The McDonald's Logo (Parabolas)

Pretty much two parabolas side-by-side. It would be a -x^2, that will be explained at a later time.

So many people did ellipses. O.o Ehhhh.....

Parabolas have only one quadratic, be it x^2 or y^2. (y=1/4x^2, x=1/4x^2).

If the equation is in standard form: y+h=a(x+k)^2 or x+h=a(y+k)2, you need to foil it out and solve for y or x.

Steps:

1) The first thing you can look at is the coefficient of the quadratic term. If it is a positive x^2, the parabola will open up. If it is a negative x^2, it will open down, like the McDonald's Logo! If it is a positive y^2, it will open to the right. If it is a negative y^2, it will open to the left.

2) The axis of symmetry is the line intersecting the vertex where if you “fold” the parabola in half, it would match up perfectly. The equation should be in the form y=ax^2+bx+c or x=ay^2+by+c. To find the axis of symmetry, you would find x=-b/2a or y=-b/2a. It is a line.

3) To find the vertex, you would take what x equals in the axis of symmetry for your x-coordinate, and plug that number into the function to find your y-coordinate. Vertex: (-b/2a, f(-b/2a)) The opposite would be true for y^2. Vertex: (f(-b/2a), -b/2a)

To find the focus and dirextrix, you need to find p using the formula 1/4p = your leading coefficient.

4) Your focus is a point.

For x^2, the x-coordinate stays the same as your vertex. Your y-coordinate is the vertex's y-coordinate + p.

If it is y^2, the y coordinate would stay the same and you would add p to the x-coordinate.

5) If it is x^2, y = the number you get from subtracting p from your vertex's y-coordinate.

If it is y^2, it would be x = the number you get from subtracting p from your vertex's x-coordinate.

Ex. 1

Sketch the graph of y=1/8x^2.

1. Opens up because it is positive.

2. Axis of symmetry: x=-b/2a
x=-0/2(1/8)
x=0
b is zero because there is no linear value.

3. Vertex: (0,(1/8*0^2) x-coordinate is 0, plug into y=1/8x^2 to get your y.
(0,0)

1/4(p) = 1/8, so p=2.

4. x-coordinate stays the same, add p=2 to y-coordinate.
Focus: (0,2)

5. Directrix: y=0-2
y=-2

To finish graphing, graph on the vertex a parabola opening up. Graph the axis of symmetry as a dashed line at x=0. Plot the focus at (0,2). Draw a dotted line for the directrix at y=-2. You directrix and focus should be equidistant from your vertex.