Lesson 9.1Parabolas

Write any parts of a parabola that you know:

Conic Sections

Many shapes and curves can be classified as a conic section

These shapes can be written algebraically as

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

or in relation to a locus (collection) of points

(x – h)2 + (y – k)2 = r2

Parabolas (Locus Definition)

Definition: Set of all points equidistant from a fixed point (focus) and line (directrix).

The midpoint between the focus and directrix is the vertex

focus

directrixvertex

Standard Equation of a Parabola (Locus)

Here, p is the distance from the focus to the vertex

khxay 2Traditional →

Visual → 2hxaky

p = ¼ →

p = larger →

p = smaller →

I like → 24

1hx

pky

Same width as y = x2Wider than y = x2Narrower than y = x2

Or, we can look at the 4p

4●p is the focal length:the width of the curve through the focus

Some things to know:

1) A parabola can go Up/Down or Left/Right

2) Why do “I like” the last equation?

You can see the effects of p or 4p on the “slope” – a

2hxaky 24

1hx

pky

Large p → large 4p → small a → low slope or flatter parabolaSmall p → small 4p → large a → high slope or steeper parabola

x-quantity squared → Up/Down

y-quantity squared → Left/Right

Example Find a standard form equation of a parabola with a vertex at the origin and focus at (0, 8)

Example Find the vertex, focus, and equation of the directrix of the parabola.

9822 yxxy

Click for moreexamples

At your tables draw and label the following:

1. A large parabola with vertex, V2. The focus, F and directrix3. The axis of symmetry4. A tangent line to the parabola at point, P5. A line passing through P and F6. An angle, , formed by the tangent and 7. An angle, , formed by the tangent and the axis

Compare your drawing with your neighbors to make a conjecture about the angles and .

Reflective Property of Parabolas

A tangent to a parabola at point P makes equal angles to:

1) A line passing through P and the focus and

2) The axis of the parabola

If these angles are congruent…

Then these sides are congruent

Example Find the equation of the tangent line to y = x2 at the point (2, 4).