Warm Up
•
•
•
•
What is the standard form of a parabola?
What is the standard form of a circle?
What is the standard form of a ellipse?
What is the standard form of a hyperbola?
Algebra 3
Chapter 10: Quadratic
Relations and Conic
Sections
Lesson 6: Graphing and
Classifying Conics
VOCAB
• Conics or Conic Sections – parabolas,
circles, ellipses, and
hyperbolas…basically all curves that
are formed by the intersections of a
plane and a double-napped cone
• Discriminant – an equation that can
tell what type of conic you have
Classifying – way 1
• Today we are going to learn one way to
classify a conic section. This way is to put it in
a normal formula.
Formulas
• Parabola
• 𝑥 2 = 4𝑝𝑦 or 𝑦 2 = 4𝑝𝑥
• Circle
• 𝑥2 + 𝑦2 = 𝑟2
• Ellipse
•
𝑥2
𝑎2
𝑦2
+ 2
𝑏
= 1 or
𝑥2
𝑏2
+
𝑦2
𝑎2
=1
• Hyperbola
•
𝑥2
𝑎2
𝑦2
− 2
𝑏
= 1 or
𝑦2
𝑎2
𝑥2
− 2
𝑏
=1
Directions
• Look at the powers of x and y
– If ONLY one of them is squared…parabola
• Get x and y on the same side
• Divide by the number
– If it is SUBTRACTION…Hyperbola
– If it is ADDITION
• Denominators are the same…Circle
• Denominators are different…Ellipse
I DO (Classifying)
•Classify the conic section
2
2
•1. 9𝑥 − 4𝑦 = 36
2
2
•2. 9𝑥 + 4𝑦 = 36
2
2
•3. 𝑥 + 𝑦 = 15
2
•4. 𝑥 + 4𝑦 = 0
WE DO (Classifying)
•Classify the conic section
2
•1. 𝑥 − 12𝑦 = 0
2
2
•2. 2𝑥 + 3𝑦 = 12
2
2
•3. 𝑥 = 12 + 3𝑦
2
2
2
•4. 𝑥 + 4𝑦 = 6 − 3𝑥
YOU DO (Classifying)
•Classify the conic section
2
•1. 𝑥 +
2
2
𝑦
3
=1
2
•2. 3𝑥 + 3𝑦 = 30
2
•3. 2x = 𝑦
2
2
2
•4. 𝑥 + 4𝑦 = 8 + 3𝑥
Review
• What did you learn today?
Homework
• NONE
Warm Up
• Name the 4 types of conic sections
• Explain how to classify a conic section
Algebra 3
Chapter 10: Quadratic
Relations and Conic
Sections
Lesson 6: Graphing and
Classifying Conics
Classifying – discriminant
• Today we are going to learn one way to
classify a conic section. This way is to find the
discriminant
Formulas
• General Equation
• 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
• Discriminant
• 𝐵2 − 4𝐴𝐶
KNOWLEDGE
• Discriminant
–Less than zero
• B = 0 and A = C …it’s a circle
• B ≠ 0 or A ≠ C … it’s an ellipse
–Equal zero
• It’s a parabola
–Greater than zero
• It’s a hyperbola
DIRECTIONS
• Find a, b, c
• Find the discriminant
• Classify the conic
I DO (Classifying)
•
•
•
•
•
Classify the conic section
2
2
1. 9𝑥 + 4𝑦 + 36𝑥 − 24𝑦 + 36 = 0
2. 4𝑥 2 + 2𝑥𝑦 − 9𝑦 2 + 18𝑦 + 3𝑥 = 0
2
2
3. 36𝑥 + 16𝑦 − 25𝑥 + 22𝑦 + 2 = 0
2
2
4. 9𝑦 − 3𝑥𝑦 − 𝑥 + 2𝑥 + 54𝑦 + 6 = 0
WE DO (Classifying)
•
•
•
•
•
Classify the conic section
2
1. 𝑥 + 𝑥𝑦 − 2𝑥 + 8𝑦 + 9 = 0
2. 12𝑥 2 + 20𝑦 2 − 12𝑥 + 40𝑦 − 37 = 0
2
2
3. 𝑥 + 4𝑥𝑦 + 𝑦 − 4𝑥 − 2𝑦 − 4 = 0
2
2
4. 16𝑦 − 𝑥 + 2𝑥 + 64𝑦 + 63 = 0
YOU DO (Classifying)
•
•
•
•
•
Classify the conic section
2
2
1. 𝑥 − 4𝑦 + 3𝑥 − 26𝑦 − 30 = 0
2. 𝑥 2 − 𝑥𝑦 + 𝑦 2 − 10𝑥 − 2𝑦 + 10 = 0
2
2
3. 4𝑥 + 2𝑥𝑦 + 4𝑦 − 6𝑥 + 4𝑦 − 6 = 0
2
2
4. 25𝑦 + 16𝑥 − 18𝑥 − 20𝑦 + 8 = 0
Review
• What did you learn today?
HOMEWORK
• Worksheet
– 10.6B (9 – 14)
Warm Up
• Classify the conic
• 1. 3𝑥 2 + 4𝑦 2 + 3𝑥 − 26𝑦 − 30 = 0
• 2. 2𝑥 2 − 5𝑥𝑦 + 𝑦 2 − 10𝑥 − 2𝑦 + 10 = 0
Algebra 3
Chapter 10: Quadratic
Relations and Conic
Sections
Lesson 6: Graphing and
Classifying Conics
TODAY
• Today we are going to learn how to write
equations of conics that are NOT in the center
of a graph
Formulas
• Parabola
• (𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘) or(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)
• Circle
• (𝑥 − ℎ)2 +(𝑦 − 𝑘)2 = 𝑟 2
• Ellipse
•
(𝑥−ℎ)2
𝑎2
+
(𝑦−𝑘)2
𝑏2
= 1 or
(𝑥−ℎ)2
𝑏2
+
(𝑦−𝑘)2
𝑎2
=1
• Hyperbola
•
(𝑥−ℎ)2
𝑎2
−
(𝑦−𝑘)2
𝑏2
= 1 or
(𝑦−𝑘)2
𝑎2
(𝑥−ℎ)2
− 2
𝑏
=1
CENTER
•
•
•
•
Center of all shapes is
(h , k)
A is the distance from the vertex to the center
C is the distance from the focus to the center
Directions
• Label what you know
• Find what your missing
– A, b, c, p, h, k
• Plug into the
I DO (Equations)
•
•
•
•
•
Write the equation of the conic section
1. Parabola … V (-2, 1)
F (-3, 1)
2. Circle … Center (3, -2)
r=4
3. Ellipse … F (3, 5) (3, -1)
V (3, 6) (3, -2)
4. Hyperbola … V (5, -4) (5, 4) F (5, -6) (5, 6)
WE DO (Equations)
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•
•
•
•
Write the equation of the conic section
1. Parabola … V (1, -2)
F (1, 1)
2. Circle … Center (9, 3)
r=4
3. Ellipse … V(2, -3) (2, 6)
F (2, 0) (2, 3)
4. Hyperbola … V (-4, 2) (1, 2) F (-7, 2) (4, 2)
YOU DO (Equations)
•
•
•
•
•
Write the equation of the conic section
1. Parabola … V (-3, 1)
directrix x = -8
2. Circle … Center (-4, 2)
r=3
3. Ellipse … F (-2, 2) (4, 2)
CV (1, 1 (1, 3)
4. Hyperbola … V (8, -4) (8, 4) F (8, -6) (8, 6)
Review
• Today you learned how to write the equation
of a translated conic
HOMEWORK
• Worksheet
– 10.6B (1 – 4)