Identifying Conic Sections

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Identifying Conic Sections
How do I determine whether the graph of an
equation represents a conic, and if so, which
conic does it represent, a circle, an ellipse, a
parabola or a hyperbola?
Created by K. Chiodo, HCPS
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General Form of a Conic Equation
where A, B, C, D, E and F are integers and A, B
and C are NOT ALL equal to zero.
Note You may see some conic equations solved
for y, but if the equation can be re-written into
the form above, it is a conic equation!
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Please Note
A conic equation written in General Form doesnt
have to have all SIX terms! Several of the
coefficients A, B, C, D, E and F can equal zero,
as long as A, B and C dont ALL equal zero.
If A, B and C all equal zero, what kind of
equation do you have?
T
H
I
N
K...
...
Linear!
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So, its a conic equation if...

• the highest degree (power) of x and/or y is 2
  (at least ONE has to be squared)

• the other terms are either linear, constant, or
  the product of x and y

• there are no variable terms with rational
  exponents (i.e. no radical expressions) or
  terms with negative exponents (i.e. no rational
  expressions)

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What values form an Ellipse?
The values of the coefficients in the conic
equation determine the TYPE of conic.
What values form a Hyperbola?
What values form a Parabola?
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Ellipses...
where A C have the SAME SIGN
NOTE There is no Bxy term, and D, E F may
equal zero!
For example
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Ellipses...
The General, or Implicit, Form of the equations
can be converted to Graphing Form by completing
the square and dividing so that the constant 1.
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Ellipses...
In this example, x2 and y2 are both negative
(still the same sign), you can see in the final
step that when we divide by negative 4 all of the
terms are positive.
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Ellipsesa special case!
When A C are the same value as well as the same
sign, the ellipse is the same length in all
directions
Circle!
it is a ...
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Hyperbola...
where A C have DIFFERENT signs.
NOTE There is no Bxy term, and D, E F may
equal zero!
For example
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Hyperbola...
The General, or Implicit, Form of the equations
can be converted to Graphing Form by completing
the square and dividing so that the constant 1.
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Hyperbola...
In this example, the signs change, but since the
quadratic terms still have different signs, it is
still a hyperbola!
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Parabola...
A Parabola can be oriented 2 different ways
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Parabola Vertical
The following equations all represent vertical
parabolas in general form they all have a
squared x term and a linear y term
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Parabola Vertical
To write the equations in Graphing Form, complete
the square for the x-terms. There are 2 popular
conventions for writing parabolas in Graphing
Form, both are given below
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Parabola Vertical
In this example, the signs must be changed at the
end so that the y-term is positive, notice that
the negative coefficient of the x squared term
makes the parabola open downward.
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Parabola Horizontal
The following equations all represent horizontal
parabolas in general form, they all have a
squared y term and a linear x term
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Parabola Horizontal
To write the equations in Graphing Form, complete
the square for the y-terms. There are 2 popular
conventions for writing parabolas in Graphing
Form, both are given below
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Parabola Horizontal
In this example, the signs must be changed at the
end so that the x-term is positive notice that
the negative coefficient of the y squared term
makes the parabola open to the left.
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What About the term Bxy?
None of the conic equations we have looked at so
far included the term Bxy. This term leads to a
hyperbolic graph
or, solved for y
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Summary ...
General Form of a Conic Equation
Identifying a Conic Equation
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Practice ...
Identify each of the following equations as
a(n) (a) ellipse (b) circle (c)
hyperbola (d) parabola (e) not a conic See if
you can rewrite each equation into its Graphing
Form!
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Answers ...
(a) ellipse (b) circle (c) hyperbola (d)
parabola (e) not a conic
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Conic Sections !
C
E
Created by K. Chiodo, HCPS