Equations of Parabolas

A parabola is a set of points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus.

For any point Q that is on the parabola, d2 = d1

Directrix

FocusQ

d1

d2The distance from the vertex to the focus is p.

The distance from the vertex to the directrix is also p.

The distance from the focus to the directrix is 2p.

Vertexp

p

V

Things you should already know about a parabola.

Forms of equations

y = a(x – h)2 + k

opens up if a is positive

opens down if a is negative

vertex is (h, k)

y = ax2 + bx + c

opens up if a is positive

opens down if a is negative

vertex is , f( )-b 2a

-b 2a

Thus far in this course we have studied parabolas that are vertical - that is, they open up or down and the axis of symmetry is vertical

In this unit we will also study parabolas that are horizontal – that is, they open right or left and the axis of symmetry is horizontal

In these equations it is the y-variable that is squared.

V

x = a(y – k)2 + h

x = ay2 + by + c

or

Horizontal Parabola

Vertical Parabola

Vertex: (h, k)

If a > 0, opens right

If a < 0, opens left

The directrix is vertical the vertex is midway between the focus and directrix

X = a(y – k)2 + h Y = a(x – h)2 + k)

Vertex: (h, k)

If a > 0, opens up

If a < 0, opens down

The directrix is horizontal and the vertex is midway between the focus and directrix

Remember: |p| is the distance from the vertex to the focus

The vertex is midway between the focus and directrix, so the vertex is (-1, 4)

Equation: x = (1/12)(y – 4)2 – 1

|p| = 3

Find the vertex form of the equation of the parabola given:the focus is (2, 4) and the directrix is x = - 4

The directrix is vertical so the parabola must be horizontal and since the focus is always inside the parabola, it must open to the right

F

Equation: x = a(y – k)2 + h

V

The vertex is midway between the focus and directrix, so the directrix for this parabola is y = -1

Equation: y = (-1/8)(x – 2)2 – 3

|p| = 2

Find the vertex form of the equation of the parabola given:the vertex is (2, -3) and focus is (2, -5)

Because of the location of the vertex and focus this must be a vertical parabola that opens down

F

Equation: y = a(x – h)2 + k

V

(1/4)(y + 3)2 = x + 1 Find the vertex, focus and directrix. Then graph the parabola

Vertex: (-1, -3)

The parabola is horizontal and opens to the right

1/4p = 1/4p = 1

F

V

Focus: (0, -3)

Directrix: x = -2

x = ¼(y + 3)2 – 1

x y -1

0-50

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