8.2 Parabolas 12/15/09

Conic Sections

Definition of a conic section: A conic section is the intersection of a plane and a double napped cone.

By changing the angle and the location of intersection, a parabola, circle, ellipse or hyperbola is produced.

Conic Sections

Parabola When a plane intersects a double-napped cone and is parallel to the side of the cone, a parabola is formed.

Parabolas Parabola: the set of points in a plane that are the same distance from a given point called the focus and a given line called the directrix. The cross section of a headlight is an example of a parabola... The light source is the Focus Directrix

Here are some other applications of the parabola...

Notice that the vertex is located at the midpoint between the focus Diagram of parabola d2 d1 Focus d2 d3 d1 d3 Vertex Directrix Notice that the vertex is located at the midpoint between the focus and the directrix... Also, notice that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix... We can determine the coordinates of the focus, and the equation of the directrix, given the equation of the parabola....

A parabolic equation could help find the length of a cable in a _________ / ____________. (Hint: I could sell you one from Brooklyn.) suspension bridge

A parabola is formed when a plane intersects a cone and is ___________ to the side of the cone. parallel

A sharpshooter on a SWAT team uses parabolas because they are used in ______________, the study of the movement of a body under the force of gravity. ballistics

Galileo made this discovery about the path of a projectile in the 17th century. Who did this help to figure out how to hurtle their weapons? cannoneers

Television addicts love parabolas because they are used in _________ / _________. satellite dishes reflecting antennas

One of the practical uses of parabolas is their reflection property One of the practical uses of parabolas is their reflection property. The part of a car that uses this is the ____________. headlights

The largest parabolic mirror in existence is located in __________ The largest parabolic mirror in existence is located in __________. It is used to collect light and radio waves from outer space. Russia

Graph: and x y -3 -9 -2 -4 -1 1 2 3 x y -3 9 -2 4 -1 1 2 3

Graph: and x y -1 error 0,0 1 1,-1 4 2, -2 9 3, -3 16 4, -4 25 5, -5 x 0,0 1 1,-1 4 2, -2 9 3, -3 16 4, -4 25 5, -5 x y 1 error 0, 0 -1 1, -1 -4 2, -2 -9 3, -3 -16 4, -4 -25 5, -5

Stop

Standard Equation of a Parabola: (Vertex at the origin) Equation Focus Directrix x2 = 4py (0, p) y = –p (If the x term is squared, the parabola is up or down) Equation Focus Directrix y2 = 4px (p, 0) x = –p (If the y term is squared, the parabola is left or right)

Quick Examples a. Identify the focus and directrix of the parabola Directrix: y = -6 Equation Focus Directrix x2 = 4py (0, p) y = –p

Under what condition is the axis of symmetry horizontal Under what condition is the axis of symmetry horizontal? when the y is squared

If the number in front of the x squared term is negative, the parabola will open in what direction? down

Quick Examples b. Equation Focus Directrix y2 = 4px (p, 0) x = –p Directrix: x = -5 Equation Focus Directrix y2 = 4px (p, 0) x = –p

How do you find the value for ‘p’ which will lead you to the focus and directrix? Solve for the squared term 2. Divide the coefficient of the linear term by 4 to yield ‘p’

Since x is squared, the parabola goes up or down… Ex1: Determine the focus and directrix of the parabola 4x2 = y, then graph : Since x is squared, the parabola goes up or down… Solve for x2 x2 = 4py y = 4x2 4 4 x2 = 1/4y 4py = 1/4y Solve for p 4p = 1/4 p = 1/16 Focus: (0, p) Directrix: y = –p Focus: (0, 1/16) Directrix: y = –1/16 : Coefficient of linear term General form: match up your equation to find ‘p’ Let’s see what this parabola looks like...

Ex2: Determine the focus and directrix of the parabola –3y2 – 12x = 0, then graph: Since y is squared, the parabola goes left or right… Solve for y2 y2 = 4px –3y2 = 12x –3y2 = 12x –3 –3 y2 = –4x Solve for p 4p = –4 p = –1 Focus: (p, 0) Directrix: x = –p Focus: (–1, 0) Directrix: x = 1 : Coefficient of linear term General form: match up your equation to find ‘p’ Let’s see what this parabola looks like...

Equation Focus Directrix x2 = 4py (0, p) y = –p Ex3: Write the standard form of the equation of the parabola with focus at (0, 3) and vertex at the origin. x, y Since the focus is on the y axis,(and vertex at the origin) the parabola goes up or down… x2 = 4py Since p = 3, the standard form of the equation is x2 = 12y Equation Focus Directrix x2 = 4py (0, p) y = –p

Since p = 1, the standard form of the equation is y2 = 4x Ex4: Write the standard form of the equation of the parabola with directrix x = –1 and vertex at the origin. Since the directrix is parallel to the y axis,(and vertex at the origin) the parabola goes left or right… y2 = 4px Since p = 1, the standard form of the equation is y2 = 4x Equation Focus Directrix y2 = 4px (p, 0) x = –p