's when x=0. plug that value in and you get y=(-1/2)(0-6)²+5. or y=-13, the point (0,-13)

Sketch a smooth curve through the y-intercept (0,-13) through the first x-intercept (2.84,0) then rounding

through the maximum point, the vertex, (6,5), then back down through the 2nd x-intercept, (9.16,0)

the parabola graph is symmetric about x=6, so the right side should be a mirror image of the left side of x=6.

One way to find the parabola equation is from the definition of a parabola: It's the locus or set of all points (x,y) such that the distance from (x,y) to the focus is the same as the distance from (x,y) to the directrix.

The distance to the directrix from any point (x,y) on the parabola is (11/2)-y It's just the vertical distance from the point to the line. Take the x-intercept point (2.84,0), it's distance to the directrix is 5.5-0=5.5. Square it and you have 5.5² = 30.25

That 30.25 will also equal the distance squared from the x-intercept to the focus (6,9/2)

the distance squared = (difference in x values)² + (difference in y values)²

30.25 = (2.84-6)²+(0-4.5)² = 10+20.25 = 30.25

the same holds in general for any point (x,y) on the parabola

(y-5.5)²= (x-6)²+(y-4.5)²

expand the above and simplify gives

y=(-1/2)x²+6x-13

or y=(-1/2)( x²-6)²+5 in the y=a(x-h)²+k general form using the vertex point (6,5)