Unit 9: Conic Sections
Name _______________________________________________________ Per ________
1/6
HOLIDAY
(*)Pre AP Only
1/13
Parabolas
HW: Part 4
1/7
General Vocab
Intro to Conics
Circles
1/8-9
More Circles
Ellipses
1/10
Hyperbolas
HW: Part 1
1/14
Identifying conics in
standard form
Parameter changes
*Real World Applications
HW: Part 2
1/15-16
Review
HW: Part 3
1/17
Test Part 1
Test Part 2
HW: Part 5
Objectives:
Describe the conic section formed by the intersection of a double right cone and a plane.
Determine the vertex form of a quadratic given the standard form
Recognize how parameter changes affect the sketch of a conic section.
Identify symmetries of conic sections
Identify the conic section from an equation
Graph and write the equation of the conic section
Use the method of completing the square to change the form of the equation
Determine the effects of changing a, h, or k on graphs of quadratics (horizontal and vertical)
*Graph and write equations of conic sections in real world applications
Essential Questions:
Compare and contrast the conic equations.
How do you know if the conic is vertically or horizontally orientated?
General Vocabulary:
• Know the vocabulary listed below in context to the material learned throughout this unit
Double ended right cone
Base of a cone
Vertical
Plane
Side of a cone
Horizontal
Conic (conic section)
Parallel
Orientation
Intersection
Perpendicular
Symmetry
Part 1 - Intro to Conics
Describe the conic section and its symmetries formed by the intersection of a double right cone and a plane.
Name
Conic
Equation
Symmetry:
(AOS)
Plane
Intersection:
Real World
Examples:
Part 1 - Circles:
( − ℎ) + ( − ) = Graph a circle given the equation:
1. Determine the center from the conic form of
the equation (watch for signs +/- h is
opposite) (h, k)
2. Find the radius r
3. Graph center and 4 points
Examples:
1. Graph: (x+5)2 + (y-4)2 = 25
4.
x2 + y2 -2x - 4y – 4 = 0
Write the equation of a circle given the graph/info:
1. Find the center (watch signs +/-) (h, k)
2. Find the radiusr
3. Plug (h, k) & r into conic form of circle equation
and simplify
Write the equation of a circle in Conic form (C.T.S.):
1. Look for squared variable and group in sets
2. Move everything else to the other side of the =
3. C.T.S. for any sets of variables of the left
2. Write the equation: Center at
(0,4) radius of 3
5.
3. Write the equation:
x2 + y2 – 6x + 4y + 9 = 0
Part 2 - Ellipses:
()
+
()
=1
• Minor Axis – shorter diameter of the ellipse
• Major Axis – longer diameter of the ellipse
• Horizontal Orientation – horizontal major axis/ is larger/horz. symm.
• Vertical Orientation – vertical major axis/ is larger/vert. symm.
• Vertices – the points where the major axis touches the ellipse
• Co-vertices – the points where the minor axis touches the ellipse
• Focus (foci) – 2 fixed points on an ellipse ( ±c , 0) using = − Graph an ellipse given the equation:
Write the equation of an ellipse given the graph/info:
1. Determine the center from the conic form of the
1. Find the center (h, k) – plug it into the equation
equation (watch signs +/- h is opposite)(h, k)
2. Find the a value (always 1st/under x value) – count
2. Find the square root of and spaces from center left /right – in equation
3. Count ‘a’ spaces from center left/right - make point
3. Find the b value (always 2nd/under y value) – count
4. Count ‘b’ spaces from center up/down – make point
spaces from center up/down – in equation
5. Connect those 4 points in an elliptical shape
4. Always ‘+’ in between terms and always = 1
Examples:
2. Find all parts to the ellipse and graph:
1. Find all parts to the ellipse and graph:
( x − 5)2 ( y + 2)2
+
=1
16
9
( x + 3) 2 y 2
+
=1
4
25
Center: __________________________
Center: __________________________
a = _______, b = _______, c = _______
a = _______, b = _______, c = _______
Foci: ____________________________
Foci: ____________________________
Vertices: _________________________
Vertices: _________________________
Co-Vertices:_______________________
Co-Vertices:_______________________
3. Write the equation from graph:
*Pre AP Write the equation of an ellipse in Conic form:
1. Group sets of variables on left of =
2. C.T.S. for both sets variables
3. Pull out GCF /apply to blanks on right side
*Example: 9x2 + 4y2 - 54x - 8y – 59 = 0
Part 3 - Hyperbolas:
()
−
()
=1
•
•
•
•
•
Horizontal Orientation – hyperbola opens left and right/ is larger/horizontal symmetry
Vertical Orientation – hyperbola opens up and down/ is larger/ vertical symmetry
Vertices – the points hyperbola is drawn through
Focus (foci) – 2 fixed points inside the hyperbola curves ( ±c , 0) using = − Asymptotes – lines where the graph does not exist – the graph cannot cross these lines at
any time (used to help guide the drawing of the graph)
•
Equations of Asymptotes – Horizontal: = Vertical: = Graph a hyperbola given the equation:
1. Determine the center from the conic form of the equation (watch signs +/- h is
opposite)(h, k)
2. Find the square root of and 3. Count ‘a’ spaces from center left/right - make point
4. Count ‘b’ spaces from center up/down – make point
5. Make a dotted box using the a and b spacing and connect corners to draw in asymptotes
6. Draw in ‘parabola’ graphs through vertex and approaching asymptotes
Examples:
1. Find all parts to the hyperbola and graph:
2. Find all parts to the hyperbola and graph:
( x − 3) 2 ( y + 2)2
−
=1
25
9
y 2 ( x − 1) 2
−
=1
16
4
Center: _______
Center: ______
a = _______
a = _______
b = _______
b = _______
*Pre AP ONLY* Write the equation of a hyperbola given the graph/info:
1. Find (h, k) where the asymptotes intersect – equidistant from vertices –plug it into the equation
2. Find the a value (always 1st/under x value) – count spaces from center left /right – in equation
3. Find the b value (always 2nd/under y value) – use asymptote equations to solve for b (ie: top or bottom of slope at
given a value) – in equation
4. Always ‘–’ in between terms and always = 1
Examples:
3. Write the equation from graph:
4. Write the equation
from graph:
Vertical
Part 4 - Parabolas:
•
•
•
•
•
•
•
•
Center – Vertex of the parabola – (h, k)
p – the distance between the vertex and the focus/directrix – determines how the
parabola opens
Directrix – line perpendicular to the axis of symmetry for the parabola – p units
from vertex
Focal Width - length of vertical or horizontal line that passes through the focus
and touches parabola on each end
Focus - a point on inside the parabola p units from the vertex used to define curve
Horizontal Orientation – y term is squared – opens left/right
Vertical Orientation – x term is squared – opens up/down
EOLR – points 2p units up/down from the focus and on the parabola
Horizontal
*Note: x and h always stay together & y and k stay together
Graph a vertical parabola given the equation:
( − ℎ) = 4( − )
1. Determine if the parabola is vertical or horizontal:
a. x term is squared : opens up/down
b. p is + opens up : p is – opens down
2. Determine the center from the conic form of the
equation (watch signs +/- h is opposite)(h, k)
3. Find p (divide constant by 4)
4. Count ‘p’ spaces from center up/down - make point
(focus) – draw in directrix
5. Count 2p spaces right/left from focus – draw ELOR’s
6. Draw ‘parabola’ through vertex and ELOR’s
Graph a horizontal parabola given the equation:
( − ) = 4( − ℎ)
1. Determine if the parabola is vertical or horizontal:
a. y term is squared : opens right/left
b. p is + opens right : p is – opens left
2. Determine the center from the conic form of the
equation (watch signs +/- h is opposite)(h, k)
3. Find p (divide constant by 4)
4. Count ‘p’ spaces from center left/right - make point
(focus) – draw in directrix
5. Count 2p spaces up/down from focus – draw ELOR’s
6. Draw ‘parabola’ through vertex and ELOR’s
Examples:
1. Find all parts to the parabola and graph:
2
( y + 2) = −16 x
2. Find all parts to the parabola and graph:
( x − 2) 2 = 20( y + 1)
Vertex: _________
Vertex: _________
P = _______
P = _______
Focus: __________
Focus: __________
Directrix: ________
Directrix: ________
Focal width: _____
Focal width: _____
EOLR: __________
EOLR: __________
3.
Write in conic form: y 2 + 6 y − x + 8 = 0
4. *Pre-AP* Write the Equation given limited information a. Focus: (0, 3) Directrix: y = -3
b. Focus: (0, 4) Directrix: x = 3
Part 5 – Determine the conic from standard form
Standard form: + +
Use % − &'( to determine the conic:
Circle: − 4 ˂ 0 and b = 0 and a = c
Ellipse: − 4 ˂ 0 and b ≠ 0 OR a ≠ c
Hyperbola: − 4 ˃ 0
Parabola: − 4 = 0
+ ! + " + # = $
Easy hints:
Circle: A = C
Ellipse: A ≠ C, A & C have same signs
Hyperbola: A ≠ C, A & C have opposite signs
Parabola: A = 0 or C = 0, NOT both (just an x2 or just a y2)
Examples:
1. Determine the conic and explain your reasoning: 6x2 + 9y2 + 12x – 15y – 25 = 0
2. Determine the conic and explain your reasoning: x2 + y2 – 6x – 7 = 0
3. Determine the conic and explain your reasoning: 3y2 – x2 – 9 = 0
4. Determine the conic and explain your reasoning: y2 – 2x – 4y + 10 = 0
Part 5 – Parameter changes with conics
For ALL
• h shifts left/right
• k shifts up/down
Circle
• r changes radius
Ellipse
• a & b change size and direction
• a2 ˃ b2
b2 ˃ a2
Parabola
• p determines whether the parabola opens
up/down/left/right
• p determines wider or narrower
Examples:
1. If the center of an ellipse if shifted to the right by 4 which value is changed?
2. If the size a circle increases, what value is changed?
3. If a2 and b2 are switched in the equation of an ellipse, how is the graph changed?
4. If a2 and b2 are equal in the equation of an ellipse, how is the graph changed?
5. If p changes from + to – in a vertical parabola, how is the graph changed?
6. If p changes from + to – in a horizontal parabola, how is the graph changed?
Part 5 - *PreAP* Real World Applications
1. A parabolic reflector is in the shape made by revolving an arc of a parabola, starting at the vertex, about the
axis of the parabola. If the focus is 9 inches from the vertex, and the parabolic arc is 16 inches deep, how
wide is the opening of the reflector?
2. The face of a one-lane tunnel is a square with a semi-circle above it. The semi-circle has a diameter of 18 ft.
A truck that is 15 ft wide and 22 ft tall tries to drive through the tunnel.
a. Will the truck fit?
b. By how much? (over or under)
Conics Units ALL Assignments
Part 1
Which conic section is formed by cutting a cone:
3) If you cut a Double cone Perpendicularly
1) Diagonally by not cutting through the base
4) Diagonally through base
2) Parallel to the base
Write each type of symmetry that the listed conic has: horizontal, vertical, or diagonal
5) Circles
7) Vertical Parabolas
9) Vertical Hyperbolas
6) Ellipses
8) Horizontal Parabolas
10) Horizontal Hyperbolas
Find the center and radius, and then graph the circle.
2
2
2
2
13) (x - 2) + (y+1) = 16
11) x + y = 81
2
2
2
2
12) (x - 3) + (y- 5) = 4
14) (x - 2) + (y- 3) = 9
Write the equation of the circle.
17)
15) Center (3, 3) and radius of 4
18)
16) Radius of 5 at center of (-1, 1)
Part II
Answer each question about ellipses. Sketch a picture to support your answer.
1) What happens when a2 and b2 are the same?
2) What happens when a2 and b2 are switched?
Find the parts and then graph (see notes).
2
2
3) x + y = 1
9
4)
5)
25
( x + 3) 2 ( y + 2) 2
+
=1
25
16
**Pre-AP #7-8 also
7) 16x2 + 9y2 = 144
8) 16x2 + 36(y – 1)2 = 576
6) x2 + y2 = 64
( x − 1) 2 ( y − 2) 2
+
=1
4
1
Write the equation.
10)
9)
11)
Use complete the square to change from General Form to Conic Form
12) x2 + y2 + 6x – 2y + 9 = 0
**Pre-AP #15-16 also
13) x2 + y2 – 16x + 10y + 53 = 0
15) 16x2 + 9y2 – 128x + 108y + 436 = 0
14) x2 + y2 + 8x – 6y – 15 = 0
16) 4x2 + 9y2 – 48x + 72y + 144 = 0
Part III
Answer each question about hyperbolas. Sketch a picture to support your answer.
1) What happens when the y2 and x2 terms are switched?
Find the parts and then graph (see notes).
2
2
2
2
2) ( y + 3) − ( x + 1) = 1
4) ( x − 2) − ( y + 5) = 1
6) (x-1)2 + y2 = 9
25
64
4
9
2
2
2
2
**Pre-AP #7 also
5) ( x + 2) + ( y − 5) = 1
3) ( y + 3) − ( x − 1) = 1
7) 4x2 – y2 = 4
4
25
4
16
Write the equation.
8)
9)
10)
Part IV
Answer each question about parabolas. Sketch a picture to support your answer.
1) What happens when you have a horizontal orientation and the “p” becomes negative?
2) What happens when you have a vertical orientation and the “p” becomes negative?
3) What happens if the x is squared instead of the y being squared?
Find the parts and then graph(see notes).
2
2
7) y2 = -8(x + 2)
4) (y – 3)2 = 8(x + 2)
9) ( x − 4) + ( y − 2) = 1
2
2
9
16
5) (x –2) = 4(y – 1)
8) x = -16(y – 2)
2
2
10)
(x
+
4)
+
(y + 1)2 = 49
6) x = 12y
Use complete the square to change from General Form to Conic Form
11) x2 - 10x - y + 21 = 0
14) 2y2 - x + 20y + 49 = 0
16) x2 + y2 + 2x + 8y + 8 = 0
2
2
12) x + 2x + y – 1 = 0
15) 3x + 30x + y + 79 = 0
13) y2 - x - 8y + 17 = 0
**Pre-AP only Write the equation.
18) Focus (0,2) directrix x = 2
19) Focus (0,1) directrix x = 2
17) Focus (0,0) directrix y = 4
Part V
Determine which conic section each equation represents.
2
2
1) y2 = x + 13
11) x2 – y2 – 2x – 8 = 0
6) ( x + 5) - ( y+2) = 1
2
2
25
4
2) x + y – 2y – 53 = 0
12) x2 – 1x - y2 – 4y = -12
2
2
2
7) 2y + 28y + 27 + 1 = 0
3) 6x2 – 38x – 97 – 11 = 0
13) 2 x + 7 y − 5 x − 1y − 13 = 0
2
2
2
2
8) x + 3y + 4y + 5x + 6 = 0
4) (x - 2) + (y + 5) = 49
2
2
2
2
9) 5 x2 − 72y + 7 x + 9 y = 3
5) ( x - 1) + (y+5) = 1
10) -9x + y – 72x – 153 = 0
64
1
14) Label each conic section formed by the intersection of a plane with a cone.
Answer the questions about parameter changes.
14) What happens when a2 and b2 are the same in an ellipse?
15) What happens when a2 and b2 are switched in an ellipse?
16) What happens when a2 and b2 are switched in a circle?
17) Explain how p affects the graph of a parabola.
*Pre-AP only* Solve the following application problems.
15) How high is a parabolic arch, of span 24ft and height 18ft, at a distance 8ft from the center of the span?
16) The face of a one-lane tunnel is a square with a semi-circle above it. The semi-circle has a diameter of 5m.
A truck that is 4.5m wide and 6.5 m tall tries to drive through the tunnel. Will the truck fit?
Graphs: