Introduction to Conics: Parabolasbanach.millersville.edu/~bob/math160/Parabolas/main.pdf · Geometric Property of a Parabola Deﬁnition A parabola is the set of all points (x;y)

Introduction to Conics: ParabolasMATH 160, Precalculus

J. Robert Buchanan

Department of Mathematics

Fall 2011

J. Robert Buchanan Introduction to Conics: Parabolas

Objectives

In this lesson we will learn to:recognize a conic as the intersection of a plane and adouble-napped cone,write the equations of parabolas in standard form andgraph parabolas,use the reflective property of parabolas to solve real-worldproblems.


Double-Napped Cone


Circle


Ellipse


Parabola


Point


Hyperbola


Line


Two Intersecting Lines


Geometric Property of a Parabola

DefinitionA parabola is the set of all points (x , y) in a plane that areequidistant from a fixed line (called the directrix) and a fixedpoint (called the focus) not on the line.

Remarks:1 The midpoint between the focus and the directrix is called

the vertex.2 The line passing through the focus and the vertex is called

the axis of the parabola.


Illustration

Vertex

Focus

Directrix

Axis

d1

d1

d2

d2


Standard Form

Standard Equation of a ParabolaThe standard form of the equation of a parabola with vertexat (h, k) is as follows:

(x − h)2 = 4p(y − k), p 6= 0Vertical axis, directrix: y = k − p

(y − k)2 = 4p(x − h), p 6= 0Horizontal axis, directrix: x = h − p

The focus lies on the axis p units (directed distance) from thevertex. If the vertex is at the origin (0, 0), the equation takes onone of the following forms.

x2 = 4py Vertical axisy2 = 4px Horizontal axis


Illustration

(x − h)2 = 4p(y − k) : p > 0

Vertex : Hh ,k L

Focus: Hh ,k+p L

Directrix : y =k-p

Axis: x =h


Example

Find the standard form of the equation of a parabola whosevertex is at the origin and whose focus is at (−3/2, 0).

The axis of the parabola is horizontal, thus we seek anequation of the form:

(y − k)2 = 4p(x − h).

The vertex is at (0, 0) = (h, k) and the focus is at(−3/2, 0) = (h + p, k) which implies p = −3/2. Therefore

y2 = −6x .


Example

Find the standard form of the equation of a parabola whosevertex is at the origin and whose focus is at (−3/2, 0).

The axis of the parabola is horizontal, thus we seek anequation of the form:

(y − k)2 = 4p(x − h).

The vertex is at (0, 0) = (h, k) and the focus is at(−3/2, 0) = (h + p, k) which implies p = −3/2. Therefore

y2 = −6x .


Example

Find the vertex, focus, and directrix of the parabola:

y2 − 4y − 4x = 0

We must write the equation of the parabola in standard form.

y2 − 4y = 4xy2 − 4y + 4 = 4x + 4

(y − 2)2 = 4(x + 1)

Note that h = −1, k = 2, and p = 1.Vertex: (h, k) = (−1, 2)

Focus: (h + p, k) = (0, 2)

Directrix: x = h − p = −2


Example

Find the vertex, focus, and directrix of the parabola:

y2 − 4y − 4x = 0

We must write the equation of the parabola in standard form.

y2 − 4y = 4xy2 − 4y + 4 = 4x + 4

(y − 2)2 = 4(x + 1)

Note that h = −1, k = 2, and p = 1.Vertex: (h, k) = (−1, 2)

Focus: (h + p, k) = (0, 2)

Directrix: x = h − p = −2


Example

Find the standard form of the equation of the parabola withvertex at (1, 2) and directrix y = −1. Also write the quadraticequation for the parabola.

Let (h, k) = (1, 2) and y = k − p = −1 which implies p = 3.

(x − h)2 = 4p(y − k)

(x − 1)2 = 4(3)(y − 2)

(x − 1)2 = 12(y − 2)

x2 − 2x + 1 = 12y − 2412y = x2 − 2x + 25

y =112

x2 − 16

x +2512


Example



(x − h)2 = 4p(y − k)

(x − 1)2 = 4(3)(y − 2)

(x − 1)2 = 12(y − 2)

x2 − 2x + 1 = 12y − 2412y = x2 − 2x + 25

y =1

12x2 − 1

6x +

2512


Example



(x − h)2 = 4p(y − k)

(x − 1)2 = 4(3)(y − 2)

(x − 1)2 = 12(y − 2)

x2 − 2x + 1 = 12y − 2412y = x2 − 2x + 25

y =1

12x2 − 1

6x +

2512


Application: Reflective Property

Rays parallel to the axis of the parabola will be reflected off thesurface of the parabola toward the focus. Rays starting at thefocus in any direction will be reflected off the parabola and willexit parallel to the axis of the parabola.


Homework

Read Section 6.2.Exercises: 1, 5, 9, 13, . . . , 53, 57


Introduction to Conics: Parabolasbanach.millersville.edu/~bob/math160/Parabolas/main.pdf · Geometric Property of a Parabola Deﬁnition A parabola is the set of all points (x;y)

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