### How to Teach Conic Sections Without All of *Those* Formulas!

```An Introduction to Conics
Let’s try some math aerobics!
Stand up and here we go!
Establish the x-axis and y-axis.
2
x
What does
do
to the graph?
It creates a
CURVE.
Hold up your curve!
2
y
What would
do
to a graph?
It also creates a
CURVE.
Which way does your curve go?
2
Think about it: y   x so y
Hold up your curve!
x
PARABOLAS
Show me a parabola with your hand.
How many curves?
ONE
So, what do you know about the variables?
Only ONE is SQUARED
When it opens UP or DOWN, what is squared?
The x .
Let’s look at these parabola equations:
y  2x
2
y  4(x  1)  3
2
y  5  4x
2
Explain how we know these equations are
parabolas.
Do these parabolas open up or down?
How do you determine if these parabolas
open up or down?
PARABOLAS
Show me another parabola with your hand.
What makes it open to the LEFT or RIGHT?
When the Y is SQUARED .
Let’s look at these parabola equations:
x  3y
2
x  4(y  1)  3
2
x  4  5y
2
Explain how we know these equations are
parabolas.
Do these parabolas open left or right?
How do you determine if these parabolas
open left or right?
What happens when you
ADD x2 and y2 ?
The
CURVES
go
TOGETHER!
Hold up your curves!
CIRCLES
Show me a circle with your hands.
How many curves? TWO
So what do you know about x and y?
BOTH SQUARED
Which way do the curves go?
TOGETHER
So what do you know about the equation?
It has x2 and y2 ADDED.
ELLIPSES
Show me an ellipse with your hands.
OR
How many curves?
TWO
So what do you know about x and y?
BOTH SQUARED
Which way do the curves go?
TOGETHER
So what do you know about the equation?
It has x2 and y2 ADDED.
What do you notice about the circle and
ellipse equations below?
2x  2y  32
3x  4y  12
4x  5y  100
5x  5y  100
2
2
2
2
2
2
2
2
How can we tell circle and ellipse
equations apart?
CIRCLES: the coefficients of both x2 and y2
are the same
ELLIPSES: the coefficients of both x2 and y2
have a different number but the same sign
What happens when you
SUBTRACT x2 and y2 ?
The
CURVES
go APART!
Hold up your curves!
Hyperbolas
Show me an hyperbola with your hands.
How many curves?
TWO
So what do you know about x and y?
BOTH SQUARED
Which way do the curves go?
APART
So what do you know about the equation?
It has x2 and y2 SUBTRACTED.
NOW, you are ready to work
with Conic Sections!
Circles
Ellipses
Hyperbolas
Parabolas
Which Conic Section is it?
x  2y
2
Horizontal parabola
3x  2y  24
2
2
3x  2y  8
2
2
y  3(x  1)
2
2x  2y  32
2
2
Opens Right
Hyperbola
Ellipse
Vertical parabola
Opens downward
Circle
Introduction to
Conic Sections
In geometry, you learned about cones. In conic
sections, we are learning about a “doublenapped” cone. Unlike a regular cone, the
double-napped cone used to generate conic
sections has no bases; each cone is infinite. In
a drawing it appears that the cone ends, but
imagine that it extends infinitely in both
directions.
axis
element
apex
DoubleRegular Cone napped cone
Lateral
Surface
Nappe
Slant Height
Element
Vertex
Apex
Circles, ellipses, parabolas, and hyperbolas
are called conic sections because they are the
cross sections formed when a double-napped
cone is sliced by a plane.
Match the description of each
conic section with its name.
- A plane intersects exactly one nappe
and is perpendicular to the axis
CIRCLE
- A plane intersects both nappes parallel
to the axis
HYPERBOLA
- A plane intersects one nappe parallel
to an element
PARABOLA
- A plane intersects exactly one nappe
and is NOT perpendicular to the axis
ELLIPSE
Shapes with Double Napped Cone
Circle
Ellipse
Parabola
Hyperbola
Other Creations with
Double-Napped Cone
Point
Line
Double Line
The equation of any conic section can be
written in the General form
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
where A and C are not both 0.
What would happen if A and C were both 0?
Answer: the equation could represent a
point, a line, or a pair of intersecting lines.
Classification of a Conic Section:
You can determine the type of conic by
looking at the general form. Look at the A
and C terms.
If:
Then the conic
is a(n):
A = C, A  0, C  0
Circle
AC > 0 (A and C have the same sign
and AC)
Ellipse
AC = 0 (A=0 or B=0, but not both).
Parabola
AC < 0 (A and C have different signs)
Hyperbola
Conics Identification Flow Chart
Is either x2 or y2 missing?
yes
Parabola
no
Is either x2 or y2 negative?
yes
Hyperbola
no
Do x2 and y2 have the
same coefficient?
yes
Circle
no
Do x2 and y2 have
different coefficients?
yes
Ellipse
Identify the type of conic by looking at
its equation in general form.
1. x  y  9  0
Circle
2. 4x  y  8x  4 y  4  0
Ellipse
3. y  8x  6 y  1  0
Parabola
4. 2x  y 16  0
Ellipse
5. 9x  y 18x  6 y  9  0
Hyperbola
2
2
2
2
2
2
2
2
2
Identify the type of conic by looking at
its equation in general form.
6. x  6 x  8 y  7  0
Parabola
7. 2x  2 y  4x  0
Circle
2
2
2
Ellipse
9
x

25
y

36
x

150
y

36

0
8.
2
2
9. 4 x  y  4  0
Ellipse
10. 16 x  4 y  64  0
Hyperbola
2
2
2
2
Identify the type of conic by looking at
its equation in general form.
4
x

4
y

8
x

8
y

7

0
11.
Circle
x

y

4
x

2
y

2

0
12.
Hyperbola
13. x  9 y  6x 18 y  9  0
Ellipse
2
y
 8x  4 y  4  0
14.
Parabola
15. x  y  4  0
Hyperbola
y

12
x

6
y

3

0
16.
Parabola
2
2
2
2
2
2
2
2
2
```