Chapter 9 Topics in Analytic Geometry What You ll Learn: 9.1 Introduction to Conics: Parabolas 9.2 Ellipses 9.3 Hyperbolas 9.5 Parametric Equations 9.6 Polar Coordinates 9.7 Graphs of Polar Equations
9.1 Introduction to Conics: Parabolas What You ll Learn: #147 - Recognize a conic as the intersection of a plane and a double-napped cone. #148 - Write equations of parabolas in standard form. #149 - Use the reflective property of parabolas to solve real-life problems.
Conics A conic section is the intersection of a plane and a double-napped cone.
Quick Algebra Review Completing the Square Examples x 2 + 6x = 5 x 2 + 6x + = 5 x 2 + 6x + 9 = 5 + 9 Add half of middle term squared to both sides x + 3 2 = 14 Factor the trinomial to a squared binomial x 2 + y 2 + 5x 8y = 1 x 2 + 5x + +y 2 8y + = 1 Rearrange equation to group like variables x 2 + 5x + 25 4 + y2 8y + 16 = 1 + 25 4 + 16 Add 1 2 middle term sqrd with both variables x + 5 2 2 + y 4 2 = 93 4 Factor both trinomials to squared binomials
Conic Sections - Algebra All conic sections can be displayed as a general algebraic equation: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 Each conic section can also be defined as a locus (collection) of points satisfying a certain geometric property. For example: (a standard circle equation) x h 2 + y k 2 = r 2
Parabolas f x = ax 2 + bx + c This is a standard quadratic equation of a parabola that opens up or down. A Parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line. The midpoint between the focus and the directrix is the vertex, and the line passing through the focus and the vertex is the axis of the parabola. *picture on next slide*
Parts of a Parabola
Equations of Lines Quick Algebra Review Page 11 #1,2,5,17,33,37
Standard Equation of a Parabola The standard form of an equation of a parabola with vertex at (h, k) is x h 2 = 4p y k, p 0 vertical axis; directrix at y = k p y k 2 = 4p x h, p 0 horizontal axis; directrix at x = h p The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0,0), the equation takes one of the following forms: x 2 = 4py y 2 = 4px Vertical Axis Horizontal Axis
Example 1 Find the standard form of the equation of the parabola with vertex at the origin and focus (0,4). x 2 = 4py
Example 2 Find the focus of the parabola given by y = 1 2 x2 x + 1 2 x h 2 = 4p y k
Example 3 Find the standard form of the equation of the parabola with vertex (1,0) and focus at (2,0). y k 2 = 4p x h
Example 4 Find the vertex, focus, and directrix of the parabola and sketch its graph. x 2 + 10y = 0 x 2 = 4py y 2 + 8y 6x = 2 y k 2 = 4p x h
Reflective Property of Parabolas
Parabolas at Work
Example 6 - Application A drinking fountain shoots water 6 inches into the air and lands 8 inches away. Find the standard equation of the parabola the water creates.
Tangent Line The tangent line to a parabola at a point P makes equal angles with the following two lines. 1. The line passing through P and the focus 2. The axis of the parabola
Example 5 Finding the Tangent Line Find the equation of the tangent line to the parabola given by y = x 2 at the point (1,1). Graph the two equations to check your solution.
Homework Page 637 #7-16 #25-31 odd #37-41 odd #49 *tangent line #56, 59 *applications
9.2 - Ellipses What You ll Learn: #150 - Write equations of ellipses in standard form. #151 - Use properties of ellipses to model and solve real-life problems. #152 - Find eccentricities of ellipses.
Definition of an Ellipse An ellipse is the set of all points (x, y) in a plane, the sum of whose distances from two distinct fixed points (foci) is constant.
Drawing an Ellipse +3 pencils +1 blank paper +1 piece of string +1 ruler (knotted loops about 5 inches apart)
Standard Equation of an Ellipse The standard equation with center (h, k) and major and minor axes of lengths 2a and 2b, respectively, where 0 < b < a, is x h 2 a 2 + y k 2 b 2 = 1 Major axis is horizontal x h 2 b 2 + y k 2 a 2 = 1 Major axis is vertical The foci lie on the major axis, c units from the center, with c 2 = a 2 b 2. If the center is at the origin (0,0), the equation takes one of the following forms. x 2 a 2 + y2 b 2 = 1 x 2 b 2 + y2 a 2 = 1 Major axis is horizontal Major axis is vertical
Major or Minor?
Example 1 Find the center, vertices, foci, and sketch its graph. x 2 9 + y2 25 = 1
Example 2 Find the center, vertices, foci, and sketch its graph. x+1 2 25 + y 3 2 12 = 1
Example 3 Finding the Standard Equation Find the standard form of the equation of the ellipse having foci at (0,1) and (4,1) and a major axis of length 6.
Example 4 Analyzing an Ellipse Find the center, vertices, and foci of the ellipse. 9x 2 + 4y 2 = 36
Example 5 Sketching an Ellipse Sketch the graph of the ellipse. x 2 + 4y 2 + 6x 8y + 9 = 0
Example 6 Analyzing an Ellipse Find the center, vertices, and foci of the ellipse. 4x 2 + y 2 8x + 4y 8 = 0
Whispering Rooms
Eccentricity The eccentricity of an ellipse is the ovalness. The eccentricity e of an ellipse is given by the ratio e = c a.
Homework Page 646 #1-10 #13-16 #23,25,33
9.3 - Hyperbolas What You ll Learn: #153 - Write equations of hyperbolas in standard form. #154 - Find asymptotes of hyperbolas. #155 - Use properties of hyperbolas to solve real-life problems. #156 - Classify conics from their general equations.
Conic Sections
Definition of a Hyperbola A hyperbola is the set of all points (x, y) in a plane, the difference of whose distances from two distinct fixed points (foci) is a positive constant.
Standard Equation of a Hyperbola The standard form of the equation of a hyperbola with center at (h, k) is x h 2 a 2 y k 2 a 2 y k 2 b 2 = 1 Transverse axis is horizontal x h 2 b 2 = 1 Transverse axis is vertical The vertices are a units from the center, and the foci are c units from the center. Moreover, c 2 = a 2 + b 2. If the center of the hyperbola is at the origin (0,0), the equation takes one of the following forms. x 2 a y 2 a 2 y2 b 2 x2 b 2 = 1 2 = 1 Horizontal transversal Vertical transversal
Horizontal vs Vertical
Example 1 Finding the Equation Find the standard form of the equation of the hyperbola with foci ( 1,2) and (5,2) and vertices (0,2) and (4,2).
Asymptotes of a Hyperbola Each hyperbola has two asymptotes that intersect at the center of the hyperbola. Asymptotes may be found by using the following formulas: y = k ± b a (x h) and y = k ± a b (x h) Horizontal Vertical
Example 2 Sketch a graph Sketch the hyperbola whose equation is 4x 2 y 2 = 16. y = k ± b (x h) a and y = k ± a (x h) b
Example 3 Finding the Asymptotes Sketch the hyperbola given by 4x 2 3y 2 + 8x + 16 = 0 and find the equation of the asymptotes.
Example 4 Using asymptotes Find the standard form of the equation of the hyperbola having vertices (3, 5) and (3,1) and having asymptotes at y = 2x 8 and y = 2x + 4
Applications The hyperbolic shape of a sonic boom
Applications Comets that pass through our solar system once follow a hyperbolic path with the sun as a focus.
Applications
Science Center Planetarium St. Louis TWA building at Kennedy Airport New York City
Example 7 Application Two microphones, 1 mile apart, record an explosion. Microphone A receives the sound 2 seconds before microphone B. Where did the explosion occur? 1 mile = 5280 feet speed of sound = 1100 ft/sec Answer: x 2 1210000 y 2 5759600 = 1 somewhere on left branch
Eccentricity of a Hyperbola As with ellipses, the eccentricity of hyperbolas is e = c a If the eccentricity is large, the branches of the hyperbola are nearly flat. If the eccentricity is close to 1, then the branches of the hyperbola are more pointed.
General Equations of Conics Classifying a Conic from its General Equation The graph of Ax 2 + Cy 2 + Dx + Ey + F = 0 is one of the following. 1. Circle A = C A 0 2. Parabola AC = 0 A = 0, or C = 0, but not both 3. Ellipse AC > 0 A and C have like signs 4. Hyperbola AC < 0 A and C have unlike signs
Example 6 Classifying Conics Classify the graph of each equation. 1. 4x 2 9x + y 5 = 0 2. 4x 2 y 2 + 8x 6y + 4 = 0 3. 2x 2 + 4y 2 4x + 12y = 0 4. 2x 2 + 2y 2 8x + 12y + 2 = 0 Circle A = C A 0 Parabola AC = 0 A = 0, or C = 0, but not both Ellipse AC > 0 A and C have like signs Hyperbola AC < 0 A and C have unlike signs
Homework Page 656 #1-7 #11-23 odd, not 21 #25,33 #47-55 odd
Interactive Animations Parabola Ellipse Hyperbola Eccentricity
Mid-Chapter 9 Test Review Page 699 #1-7,9 #11,14,17 #21,25,26 #31-34 *Parabolas *Ellipses *Hyperbolas *Classifying
9.5 Parametric Equations What You ll Learn: Evaluate sets of parametric equations for given values of the parameter. Graph curves that are represented by sets of parametric equations. Rewrite sets of parametric equations as single rectangular equations by eliminating the parameter. Find sets of parametric equations for graphs.
Definition of a Plane Curve In addition to the (x, y) variable, we will have a third variable, t (usually used to represent time) called a parameter. x = f(t) y = g(t) The (x, y) are dependent on the parameter. Instead of x, y, its (f t, g t ) The x and y coordinates are functions with respect to t
Sketching a Plane Curve Example 1 x = t 2 4 y = 1 2 t 2 t 3 t 2 1 0 1 2 3 x y
Example 2 Parametric Mode Use a graphing utility to graph the curves in parametric mode. 1. x = t 2, y = t 3 2. x = t, y = t 3 3. x = t 2, y = t
Eliminating the Parameter Many graphs that are represented in parametric form can also be represented by rectangular equations (in x and y). Steps to eliminate the parameter: #1 solve for t in either equation (x or y) #2 substitute equation found in step 1 into the other equation You may have to alter the domain of the resulting equation to match the graph of the original parametric equation.
Example Eliminate the Parameter x = t 2 4 y = 1 2 t Use substitution to re-write as an x and y equation with no t.
Example Eliminate the Parameter Identify the curve and eliminate the parameter. x = 1 t+1 and y = t t+1
Example Eliminate the Parameter Sketch the curve represented by x = 3 cos θ and y = 4 sin θ, 0 θ 2π, by eliminating the parameter.
Example Eliminate the Parameter Sketch the curve represented by x = 3 sec θ and y = 4 tan θ, 0 θ 2π, by eliminating the parameter.
Homework Page 673 #1-8 #9-23 odd
9.6 Polar Coordinates What You ll Learn: Plot points and find multiple representations of points in the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and vice versa.
The Polar Grid
Polar Coordinate System Just like the (x, y) rectangular coordinates, we have the polar coordinates (r, θ). r = Radius (distance from O to P) θ = Directed Angle (counterclockwise angle from polar axis) P O
Example 1 Plotting Points in the PCS 1. (5, π ) 3 2. (3, π ) 6 3. ( 4, π ) 2 4. (4, 3π ) 2
Example 2 Multiple Representations Plot the point (6, 3π ) and find three additional polar 4 representations of this point, using 2π < θ < 2π.
Coordinate Conversion (r, θ) can be converted to rectangular coordinates (x, y) and vice versa. To convert polar rect x = r cos θ y = r sin θ To convert rect polar tan θ = y x or θ = tan 1 y x x 2 + y 2 = r 2
Example 3 Polar to Rectangular Convert each point to rectangular coordinates. A. (2, π) B. ( 3, π 6 )
Example 4 Rectangular to Polar Convert each point to polar coordinates. A. ( 1,1) B. 0,2 C. ( 3,2) If your plotted x, y point is in Quadrant II or III (left side), then +π to θ to correct the calculator s applied inverse tangent restriction.
Equation Conversion Just like rectangular (x, y) equations, we have polar (r, θ) equations. Examples (convert to polar equation) 1. y = x 2 2. x 2 + y 2 = 4 Use the following: y = r sin θ and x = r cos θ
Example 5 Converting Equations Describe the graph of each polar equation and find the corresponding rectangular equation. 1) r = 2 2) θ = π 3 3) r = sec θ 4) r = 4 sin θ
Homework Page 680 #1-7,13,14,19,21,25,27,30,33,35 *coordinates #39,41,43,57,59,61 *equations
9.7 Graphs of Polar Equations What You ll Learn: Recognize special polar graphs Graph polar graphs
Polar Graphs Straight line through the origin θ = π 4 θ = π 6 θ = 2π 3
Polar Graphs Circles with a given radius centered at the origin r = 4 r = 6
Polar Graphs Circles with center not at origin r = a cos θ r = a sin θ cos θ is symmetrical about the x axis sin θ is symmetrical about the y axis Positive on positive axis, negative on negative axis
Graphing Calculator Graph each of the following: 1. r = 4 + 2 cos θ 2. r = 2 + 6 cos θ 3. r = 5 + 5 cos θ 4. r = 4 + 2 sin θ 5. r = 2 6 sin θ 6. r = 5 5 sin θ
Polar Graphs - Limacons Lima beans r = a ± b cos θ r = a ± b sin θ If a < b, then it has an inner loop If a = b, then it is a cardioid (heart-shaped) If a > b, then it has no inner loop
Graphing Calculator Graph each of the following: 1. r = 5 cos 2θ 2. r = 5 cos 3θ 3. r = 3 cos 4θ 4. r = 4 sin 2θ 5. r = 4 sin 3θ 6. r = 8 sin 4θ
Polar Graphs - Roses Rose Curves r = a cos(nθ) or r = a sin(nθ) n petals if n is odd 2n petals if n is even cosθ roses are symmetrical about the x axis sin θ roses are symmetrical about the y axis
Graphing Calculator Graph each of the following: 1. r = θ 2. r = 2θ 3. r = 3θ 4. r = θ 1/2 5. r = θ 1/3 6. r = θ 1/5
Polar Graphs - Swirls Spirals r = aθ n n determines the sharpness of the curves As n decreases less than one it becomes denser As n increases greater than one it spreads out The value of a can either make it larger or smaller
Polar Graph Project Create a design using at least 4 total polar graphs You must use at least 2 different polar graphs (example: limacon, rose, lines, circles, swirls) Display all equations of polar graphs used Any polar equation may be used to make the design (not your required min of two) Download/Print on printer paper with color (color with computer program or with crayons, etc.) 50 pts (T/Q pts)
Review Chapter 9 Sections 9.6 & 9.7 Page 701 #59-82 Review Polar Graphs Matching: Polar Equations Polar Graphs