CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS

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1 CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS Big IDEAS: 1) Writing equations of conic sections ) Graphing equations of conic sections 3) Solving quadratic systems Section: Essential Question 8-1 Apply the Distance and Midpoint Formulas How can you find the center, radius, or diameter of a circle when given the endpoints of the diameter? Use the Pythagorean Theorem to find the distance between A and B. Key Vocabulary DISTANCE FORMULA MIDPOINT FORMULA Ex 1) Write the distance between 5,1 and 3,? Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #1

2 Ex ) Classify DEF as scalene, isosceles, or equilateral. Ex 3) Find the midpoint of the segment joining, 3 and 4,. Ex 4) Write an equation for the perpendicular bisector of the line segment joining A 5, 4 and B 1, 6. Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #

3 Ex 5) A circular swimming pool has a slide at B and stairs at A and C. Use the three labeled points to estimate the diameter of the pool. (Each unit in the coordinate plane represents 1 meter.) Closure How do you find the center of the circle when given 3 points on the circle? Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #3

4 Section: Essential Question 8- Graph and Write Equations of Parabolas How do you graph equations of parabolas? How do you write equations of parabolas? Conic Sections Conic sections are considered to be two dimensional shapes formed by cutting a double cone by a plane region. Key Vocabulary a PARABOLAS Analyzing Parabolas y k ax h where 1 a 4c and c is the distance from: *vertex to focus *vertex to directrix x h a y k where 1 a 4c and c is the distance from: *vertex to focus *vertex to directrix a 0 Parabola opens Parabola opens a 0 Parabola opens Parabola opens VERTEX, b b hk or, f a a b b hk, or, f a a FOCUS h, k c h c, k DIRECTRIX Horizontal Line y k c Vertical Line x h c AXIS OF SYMMETRY (Stretched on the squared term!) x h or b x y k or a y f b a Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #4

5 Ex 1) A parabola has focus 4, 3 and directrix y 1, find the vertex. Option 1: Graph the given information to determine the vertex. Vertex = Option : Use algebra to determine the vertex. a. What do you notice about the relationship between the focus, directrix, and vertex? b. How does this parabola open? Why? Ex ) A parabola has vertex, 3 and directrix y 5, find its focus. Option 1: Graph the given information to determine the focus. Focus = Option : Use algebra to determine the focus. How does this parabola open? Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #5

6 Ex 3) Graph x y. Identify the focus, directrix, and axis of symmetry. Ex 4) Graph y 0x. Identify the focus, directrix, and axis of symmetry. Ex 5) Graph x x y 0. Identify the focus, directrix, and axis of symmetry. Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #6

7 Ex 6) Write an equation of the parabola shown. Ex 7) Write the standard form of the equation of the parabola with vertex at 0, 0 and the given directrix or focus. A. Directrix y B. Directrix x 4 C. Focus, 0 D. Focus 0, 3 Ex 8) A flashlight uses a parabolic mirror and a light source. Write an equation for the mirror s cross section. How deep is the mirror? Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #7

8 Section: Essential Question 8-3 Graph and Write Equations of Circles What information do you need to write the equation of a circle? Key Vocabulary Circle- the set of all points x, y in a plane that are from a fixed point, called the, Radius- the distance between the and any point x, y on the circle We can use the distance formula to derive the equation of a circle! EQUATION OF A CIRCLE: hk, x, y Ex 1) Find the equation of the circle with center 4, 3 and radius 6. Ex ) Find the equation of the circle with center 1, 3 and radius. Ex 3) Write the standard form of the equation of the circle that passes through 4, 7 and whose center is the origin. Ex 4) Write the standard form of the equation of the circle that passes through 5,1 and whose center is the origin. Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #8

9 Ex 5) Graph x y Identify the radius and center. Ex 6) Graph x 4 y. Identify the radius and center. Ex 7) Write an equation of the line tangent to the circle x y 61 at 5, 6. Ex 8) A business owner wants to buy a property within a 30 mile radius of the airport. She finds a property 10 miles west and 4 miles south of the airport. Is the property within a 30 mile radius of the airport? Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #9

10 If the graph of the given equation is a circle, find its center and radius. If the equation has no graph say so. Ex 9) x y y 8 Ex 10) x y 4x 10y 7 Ex 11) x y 8x y 0 Ex 1) x y x y 8 0 Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #10

11 Section: Essential Question 8-4 Graph and Write Equations of Ellipses What information do you need to write an equation of an ellipse? Key Vocabulary Ellipse- The set of all points P in a plane such that the sum of the distances from P to two fixed points F 1 and F is a constant. In other words, D1 D= constant. D 1 F 1 hk, P x, y F D Analyzing An Ellipse x h y k 1 a b where 0 b a 1 b a x h y k Vertex ha, Center hk, b c Co-vertex bk, Major Axis Minor Axis Length of a Length of b Co-vertex bk, hk, Foci Distance c from the center along the major axis, where. a Co-vertex hb, Vertex h, a Vertex ak, c b hk, Co-vertex h, b a Vertex ak, Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #11

12 Steps for Graphing an Ellipse! 1. Rewrite the equation into the general form of an ellipse.. Find the x-intercepts by substituting zero in for y and solving for x. 3. Find the y-intercepts by substituting zero in for x and solving for y. 4. Find the foci by moving a distance c away from the center along the major axis. (Find c by using the formula c a b.) Graph the ellipse. Identify the vertices, co-vertices, and foci of the ellipse. Ex 1) 5x 4y 100 Ex ) 9x 4y Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #1

13 Graph the ellipse. Identify the vertices, co-vertices, and foci of the ellipse. Ex 3) x 4y 16 0 Ex 4) Write an equation of the ellipse that has a vertex at 4, 0, and center at 0, 0. 0, 7, a co-vertex at Ex 5) Write an equation of the ellipse that has a vertex at 5, 0, a focus at 3, 0, and center at 0, 0. Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #13

14 Section: Essential Question 8-5 Graph and Write Equations of Hyperbolas How do you graph hyperbolas? How do you write equations of hyperbolas? Key Vocabulary Hyperbola- the set of all points P in a plane such that the of the distances between P and the foci is a constant Branches- the two graphical sections of a hyperbola Transverse Axis- the line connecting the vertices of the hyperbolic branches having the as its Asymptote- a line that a graph but touches A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by every one in its path. Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #14

15 Analyzing Hyperbolas x h y k a or b 1 y k x h a 1 b Center Transverse Axis Always a length of a notice in a hyperbola the transverse axis is horizontal when the term is. A vertical transverse axis occurs when the term is. Vertices Foci The Box and the Oblique/Slant Asymptotes A hyperbola has a at each end of the transverse axis which is a distance a from the center. The ordered pairs are determined by adding and subtracting a to the appropriate element of the point hk,. Distance c from the center the transverse axis, where c a b. Build a box using the a and b values around the center hk,. The hyperbolic asymptotes pass through the of this box and through the center. The equations for these asymptotes are easily determined using the meaning of slope and the point slope form of a linear y y m x x equation 1 1 OR Use the following formulas to find the equations of the slant asymptotes. (These equations can only be used if the center is 0, 0!!!) If the transverse axis is on the x-axis (making the hyperbola horizontal ) use b y x a If the transverse axis is on the y-axis (making the hyperbola vertical ) use a y x b Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #15

16 Ex 1) Sketch the graph of the hyperbola asymptotes of the hyperbola. x y Identify the vertices, foci, and a. Find the center hk,. b. Find the x and y intercepts. c. Determine a, b, and c. d. Find the vertices. (add a to the center) e. Find the foci. f. Find the equations of the slant asymptotes. Ex ) Sketch the graph of the hyperbola asymptotes of the hyperbola. y 9x 9. Identify the vertices, foci, and Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #16

17 Ex 3) Sketch the graph of the hyperbola asymptotes of the hyperbola. 16x 9y 144. Identify the vertices, foci, and Ex 4) Write an equation of the hyperbola with foci at 8, 0 and 8, 0 and vertices at 5, 0 and 5, 0. Ex 5) Write an equation of the hyperbola with foci at 0, 10 and 0, 6 0, 6. and 0,10 and vertices at Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #17

18 Section: Essential Question 8-6 Translate and Classify Conic Sections How do you identify and graph a conic? Key Vocabulary General Second-Degree Equation- used to describe any conic section Ax Bxy Cy Dx Ey F 0 Discriminant- used to classify conics B 4AC CLASSIFYING CONICS DISCRIMINANT B 4AC 0, B 0, and A C B 4AC 0, and either B 0, or A C TYPE OF CONIC B B 4AC 0 4AC 0 If B 0, each axis of the conic is horizontal or vertical. Graph the equation. Ex 1) x y1 1 Ex ) x y Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #18

19 Write the equation of the conic section. Ex 3) Ellipse with foci at 3, 5 and and co-vertices at 1, 5, 3, 1 and Ex 4) Hyperbola with vertices at 7, 3 and 1, 3 and foci at 9, 3 and 1, 3 Classify the conic section and write its equation in standard form. Then graph the equation. Ex 5) x y x 4y 1 0 Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #19

20 Classify the conic section and write its equation in standard form. Then graph the equation. Ex 6) x y 4x 4 0 Ex 7) y y x Ex 8) 4x y 16x 4y 4 0 Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #0

21 Section: Essential Question 8-7 Solve Quadratic Systems How can you identify the points of intersection of two distinct conics? Prior Knowledge Connection: Solve each system of linear equations. 1. y x 1 and 4x 5y 3. x 3y 1 and x y 5 Key Vocabulary Quadratic Systems- systems that include one or more equations of LINEAR-QUADRATIC SYSTEMS No Solution One Solution Two Solutions Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #1

22 QUADRATIC SYSTEMS No Solution One Solution Two Solutions Three Solutions Four Solutions METHODS FOR SOLVING SYSTEMS 1. Substitution Method- Solve one of the equations for x or y and substitute the expression into the other equation. Solve the resulting equation for x or y, then back substitute to find the other variable.. Elimination Method- Write the equations in standard form, then multiply by a number that will cause a variable to eliminate when you add the columns. Solve the resulting equation for x or y, then back substitute to find the other variable. 3. Graphing Calculator- Solve both equations for y and graph them on your calculator. Select a window so that you are able to view all points of intersection, then select to locate the intersection point. Move the cursor near a point of intersection using the arrow keys and press to select the first curve, to select the second curve, and to guess. (You may need to use the arrow keys to select the correct curves.) Repeat this process for every point of intersection. Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #

23 Solve the system. Ex 1) 3x 5y 49 y 10x 34 Ex ) x y 5 y x x 6 7 Ex 3) x 8y 4 0 y x 7 Ex 4) x y x y y x y 5 Student Notes Honors Algebra II Chapter 8 Quadratic Relations and Conic Sections Page #3