Mr. Rives Name: ______Algebra 2X Unit 9 – Conic Sections Please be flexible as we will do some activities which will cause us to move away from the syllabus for that day. We will also have mini-quizzes. DA TOPIC ASSIGNMENT Y 1 CIRCLES P. 732 # 2-17, 12-17 ELLIPSES (Optional Activity – Ellipse P. 740 # 1,4,8,9,10,11 2 Construction) P. 740 # 15, 23-25, 27-30 3 ELLIPSES Study for Quiz! 4 QUIZ ON CIRCLES AND ELLIPSES TBA 5 HYPERBOLAS P. 748 # 8-15 P. 748 # 18-29 6 HYPERBOLAS Study for Quiz! 7 QUIZ ON HYPERBOLAS TBA 8 PARABOLAS P. 755 # 10-12, 22-24 P. 755 # 17-21,27-34 9 PARABOLAS Study for Quiz! 10 QUIZ ON PARABOLAS TBA 11 IDENTIFYING CONICS P. 764 # 1-6, 14 – 17, 43 P. 764 # 9, 10, 22-31 (graph the 12 IDENTIFYING CONICS odds only from 22-31 after you complete the square) P. 772 # 1,4,10,13, 32, 34, 13 SOLVING NON LINAR SYSTEMS 41,43,497 UNIT REVIEW WORKSHEET 14 REVIEW Study for Test! 15 TEST (with a notecard!)

Page 1 of 30 Day 1: Circles

Distance Formula

Warm up Using the distance formula above, find the distance between each pair of points. 1. (-2, 12) and (6, -3) 2. (1, 5) and (4, 1)

Find the slope of the line that connects each pair of points. 3. (5, 7) and (-1, 6) 4. (3, -4) and (-4, 3)

A circle is the set of points in a plane that are a ______distance, called the r______, from a fixed point called the c______.

Circle whose center is at the origin. Graph it. General: x2+ y 2 = r 2 Example: x2+ y 2 = 49

Page 2 of 30 Examples 1a. Write the equation of a circle whose center is at the origin and whose radius is 10 cm.

1b. Write the equation of a circle whose center is (-2, 8) and whose radius is 1 cm.

.

Think The center of a circle is (h, k). In example 1b identify h and k. How do h and k transform the circle from the origin (0, 0), i.e. how does it move?

Writing the equation of a circle given the center and a point. 1. Center (-4, 11); Containing the point (5, -1).

(use the distance formula to find radius)

2. Center (2, 4); Containing the point (8, 12). 3. Center (-3, 5); Containing the point (9, 10).

Page 3 of 30 Inequality and the circle

Application Jennifer and her friends are having a pizza party and will decide where to have the party based on the delivery area of the Lorenzo‘s Pizza. Lorenzo’s is located at (-1, 2) and the letters represent the homes of Jennifer and her friends. Find the homes within a 3-mile radius of Lorenzo’s.

Try finding the homes located within a 3-mile radius of Scoops Ice Cream located at (2, -1).

CLOSURE 1. What is the distance between (9, 4) and (-7, 4)? Do this one mentally.

2. How is the following circle transformed (identify h and k)?

3. What is the radius of the circle in #2?

4. Why is the following equation NOT an example of a circle? (x - 3) + (y + 7) = 25

Page 4 of 30 Day 2: Ellipses

If you pulled apart the center of a circle into 2 points, it would stretch the circle into an ellipse.

Definition: the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F1 and F2, called the foci is the constant sum d = PF1 + PF2.

(Do optional activity.)

Parts/Types of an Ellipse

Mini-Exploration

1. In mm, measure the distance called b (from the center to either co-vertex). b = _____ 2. In mm, measure the distance called c (from the center to either foci). c = _____ 3. On the diagram create a right triangle by connecting a line from b to c. What formula can we use to find the hypotenuse? Find the hypotenuse length.

4. In mm, measure the distance called a (from the center to either vertex). Notice anything?

Page 5 of 30 List 2 differences between the horizontal and vertical ellipse equations. 1.

2.

It turns out that the values a, b and c are related by the Pythagorean Theorem but just to torture us some of the values have changed places - YIKES! c2 = a2 - b2 What acts as the hypotenuse and is therefore always the longest side?

Examples Write an equation in standard form for each ellipse with center at (0, 0). First identify if the ellipse is horizontal or vertical.

1. Vertex at (6, 0); co-vertex at (0, 4). Horizontal or Vertical?

2. Co-vertex at (5, 0); focus at (0, 3). Horizontal or Vertical?

3. Vertex at (-10, 0); focus at (8, 0). Horizontal or Vertical?

4. Vertex (9, 0) and co-vertex (0, 5). Horizontal or Vertical?

Page 6 of 30 Graphing with center at (0, 0) x2 y 2 x2 y 2 1. + =1 2. + =1 4 49 49 4

H or V? H or V? a = _____; b = _____ a = _____; b = _____ c = _____ c = _____

Day 3: Ellipses (Center not at the origin.)

Warm up 1. Among a, b and c what is the greatest distance in an ellipse? Define this distance.

Using the pythagorean theorem for ellipses, find c in the following: 2. a = 13, b = 5.

3. a = 4, b = 3.

4. This time find b. a = 25, c = 17.

Reference Example for Graphing

It’s horizontal because the larger number (a squared) is under the x-squared term. a = 5, b = 3. The center is (h, k) = (2, 4).

Page 7 of 30 Graphing with center at (h, k).

1. 2. Center:______Center:______a = ______; b = ______a = ______; b = ______c = ______c = ______

Foci: Foci: ( , ); ( , ) ( , ); ( , )

3. 4. Center:______Center:______a = ______; b = ______a = ______; b = ______c = ______c = ______

Foci: Foci: ( , ); ( , ) ( , ); ( , )

Page 8 of 30 Writing Equations: Create a diagram to help write the equation of the ellipse described.

5. Write an equation of an ellipse that has a vertex at (10, 12) a co-vertex at (3, 6).

6. Write an equation of an ellipse that has a vertex at (-3, 4) and a co-vertex at (-1, -5).

7. An ellipse has a center at (-5, 0) and a co-vertex at the point (-5, 3). If the ellipse contains the point (-6, 2) find the equation of the ellipse. Show all work.

8. An ellipse has a center at (3, 5), a vertex at (3, 12), and a co-vertex at (0, 5). Determine the equation of the ellipse. Identify the lengths of the major and minor axes.

9. Write an equation of the ellipse shown.

Page 9 of 30 Day 4 Review and Quiz (Circles/Ellipses)

Circles

Ellipses – First complete any problems from Days 2 and 3.

Page 10 of 30 Day 5: Hyperbolas Part 1

Warm up: What does the equation of an ellipse look like?

 A hyperbola has the same equation except…

x2y 2 y2x 2 Either variable (x or y) can come first… - =1 or - = 1 a2 b 2 a2 b 2

 When x comes first, the hyperbola is ______(think about the x-axis).

 When y comes first, the hyperbola is ______(think about the y-axis).

A hyperbola is made from a box consisting of ______and ______. (a value: positive term) (b value: negative term)

Directions: for each hyperbola below, identify the direction, the vertices, and the co-vertices.

x2 y 2 y2 x 2 x2 y 2 1. - = 1 2. - =1 3. - = 1 36 49 4 25 9 64 direction: ______direction: ______direction: ______vertices: ______& ______vertices: ______& ______vertices: ______& ______co-v’s: ______& ______co-v’s: ______& ______co-v’s: ______& ______

Asymptotes: ______Asymptotes: ______

Page 11 of 30 The asymptotes of a graph (we’ve see these before!) are lines that the graph approaches, but never actually ______. They can be found by taking ______of the box!

Directions: Find the vertices, co-vertices, and asymptotes of the hyperbola, and then graph! x2 y 2 ( y-1)2( x + 3) 2 ( y+2)2( x - 4) 2 1. - =1 2. - =1 3. - =1 4 9 16 36 1 4 direction: ______direction: ______direction: ______center: ______center: ______center: ______vertices: ______& ______vertices: ______& ______vertices: ______& ______co-v’s: ______& ______co-v’s: ______& ______co-v’s: ______& ______

Asymptotes: ______Asymptotes: ______Asymptotes: ______

To summarize translations: The center of the hyperbola is (h, k )

( x- h)2( y - k ) 2 ( y- k)2( x - h) 2 Horizontally: - =1 Vertically: - =1 a2 b 2 a2 b 2

There are 3 additional vocabulary words that we need…

 Focus (Foci):

 Transverse Axis:

 Conjugate Axis:

Page 12 of 30 10-4 Hyperbolas 10-4 Hyperbolas

The standard form of the equation of a hyperbola The standard form of the equation of a hyperbola depends on whether the hyperbola’s transverse axis is depends on whether the hyperbola’s transverse axis is horizontal or vertical. horizontal or vertical.

Holt Algebra 2 Holt Algebra 2

Page 13 of 30 Day 6: Hyperbolas Part 2

Wamup: Vocabulary Review for Hyperbolas! 1. ___ Transverse Axis A. Hyperbola opening vertically 2. ___ Foci B. The endpoints are the vertices of the hyperbola y2x 2 C. The important points on the hyperbola. They are 3. ___ - = 1 a2 b 2 at the end of the transverse axis. 4. ___ Co-vertices D. The graph approaches these lines, but never touches them 5. ___ Conjugate Axis E. The endpoints are the co-vertices of the hyperbola 6. ___ Asymptote F. Hyperbola opening horizontally x2y 2 G. Fixed points in the interior of the branches of the 7. ___ - =1 a2 b 2 hyperbola 8. ___ Vertices H. The points that create the “other part” of the box

Today we will be working backwards from yesterday: I give you the graph (or the vertices, etc) and you tell me the equation. We will also do some problems like we did yesterday…

Directions: Write an equation in standard form for each hyperbola. Sketch? 1) Center (0, 0), vertex (5, 0), co-vertex (0, -2)

Direction: ______(remember, it must open w/ the vertex)

General equation: ______, a = ______, b = ______

Final Answer: ______

2) Center (0, 0), vertex (0, 3), co-vertex (4, 0)

Direction: ______(remember, it must open w/ the vertex)

General equation: ______, a = ______, b = ______

Final Answer: ______

Page 14 of 30 3) Center (0, 0), vertex (0, 5), focus (0, 13) *

Direction: ______(remember, it must open w/ the vertex)

General equation: ______, a = ______, b = ______

Final Answer: ______Just like with the ellipse, there is a connection between the foci, the vertices, and the co-vertices.

c: ______c2= a 2 + b 2 a: ______b: ______

4) Write an equation of a hyperbola with a focus of (5,0) and a co-vertex of (0, 3)

Sometimes we will also write the equation when given a graph. To do so, first find the center, the vertices, and the co-vertices.

5) 6)

Equation: ______Equation: ______

Page 15 of 30 Directions: Find the vertices, co-vertices, and asymptotes of the hyperbola, then graph.

2 y2 ( x + 2) - =1 16 9

Additional Practice:

Page 16 of 30 Day 7 Review and Quiz (Hyperbolas)

Hyperbola Review In problem 1, you will identify parts of a hyperbola.

1. Given the following hyperbola:

a) Draw a circle around each vertex.

b) Draw a square around each co-vertex.

c) Draw an arrow pointing at each asymptote.

2. In a hyperbola, we use to letters a, and b to refer to key parts' distances from the hyperbola's center.

a) What distance does a represent?

b) What distance does b represent?

y2 x 2 3. Given the hyperbola - = 1, 16 9 a) What are the coordinates of the hyperbola's vertices? (Which way does it open?)

b) What are the coordinates of the hyperbola's co-vertices?

c) What are the slopes of the asymptotes?

d) Graph the hyperbola.

4. Write the equation of the hyperbola with center (0,0), one vertex at (4,0), and one co-vertex at (0,6).

Day 8 10-5 Parabolas – Part 1 Page 17 of 30 Earlier in the year, we learned that the graph of a quadratic function in the form y= a( x - h )2 + k gives us a parabola.

List as many characteristics as you can recall about parabolas?

Define a, h and k.

Definition A parabola is the set of all points in a plane that are an equal distance from both a given point called the______and a given line called the______.

We will use the variable “p” to represent the distance from the vertex to the focus and vertex to directrix

Axis of Symmetry (AoS)  perpendicular to its directrix  passes through the vertex

The vertex is the ______between the focus and the directrix. 1 The graph shown is y=( x + 1)2 - 4 8 Opens:______

Vertex:______

Focus (point):______

Directrix (line):______

If a parabola opens up, where is the focus in comparison to the vertex? Where is the directrix? Page 18 of 30 If a parabola opens down, where is the focus in comparison to the vertex? Where is the directrix?

______Horizontal Parabola: opens to the right or the left; axis of symmetry is also ______.

This means that the focus must be either ______or ______of the vertex.

Also, the directrix, since it is perpendicular to the axis of symmetry, will have to be a ______line.

1 The value of “a” that we have used is now going to be defined as being equal to , 4 p where p = the distance from vertex to focus = distance from vertex to directrix. 1 Know this: a = 4 p

Page 19 of 30 1 1 1. y= - ( x )2 2. x= ( y )2 (careful this one is new) 12 12

Opens:______Opens:______

Vertex:______Vertex:______

AoS:______AoS:______p = ______p = ______

Focus:______Focus:______

Directrix:______Directrix:______

Find a total of 5 points. Choose a value of x that Don’t forget a total of 5 points for precision. yields an integer for y (if possible).

Just for fun  Select any point on your parabola  Measure the perpendicular distance from this point to the directrix.  Measure from the selected point to the focus.  Are they the same? They should be.

What if the vertex is not at the origin (0, 0)?

Page 20 of 30 Study h and k in both columns. What do you notice???

1 2 1 2 1. y= -( x - 2) + 3 2. x=( y - 2) - 1 (careful this one is cray-Z) 4 8

Opens:______Opens:______

Vertex:______Vertex:______

AoS:______AoS:______p = ______p = ______

Focus:______Focus:______

Directrix:______Directrix:______

Find a total of 5 points. Choose a value of x that Don’t forget a total of 5 points for precision. yields an integer for y (if possible).

Page 21 of 30 Day 9 10-5 Parabolas – Part 2 (Practice)

First finish day 8 graphs. Warm Up 1 x= -( y )2 + 3 12

Opens:______

Vertex:______AoS:______p = ______

Focus:______Directrix:______

Find a total of 5 points. Choose a value of y that yields an integer for x (if possible).

Writing Parabolic Equations

Type 1: vertex is at the origin (Note: the scale = 2)

Page 22 of 30 Type 2: vertex is not at the origin.

Use the given focus or directrix to find p.

#1. #2. Sketch the focus (-2, -2)

Vertex:______; p = _____; a = _____; Vertex:______; p = _____; a = _____; Equation:______Equation:______

#3.

Vertex:______; p = _____; a = _____;

Equation:______

Application Page 23 of 30 Day 10 Review/Quiz - Parabolas

For each of the following problems, find the equation and the graph of the parabola. Make sure to label your vertex (V), focus (F), and directrix (D) on the graph. Then give the coordinates or equation of the missing piece.

1. The parabola that has a vertex on (0, 0), and a directrix y=4.

Equation: ______

Focus: ______

8. The parabola that has a vertex at (2, 1), and a focus at (2, 5).

Equation: ______

Directrix: ______

Day 11 Identifying Conics Page 24 of 30 Match the equation of the parabola, circle, or ellipse with its graph. x2 y2 (x + 1)2 (y - 2)2 1 + = 1 + = 1 x + 4 = (y - 2)2 16 64 64 100 12 1 x2 y2 1 x + 1 = - (y - 2)2 + = 1 y + 3 = - (x - 3)2 4 36 81 12

1. 2. 3.

4. 5. 6.

Classifying Conic Sections

Only one squared term It’s a ______.

A squared term minus a squared term It’s a ______.

A squared term plus a squared term It’s a ______.

Here are 2 equations. One is a circle and the other an ellipse. How can you determine the difference?

x 2 y 2 x 2  y 2 100 versus  1 100 25

Classify the conic section.

Page 25 of 30 1. (x - 4)2 (y + 4)2 2. 1 + = 1 x - 5 = - (y - 1)2 25 36 36 3. -1 4. 4489 x = y2 (x - 6)2 + (y + 6)2 = 48 100 5. x2 y2 6. y2 x2 - = 1 - = 1 64 16 4 4 7. -1 8. (x + 1)2 (y - 7)2 x = y2 + = 1 44 25 9 9. 1 10. 1 x + 8 = (y + 7)2 (x + 2)2 + (y - 3)2 = 28 81 11. 1 12. y2 x2 x = y2 - = 1 4 9 36 13. 1 14. x2 + y2 = 4 x - 7 = - (y - 8)2 40 Begin Completing the Square

Circles: x2+ y 2 -6 x - 2 y + 4 = 0 You Try x2+ y 2 +16 x - 4 y - 16 = 0

Ellipses 25x2+ 16 y 2 + 150 x - 160 y = - 225 You Try 2x2+ 8 x + y 2 + 4 = 0

Parabola 2x2 - y + 20 x = - 53 You Try y2 -8 y + 8 x = - 16

Hyperbola x2+8 x - y 2 + 4 y - 2 = 0 You Try x2+4 x - y 2 - 20 y - 100 = 0

Page 26 of 30 Day 12 Identifying Conics cont… Mixed Review – Write each equation in standard form and identify the conic section. Also identify the center/vertex.

1. x2+ y 2 -6 x + 4 y + 12 = 0 2. y2- x 2 +2 y - 14 x - 57 = 0

3. y2 -10 y - 4 x + 57 = 0 4. -2y2 - 24 y + 72 x - 432 = 0

5. 36x2+ 25 y 2 - 216 x - 100 y - 476 = 0

Begin your Homework (You’ll need graph paper.)

Page 27 of 30 Day 13 Solving Nonlinear Systems

Warm-up Do you remember what is meant when we say we have a “system of equations?”

Remember we were looking for a point(s) of intersection.

Two Conic sections may have 0, 1, 2, 3, or 4 points of intersection. Construct your own sketches to demonstrate this. (Use the first graph to demonstrate 0 solutions then 1 solution, etc.)

Solve the following systems.

3x+ 4 y = 57 3x+ 4 y = 15 1. use elimination 2. use substitution 5x- 4 y = - 1 x=6 y - 6

Page 28 of 30 Solve each of the following systems of equations. Identify the conics while you’re at it and sketch a graph! Show the algebra steps.

x2+ y 2 = 25 25x2+ 9 y 2 = 225 1. 2. 2y+ 10 = x2 16x2- 9 y 2 = 144

x2- y 2 =16 2x2+ 3 y 2 = 83 3. 4. x+ y2 = 4 4x2- 2 y 2 = - 34

Page 29 of 30 Day 14 Review

Page 30 of 30