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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.
J. Robert Buchanan Introduction to Conics: Parabolas
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.
J. Robert Buchanan Introduction to Conics: Parabolas
Illustration
Vertex
Focus
Directrix
Axis
d1
d1
d2
d2
J. Robert Buchanan Introduction to Conics: Parabolas
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
J. Robert Buchanan Introduction to Conics: Parabolas
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
J. Robert Buchanan Introduction to Conics: Parabolas
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 .
J. Robert Buchanan Introduction to Conics: Parabolas
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 .
J. Robert Buchanan Introduction to Conics: Parabolas
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
J. Robert Buchanan Introduction to Conics: Parabolas
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
J. Robert Buchanan Introduction to Conics: Parabolas
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
J. Robert Buchanan Introduction to Conics: Parabolas
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 =1
12x2 − 1
6x +
2512
J. Robert Buchanan Introduction to Conics: Parabolas
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 =1
12x2 − 1
6x +
2512
J. Robert Buchanan Introduction to Conics: Parabolas
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.
J. Robert Buchanan Introduction to Conics: Parabolas