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Unit 9: Conic Sections

Name _______________________________________________________ Per ________ 1/6 HOLIDAY (*)Pre AP Only 1/7 General Vocab Intro to Conics Circles HW: Part 1 1/8-9 More Circles Ellipses HW: Part 2 1/10 Hyperbolas HW: Part 3 1/13 Parabolas HW: Part 4 1/14 Identifying conics in standard form Parameter changes *Real World Applications HW: Part 5 1/15-16 Review

Test Part 1

1/17

Test Part 2

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

Plane

Conic (conic section) Intersection Base of a cone Side of a cone Parallel Perpendicular Vertical Horizontal Orientation Symmetry

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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

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 Examples:

1. Graph: (x+5)2 + (y-4)2 = 25 2. Write the equation: Center at (0,4) radius of 3

3. Write the equation:

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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:

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. Connect those 4 points in an elliptical shape

Write the equation of an ellipse given the graph/info: 1. Find the center (h, k) – 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) – count

spaces from center up/down – in equation 4. Always ‘+’ in between terms and always = 1 Examples:

1. Find all parts to the ellipse and graph:

2 2

(

5)

(

2)

1

16

9

x

y

+

+

=

Center: __________________________ a = _______, b = _______, c = _______ Foci: ____________________________ Vertices: _________________________ Co-Vertices:_______________________

2. Find all parts to the ellipse and graph:

2 2

(

3)

1

4

25

x

+

y

+

=

Center: __________________________ a = _______, b = _______, c = _______ Foci: ____________________________ 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

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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 2

(

3)

(

2)

1

25

9

x

y

+

=

Center: _______ a = _______ b = _______

2. Find all parts to the hyperbola and graph:

2 2

(

1)

1

16

4

y

x

=

Center: ______ a = _______ 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:

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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

*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

Vertex: _________ P = _______ Focus: __________ Directrix: ________ Focal width: _____ EOLR: __________

2. Find all parts to the parabola and graph: 2

(

x

2)

=

20(

y

+

1)

Vertex: _________ P = _______ Focus: __________ Directrix: ________ Focal width: _____ 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

Vertical

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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?

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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.

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Conics Units ALL Assignments

Part 1

Which conic section is formed by cutting a cone:

1) Diagonally by

not

cutting through the base

2) Parallel to the base

3) If you cut a Double cone Perpendicularly

4) Diagonally through base

Write each type of symmetry that the listed conic has: horizontal, vertical, or diagonal

5) Circles 6) Ellipses 7) Vertical Parabolas 8) Horizontal Parabolas 9) Vertical Hyperbolas 10)Horizontal Hyperbolas

Find the center and radius, and then graph the circle.

11)

x

2

+ y

2

= 81

12)

(x - 3)

2

+ (y- 5)

2

= 4

13)

(x - 2)

2

+ (y+1)

2

= 16

14)

(x - 2)

2

+ (y- 3)

2

= 9

Write the equation of the circle.

15)Center (3, 3) and radius of 4

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 a2and b2 are the same? 2) What happens when a2and b2 are switched?

Find the parts and then graph (see notes).

3) 1 25 y 9 x2 2 = + 4) 1 1 ) 2 y ( 4 ) 1 x ( 2 2 = − + − 5) 1 16 ) 2 y ( 25 ) 3 x ( 2 2 = + + + 6) x2 + y2 = 64 **Pre-AP #7-8 also 7) 16x2 + 9y2 = 144 8) 16x2 + 36(y – 1)2 = 576

Write the equation.

Use complete the square to change from General Form to Conic Form

12)x2 + y2 + 6x – 2y + 9 = 0 13)x2 + y2 – 16x + 10y + 53 = 0 14)x2 + y2 + 8x – 6y – 15 = 0 **Pre-AP #15-16 also 15)16x2 + 9y2 – 128x + 108y + 436 = 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 y2and x2 terms are switched?

Find the parts and then graph (see notes).

2) 1 64 ) 1 x ( 25 ) 3 y ( 2 2 = + − + 3) 1 16 ) 1 x ( 4 ) 3 y ( 2 2 = − − + 4) 1 9 ) 5 y ( 4 ) 2 x ( 2 2 = + − − 5) ( 2)2 ( 5)2 1 4 25 x+ y− + = 6) (x-1)2 + y2 = 9 **Pre-AP #7 also 7) 4x2 – y2 = 4 17) 18) 9) 10) 11)

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Write the equation.

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).

4) (y – 3)2 = 8(x + 2) 5) (x –2)2 = 4(y – 1) 6) x2 = 12y 7) y2 = -8(x + 2) 8) x2 = -16(y – 2) 9) ( 4)2 ( 2)2 1 9 16 xy− + = 10)(x + 4)2 + (y + 1)2 = 49

Use complete the square to change from General Form to Conic Form

11)x2 - 10x - y + 21 = 0 12)x2 + 2x + y – 1 = 0 13)y2 - x - 8y + 17 = 0 14)2y2 - x + 20y + 49 = 0 15)3x2 + 30x + y + 79 = 0 16)x2 + y2 + 2x + 8y + 8 = 0

**Pre-AP only Write the equation.

17) Focus (0,0) directrix y = 4 18)Focus (0,2) directrix x = 2 19)Focus (0,1) directrix x = 2

Part V

Determine which conic section each equation represents.

1) y2 = x + 13 2) x2 + y2 – 2y – 53 = 0 3) 6x2 – 38x – 97 – 11 = 0 4) 2 2

(x - 2) + (y + 5) = 49

5)

(

)

2 2 x - 1 (y+5) + = 1 64 1 6)

(

x + 5

)

2 ( y+2)2 - = 1 25 4 7) 2y2 + 28y + 27 + 1 = 0 8) x2 + 3y2 + 4y + 5x + 6 = 0 9)

5

x

2

7

y

2

+

7

x

+

9

y

=

3

10)-9x2 + y2 – 72x – 153 = 0 11)x2 – y2 – 2x – 8 = 0 12)x2 – 1x - y2 – 4y = -12 13)

2

x

2

+

7

y

2

5

x

1

y

13

=

0

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 a2and b2 are the same in an ellipse?

15)What happens when a2and 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?

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References

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