1. An Introduction to Conics
Let’s try some math aerobics!
Stand up and here we go!
Establish the x-axis and y-axis.

2. What does x2 do to the graph?
It creates a
CURVE.
Hold up your curve!

3. CURVE. What would y2 do to a graph? It also creates a
Which way does your curve go?
Think about it: so
Hold up your curve!

4. 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 .

5. Let’s look at these parabola equations:
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?

6. PARABOLAS Show me another parabola with your hand.
What makes it open to the LEFT or RIGHT?
When the Y is SQUARED .

7. Let’s look at these parabola equations:
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?

8. What happens when you ADD x2 and y2 ?
The CURVES go TOGETHER!
Hold up your curves!

9. 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.

10. 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.

11. What do you notice about the circle and ellipse equations below?
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

12. What happens when you SUBTRACT x2 and y2 ?
The CURVES
go APART!
Hold up your curves!

13. 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.

14. NOW, you are ready to work with Conic Sections
NOW, you are ready to work with Conic Sections! Circles Ellipses Hyperbolas Parabolas

15. Which Conic Section is it? Hint: Use your hands to help you out…
Horizontal parabola
Opens Right
Hyperbola
Ellipse
Vertical parabola
Opens downward
Circle

16. Introduction to Conic Sections

17. In geometry, you learned about cones
In geometry, you learned about cones. In conic sections, we are learning about a “double-napped” 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.

18. Double-napped cone Regular Cone Nappe Lateral Surface Element
axis
Double-napped cone
Nappe
Element
Apex
element
apex
Regular Cone
Lateral Surface
Slant Height
Vertex

19. 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
A plane intersects both nappes parallel
to the axis
A plane intersects one nappe parallel
to an element
and is NOT perpendicular to the axis
CIRCLE
HYPERBOLA
PARABOLA
ELLIPSE

20. Shapes with Double Napped Cone
Circle
Ellipse
Parabola
Hyperbola

21. Other Creations with Double-Napped Cone
Line
Double Line
Point

22. 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.

23. 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

24. Conics Identification Flow Chart
yes
Is either x2 or y2 missing?
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

25. Identify the type of conic by looking at its equation in general form.
Circle
Ellipse
Parabola
Hyperbola 1. 2. 3. 4. 5.

26. Identify the type of conic by looking at its equation in general form.
Parabola
Circle
Ellipse
Hyperbola 6. 7. 8. 9.
10.

27. Identify the type of conic by looking at its equation in general form.
Circle
Hyperbola
Ellipse
Parabola
11.
12.
13.
14.
15.
16.