The document discusses parabolas and their geometric and algebraic properties. It includes: 1. A summary of the geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). 2. Examples of finding the equation of a parabola given its geometric features, and finding the focus and directrix from a parabola's equation. 3. A discussion of parabolas with vertical and horizontal axes of symmetry and their standard equations. 4. An application of parabolic reflectors for lamps and telescopes.

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Conic Sections
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INFORMATION - THURSDAY 30 JULY 2015
1.Homework Task 3 on par. 10.2 due
by Tuesday 4 August.
2.Find all information on uLink.
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ICTMA-17 Conference
ICTMA - The International Community of Teachers of
Mathematical Modelling and Applications
University of Nottingham, Nottingham, England
University Park Campus
24 Countries represented
138 Presentations
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Chatsworth House & Robin Hood
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ICTMA-17
10.2 Parabolas
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Objectives
►Geometric Definition of a
Parabola
►Equations and Graphs of
Parabolas
►Applications
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Geometric Definition of a Parabola
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Geometric Definition of a Parabola
The graph of the equation
y = ax2 + bx + c
is a U-shaped curve called a PARABOLA that opens
either upward or downward, depending on whether
the sign of a is positive or negative.
In this section we study parabolas from a geometric
rather than an algebraic point of view.
We begin with the geometric definition of a parabola
and show how this leads to the algebraic formula that
we are already familiar with. 10
Geometric Definition of a Parabola
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Geometric Definition of a Parabola
This definition is illustrated.
The vertex V of the
parabola lies halfway
between
the focus and the directrix,
and
the axis of symmetry is the
line that runs through the
focus
perpendicular to the
directrix.
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Geometric Definition of a Parabola
In this section we restrict our attention to
parabolas that are situated with the
vertex at the origin and that have a
vertical or horizontal axis of symmetry.
If the focus of such a parabola is the
point F(0, p) then the axis of symmetry
must be vertical, and the directrix has the
equation y = –p. The figure illustrates the
case p > 0.

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Geometric Definition of a Parabola
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Geometric Definition of a Parabola
If P(x, y) is any point on the parabola, then the
distance from P to the focus F (using the Distance
Formula) is
The distance from P to the directrix is
By the definition of a parabola these two distances
must be equal:
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Geometric Definition of a Parabola
x2 + (y – p)2 = |y + p|2 = (y + p)2
x2 + y2 – 2py + p2 = y2 + 2py + p2
x2 – 2py = 2py
x2 = 4py
If p > 0, then the parabola opens upward;
but if p < 0, it opens downward. When x is
replaced by –x, the equation remains
unchanged, so the graph is symmetric about
the y-axis.
Square both sides
Expand
Simplify
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Equations and Graphs
of Parabolas
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Equations and Graphs of Parabolas
The following box summarizes about the equation and
features of a parabola with a vertical axis.
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Example 1 – Finding the Equation of a Parabola
Find an equation for the parabola with vertex V(0, 0) and
focus F(0, 2), and sketch its graph.
Solution:
Since the focus is F(0, 2), we conclude that p = 2
(so the directrix is y = –2). Thus the equation of the
parabola is
x2 = 4(2)y
x2 = 8y
x2 = 4py with p = 2
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Example 1 – Solution
Since p = 2 > 0, the parabola opens upwards.
cont’d
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Example 2 – Finding the Focus and Directrix of a Parabola from Its Equation
Find the focus and directrix of the parabola y = –x2, and
sketch the graph.
Solution:
To find the focus and directrix, we put the given equation in
the standard form x2 = –y.
Comparing this to the general equation x2 = 4py, we see
that 4p = –1, so p = – .
Thus the focus is F(0, – ), and the directrix is y = .
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Example 2 – Solution
The graph of the parabola, together with the focus and the
directrix, is shown.
cont’d
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Equations and Graphs of Parabolas
Reflecting the graph in the figure about the diagonal
line y = x has the effect of interchanging the roles of x
and y. This results in a parabola with horizontal axis.
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Equations and Graphs of Parabolas
By the same method as before, we can prove the
following properties.
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Example 3 – A Parabola with Horizontal Axis
A parabola has the equation 6x + y2 = 0.
(a) Find the focus and directrix of the parabola and sketch
the graph.
(b) Use a graphing calculator to draw the graph.
Solution:
To find the focus and directrix, we put the given equation in
the standard form y2 = –6x.
Comparing this to the general equation y2 = 4px we see
that 4p = –6, so p = – .
Thus the focus is F(– , 0), and the directrix is x = .

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Example 3 – Solution
Since p < 0, the parabola opens to the left. The graph
of the parabola, together with the focus and the
directrix, is shown.
cont’d
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Example 3 – Solution
(b) To draw the graph using a graphing calculator, we need
to solve for y.
6x + y2 = 0
y2 = –6x
y = 
cont’d
Subtract 6x
Take square roots
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Example 3 – Solution
To obtain the graph of the parabola, we graph both
functions y = and y = – as shown.
cont’d
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Applications
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Applications: Parabolic Reflector
Parabolas have an important property that makes
them useful as reflectors for lamps and telescopes.
Light from a source placed at the focus of a surface
with parabolic cross section will be reflected in such a
way that it travels parallel to the axis of symmetry of
the parabola.
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