Circles and Parabolas Discus throwing is an ancient sport at least 3000 years old. This sport has been part of the Olympics since the first Summer Olympics in 1896. Many ancient Greek and Roman statues feature images of discus throwers..1 The Coordinate Plane Circles and Polygons on the Coordinate Plane....... 965.2 Bring On the Algebra Deriving the Equation for a Circle.................. 973.3 Is That Point on the Circle? Determining Points on a Circle.................... 989.4 The Parabola Equation of a Parabola.......................... 997.5 Simply Parabolic More with Parabolas............................ 1019 963 451440_Ch_963-1034.indd 963 21/05/ 7:37 PM
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The Coordinate Plane Circles and Polygons on the Coordinate Plane.1 Learning Goals In this lesson, you will: Apply theorems to circles on the coordinate plane. Classify polygons on the coordinate plane. Use midpoints to determine characteristics of polygons. Distinguish between showing something is true under certain conditions, and proving it is always true. Have you ever seen a discus throw? It may seem like this track event does not require a lot of skill, but any experienced discus thrower will tell you otherwise. Balance is an extremely important factor, as is rhythm. Orbit and delivery are key as well. A great discus thrower can release the discus at the right angle so that it catches the air underneath it for a longer throw. Do you think any other sports use math as important factors? 965
Problem 1 Walk the Line This lesson provides you an opportunity to review several familiar theorems while classifying polygons formed on a coordinate plane. Circles and polygons located on a coordinate plane enable you to calculate distances, slopes, and equations of lines. 1. Lauren and Jamie were practicing their discus throws. They each spun around in a circle to gain speed and then released their discus at a tangent to the circular spin. Jamie is left-handed, so she spun clockwise and released her discus at point T shown. Lauren is right-handed. She spun counterclockwise and released at point N. Both of the girls throws landed at point A. a. Use the given information to algebraically show that if two tangents are drawn from the same point on the exterior of a circle, then the tangent segments are congruent. 8 6 8 6 4 2 C (0, 0) 4 2 0 2 4 6 8 y 2 N (0, 5) 4 T (4, 3) 6 8 x A b. What conclusions can you make about Lauren s and Jamie s discus throws? 966 Chapter Circles and Parabolas
2. Line AT is tangent to circle C at point T. Secant AG intersects circle C at points N and G. AT intersects AG at point A. The coordinates of point A are (10, 0). The center of circle C is at the origin and the length of the radius is 6 units. Use the given information to algebraically show that if a tangent and a secant intersect on the exterior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the square of the length of the tangent segment. y 8 6 4 T 8 G 6 2 C (0, 0) 4 2 0 2 2 4 N 6 8 A (10, 0) x 4 6 8.1 Circles and Polygons on the Coordinate Plane 967
Problem 2 Work Your Quads 1. Four points on the circle were connected to form a square. Classify the polygon formed by connecting the midpoints of the sides of the square. y (0, y) (x, y) (0, 0) (x, 0) x 968 Chapter Circles and Parabolas
2. Choose four points on the circle to form a quadrilateral that is not a square. Provide the coordinates and labels for each. Classify the polygon formed by connecting the midpoints of the sides of the quadrilateral. y 8 6 4 2 8 6 4 2 2 0 2 4 6 8 x 4 6 8 A (0, 7) If you choose another set of four points to form a quadrilateral would you come to the same conclusion about the classification of the shape formed by connecting the midpoints?.1 Circles and Polygons on the Coordinate Plane 969
3. Determine the shape formed by connecting the midpoints of the sides of an isosceles trapezoid. a. Draw an isosceles trapezoid. Choose reasonable coordinates for each vertex. b. Label the vertices of the trapezoid. c. Determine the coordinates of the midpoint of each side. y 8 6 4 2 8 6 4 2 2 2 4 N 6 8 x 4 6 8 970 Chapter Circles and Parabolas
4. Think back about the work you did in Questions 1 through 3. Which were instances of showing something is true under certain conditions or proving something is always true? Explain your reasoning. 5. Describe the difference between showing something is true under certain conditions and proving something to be true under all conditions. Be prepared to share your solutions and methods..1 Circles and Polygons on the Coordinate Plane 971
972 Chapter Circles and Parabolas
Bring On the Algebra Deriving the Equation for a Circle.2 Learning Goals In this lesson, you will: Use the Pythagorean Theorem to derive the equation of a circle given the center and radius. Distinguish between the equation of a circle written in general form and the equation of a circle written in standard form (center-radius form). Complete the square to determine the center and the radius of a circle. You stop a friend in the hallway to ask about his or her weekend. As you are talking, another friend shows up and joins the conversation. Soon, another and then another and then another person joins in. Before long, your group will form a shape without even thinking about it. Your group will probably form a circle. Try and notice this the next time it happens either to you or in another group. Why do you think people naturally form a circle when talking? 973
Problem 1 Pythagorean Theorem Connections Recall that a circle is the set of points on a plane equidistant from a given point. If this circle is drawn on a coordinate plane, with its center point located at the origin and a point (x, y) on the circle, it is possible to write an algebraic equation for the circle. 1. Label the sides of the right triangle formed. Then, use the Pythagorean Theorem to solve for r 2. (x, y) Because your equation is true for every point on the circle, it can be used as the equation of a circle centered at the origin. r (0, 0) (x, 0) 974 Chapter Circles and Parabolas
2. Next, consider a circle with its center located at point (h, k) and a point (x, y) on the circle. Label the sides of the right triangle formed. Then, use the Pythagorean Theorem to solve for r 2. (x, y) r (h, k) 3. How does the equation you wrote in Question 2 also describe a circle with its center point at the origin? Explain your reasoning. The standard form of the equation of a circle centered at (h, k) with radius r can be expressed as (x 2 h) 2 1 (y 2 k) 2 5 r 2 4. Write an equation for: a. a circle with center at the origin and r 5 8. Pay attention to the form of the equation. The coefficients of both x and y are one. b. a circle with center (3, 25) and r 5 6..2 Deriving the Equation for a Circle 975
c. circle P. y 10 8 6 4 10 8 6 P 4 2 2 2 0 2 4 6 8 10 x 4 6 8 10 d. circle Q. y 10 8 6 4 2 10 8 6 4 2 2 0 Q 2 4 6 8 10 x 4 6 8 10 976 Chapter Circles and Parabolas
Problem 2 Transforming Equations to Identify Key Characteristics By expanding the binomial terms of the standard form of a circle, (x 2 h) 2 1 (y 2 k) 2 5 r 2, the equation can be written as x 2 2 2hx 1 h 2 1 y 2 2 2ky 1 k 2 5 r 2, or x 2 1 y 2 2 2hx 2 2ky 1 h 2 1 k 2 5 r 2 Remember, h and k are constants; they represent the center of the circle. The equation for a circle in general form is Ax 2 1 Cy 2 1 Dx 1 Ey 1 F 5 0, where A, C, D, and E are constants, A 5 C, and x fi y. In order to identify the center and the radius of a circle written in general form, it is necessary to rewrite the equation in standard form. A mathematical process called completing the square is often necessary to rewrite the equation in standard form. You can rewrite x 2 1 y 2 2 4x 2 6y 1 9 5 0 in standard form by completing the square. First, use an algebraic transformation to remove the constant term from the variable expression. Write the resulting equation grouping the x-terms together and the y-terms together using sets of parentheses. (x 2 2 4x) 1 (y 2 2 6y) 5 29 Next, complete the square within each parenthesis. To do this, examine the first two terms of each quadratic expression. Determine what the constant term would be if each expression were a perfect square trinomial. Add those constant terms to each side of the equation. Write the resulting equation. (x 2 2 4x 1 4) 1 (y 2 2 6y 1 9) 5 29 1 4 1 9 Finally, factor the left side of the equation, which should be a perfect square trinomial. Write the resulting equation. (x 2 2) 2 1 (y 2 3) 2 5 4 1. Identify the center point and radius of the circle described by the equation in the worked example..2 Deriving the Equation for a Circle 977
2. Consider the general form of the equation 2x 2 1 2y 2 2 x 1 4y 1 2 5 0. a. How do the A and B values in this equation differ from the equation in the worked example? How can you tell this equation represents a circle? b. How does this general form of a circle equation change your strategy to rewrite it in standard form? c. Rewrite the equation in standard form. 978 Chapter Circles and Parabolas
3. Determine if each equation represents a circle. If so, describe the location of the center and the radius. a. x 2 1 y 2 2 2x 1 4y 1 4 5 0 b. x 2 1 2y 2 2 x 1 10y 1 25 5 0 c. 2x 2 1 2y 2 2 5x 1 8y 1 10 5 0 d. x 2 1 y 2 2 10x 1 12y 1 51 5 0.2 Deriving the Equation for a Circle 979
980 Chapter Circles and Parabolas
Problem 3 Match Game Cut out each circle equation. Then tape each equation into the table on the following page with its corresponding center and radius. A. (x) 2 1 (y 2 2) 2 1 5 5 10 B. (x 1 1) 2 1 (y 2 2) 2 5 25 C. (x 2 1) 2 1 (y 1 2) 2 5 10 D. x 2 1 y 2 2 2y 1 4 5 100 E. (x 2 1) 2 1 (y 2 2) 2 5 25 F. x 2 1 y 2 2 2x 1 4y 5 0 G. x 2 1 y 2 2 2x 2 4y 2 95 5 0 H. x 2 1 y 2 1 2x 2 4y 5 5 I. (x 2 1) 2 1 (y 1 2) 2 5 25 J. (x) 2 1 (y 2 2) 2 5 10 K. (x 1 1) 2 1 (y 2 2) 2 5 100 L. (x 2 1) 2 1 (y 2 2) 2 5 5 M. x 2 1 y 2 2 x 2 4y 2 5 5 0 N. (x 2 1) 2 1 (y 1 2) 2 5 100 O. (x 1 1) 2 1 (y 2 2) 2 5 5 P. x 2 1 y 2 2 4y 2 21 5 0.2 Deriving the Equation for a Circle 981
982 Chapter Circles and Parabolas
Circle Equations Center at (0, 2) Center at (1, 2) Center at (1, 22) Center at (21, 22) Radius at 5 Radius at 10 Radius at 5 Radius at 10.2 Deriving the Equation for a Circle 983
Problem 4 Twice (or Thrice) As Nice 1. Circle P is represented by the equation (x 2 4) 2 1 (y 1 1) 2 5 36. a. Determine the equation of a circle that has the same center as circle P but whose circumference is twice that of circle P. b. Determine the equation of a circle that has the same center as circle P but whose circumference is three times that of circle P. 984 Chapter Circles and Parabolas
c. Determine the equation of a circle that has the same center as circle P but whose area is twice that of circle P. d. Determine the equation of a circle that has the same center as circle P but whose area is three times that of circle P..2 Deriving the Equation for a Circle 985
2. Complete the table. Then, identify any pattern(s) you notice. Equation of Circle P r 5 6 (x 2 4) 2 1 (y 1 1) 2 5 36 2 Times the Circumference of Circle P 3 Times the Circumference of Circle P 4 Times the Circumference of Circle P 5 Times the Circumference of Circle P n Times the Circumference of Circle P 3. Complete the table. Then, identify any pattern(s) you notice. Equation of Circle P r 56 (x 2 4) 2 1 (y 1 1) 2 5 36 2 Times the Area of Circle P 3 Times the Area of Circle P 4 Times the Area of Circle P 5 Times the Area of Circle P n Times the Area of Circle P 986 Chapter Circles and Parabolas
4. Circle F is represented by the equation x 2 1 (y 2 3) 2 5 25. a. Determine the equation of a circle that has the same center as circle F but whose circumference is 10 times that of circle F. b. Determine the equation of a circle that has the same center as circle F but whose area is 10 times that of circle F. Be prepared to share your solutions and methods..2 Deriving the Equation for a Circle 987
988 Chapter Circles and Parabolas
Is That Point on the Circle? Determining Points on a Circle.3 Learning Goals In this lesson, you will: Use the Pythagorean Theorem to determine if a point lies on a circle on the coordinate plane given the circle s center at the origin, the radius of the circle, and the coordinates of the point. Use the Pythagorean Theorem to determine if a point lies on a circle on the coordinate plane given the circle s center not at the origin, the radius of the circle, and the coordinates of the point. Use rigid motion to transform a circle about the coordinate plane to determine if a point lies on a circle s image given the pre-image s center, radius, and the coordinates of the point. Determine the coordinates of a point that lies on a circle given the location of the center point and the radius of the circle. Use the Pythagorean Theorem to determine the coordinates of a point that lies on a circle. Beginning in about the 1970s, people in many different countries began reporting formations formed in fields, created by flattening down crops in certain ways. These came to be known as crop circles. At first, people thought that weather or even aliens were creating these formations, but it turned out that groups of people would go into fields at night and create the crop circles themselves. Many of these formations are extremely complex and beautiful. 989
Problem 1 Identifying Points on a Circle In this problem, you will continue to explore the connection between the Pythagorean Theorem and circles. Consider circle A with its center point located at the origin and point P (5, 0) on the circle as shown. 10 8 6 4 y 2 10 8 6 4 A (0, 0) 2 2 0 2 4 P (5, 0) 6 8 10 x 4 6 8 10 1. Use the axes to plot three additional points on circle A and label the coordinates for each point. There are an infinite number of points located on circle A. To determine the coordinates of other points located on circle A, you can use the Pythagorean Theorem. 2. Use the Pythagorean Theorem to determine if point B (4, 3) lies on circle A, and then explain your reasoning. 990 Chapter Circles and Parabolas
3. Consider circle D centered at the origin with a diameter of 16 units as shown. Use the Pythagorean Theorem to determine if point H (5, 38 ) lies on circle D, and then explain your reasoning. y 10 8 6 H (5, 38) 4 10 8 6 4 2 D (0, 0) 2 2 0 2 4 6 P (8, 0) x 8 10 4 6 8 10.3 Determining Points on a Circle 991
4. Consider circle E centered at the origin with a diameter of 34 units as shown. a. Verify that point J (8, 15) lies on circle E. Explain your reasoning. y 20 16 J (8, 15) 12 8 4 20 16 12 8 4 4 0 4 8 12 16 20 x 8 12 16 20 b. Use symmetry to determine 3 more points on circle E. 992 Chapter Circles and Parabolas
5. Consider circle G with its center point located at (3, 0) and point M (3, 2) on the circle. Determine whether each point lies on circle G, and then explain your reasoning. J (4.5, 3 2 ) P (4, 3 ) 5 4 3 2 5 4 3 2 1 1 1 y 0 1 M (3, 2) G (3, 0) 2 3 4 5 x 2 3 4 5 6. Consider Elizabeth s statement about additional points on circle G. Elizabeth So, only one of those points was located on circle G. I can use that point and what I know about symmetry in a circle to identify other points on circle G. Justify Elizabeth s reasoning and identify additional points on circle G..3 Determining Points on a Circle 993
Problem 2 Oh No! The Origin Is NOT at the Center! Scientists are working on a new circular mirror for an orbiting telescope. They must install eight mounting brackets around the mirror. Once ready, the mirror will be launched into space and astronauts will be required to install the mirror with bolts. 1. Circle C with its center point located at (2, 5) and a radius of 3 units represents the telescope s new mirror. The location of one of the mounting brackets is shown. y 9 6 5 4 3 C (2, 5) There is no room for error on this job. So use exact values, not approximations. 1 3 2 1 0 1 2 1 3 4 5 6 7 x a. Determine the coordinates of the mounting bracket shown. b. Use symmetry to identify the locations of the other seven mounting brackets on the mirror. 994 Chapter Circles and Parabolas
2. Sal listens to two radio stations. Station WPOP plays Top 40 hits. Station WREQ, located 50 miles due east of Station WPOP, plays oldies. Each radio station has a circular broadcast range, and Sal lives on the very edge of both of the stations ranges. Sal s house is 24 miles north and 32 miles east of Station WPOP. a. Let Station WPOP be located at the origin. Plot the location of each station and Sal s house. Then, graph each station s broadcast range. b. Use symmetry to describe the location of the other point that is on the very edge of both of the radio stations broadcast ranges. c. What area is covered by each station s broadcast range? Show your work..3 Determining Points on a Circle 995
d. How many square miles are covered by both radio stations broadcast ranges. Show your work and explain your reasoning. Be prepared to share your solutions and methods. 996 Chapter Circles and Parabolas
The Parabola Equation of a Parabola.4 Learning Goal Key Terms In this lesson, you will: Derive the equation of a parabola given the focus and the directrix. locus of points parabola focus of a parabola directrix of a parabola general form of a parabola standard form of a parabola axis of symmetry vertex of a parabola concavity The Golden Gate Bridge, which crosses the entrance to the San Francisco Bay from the Pacific Ocean, is one of the engineering marvels of the United States with its parabola shaped suspension cables. When it was built, its 4200 foot (1280 meter) suspension span was the longest in the world. Almost 2 billion vehicles have crossed the Golden Gate Bridge since its opening in 1937. 997
Problem 1 Parabolas as a Set of Points You previously studied parabolas as quadratic functions. You analyzed equations and graphed parabolas based on the position of the vertex and additional points determined by using x-values on either side of the axis of symmetry. In this lesson, you will explore a parabola as a locus of points. A locus of points is a set of points that share a property. A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line. The focus of a parabola is the fixed point. The directrix of a parabola is the fixed line. 1. The center of the circle is represented by point X. The radius of the smallest circle with center X is 1 unit with the radius of each successive circle increasing by 1 unit. y 11 10 9 8 7 6 5 4 3 x 2 focus 1 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 2 3 directrix 4 5 6 7 x a. What is the distance from the star to point X, the focus? How do you know? b. What is the distance from the star to the given line, the directrix? How do you know? c. What is the relationship between the distance from the focus to the star and the distance from the directrix to the star? d. Graph 8 additional points such that the distance from a point to the focus is equal to the distance from the point to the directrix. e. Draw a parabola by connecting the points with a smooth curve. 998 Chapter Circles and Parabolas
Problem 2 Equations of Parabolas 1. A parabola is defined such that all points on the parabola are equidistant from the point (0, 2) and the line y 5 22. One point on the parabola is labeled as (x, y). Determine the equation of the parabola by completing the steps. y d 1 = d 2 8 8 6 4 (0, 2) 2 d 1 (x, y) d 2 6 4 2 0 2 4 6 8 2 y = 2 4 6 8 x a. Let d 1 represent the distance from (x, y) to (0, 2). Write an equation using the Distance Formula to represent d 1. Simplify the equation. b. Let d 2 represent the distance from (x, y) to the line y 5 22. Write an equation using the Distance Formula to represent d 2. Simplify the equation. c. What do you know about the relationship between d 1 and d 2? d. Write an equation for the parabola using Question 1, parts (a) through (c). Simplify the equation so that one side of the equation is x 2..4 Equation of a Parabola 999
The general form of a parabola centered at the origin is an equation of the form Ax 2 1 Dy 5 0 or By 2 1 Cx 5 0. The standard form of a parabola centered at the origin is an equation of the form x 2 5 4py or y 2 5 4px, where p represents the distance from the vertex to the focus. 2. Write the equation of the parabola from Question 1 in general form and in standard form. 3. What are the coordinates for the x-intercept(s) of the parabola? 4. What are the coordinates for the y-intercept(s) of the parabola? 5. How many points on the parabola have an x-coordinate of 4? Calculate the coordinates of each point. 6. How many points on the parabola have an x-coordinate of 24? Calculate the coordinates of each point. 7. How many points on the parabola have a y-coordinate of 4.5? Calculate the coordinates of each point. 1000 Chapter Circles and Parabolas
8. How many points on the parabola have a y-coordinate of 24.5? Calculate the coordinates of each point. 9. Sketch the parabola on the grid shown using the points from Questions 3 through 8. 10. Describe the symmetry of the parabola..4 Equation of a Parabola 1001
Problem 3 Key Characteristics of a Parabola 1. How many lines of symmetry exist for a parabola? The axis of symmetry of a parabola is a line that passes through the parabola and divides the parabola into two symmetrical parts that are mirror images of each other. The vertex of a parabola is a maximum or minimum point on the curve. The concavity of a parabola describes the orientation of the curvature of the parabola. A parabola can be concave up, concave down, concave right, or concave left, as shown. y y y y x x x x concave up concave right concave down concave left 2. Consider the parabola represented by the equation y 2 5 2x. a. What are the coordinates for the x-intercept(s) of the parabola? b. What are the coordinates for the y-intercept(s) of the parabola? c. How many points on the parabola have an x-coordinate of 8? Calculate the coordinates of each point. 1002 Chapter Circles and Parabolas
d. Sketch the parabola using the coordinates from parts (a) through (c). e. Is the axis of symmetry along the x-axis or the y-axis? What is the equation for the axis of symmetry? f. What is the relationship between the axis of symmetry and the parabola s equation? g. What are the coordinates of the vertex? h. Describe the concavity of the parabola. i. How is the concavity of the parabola related to the orientation of the parabola? 3. Consider the parabola represented by the equation x 2 5 9y. a. What are the coordinates for the x-intercept(s) of the parabola? b. What are the coordinates for the y-intercept(s) of the parabola?.4 Equation of a Parabola 1003
c. How many points on the parabola have a y-coordinate of 4? Calculate the coordinates of each point. d. Sketch the parabola using the coordinates from parts (a) through (c). e. Is the axis of symmetry along the x-axis or the y-axis? What is the equation for the axis of symmetry? f. What is the relationship between the axis of symmetry and the parabola s equation? g. What are the coordinates of the vertex? h. Describe the concavity of the parabola. i. How is the concavity of the parabola related to the orientation of the parabola? 1004 Chapter Circles and Parabolas
4. The standard form of a parabola with its vertex at the origin is an equation of the form x 2 5 4py or y 2 5 4px. a. What is the standard form of a parabola that has its axis of symmetry along the y-axis? b. What is the standard form of a parabola that has its axis of symmetry along the x-axis? c. What is the equation of the axis of symmetry for a parabola with a vertical orientation? d. What is the equation of the axis of symmetry for a parabola with a horizontal orientation? e. Is the concavity of a parabola with a vertical orientation described as concave up/down or concave left/right? f. Is the concavity of a parabola with a horizontal orientation described as concave up/down or concave left/right? 5. Consider the parabola represented by the equation x 2 5 27y. a. Does the parabola have a horizontal or vertical orientation? How can you tell? b. Is the axis of symmetry along the x-axis or the y-axis? What is the equation for the axis of symmetry?.4 Equation of a Parabola 1005
c. Describe the concavity of the parabola. How can you tell? d. What are the coordinates for the x-intercept(s) of the parabola? e. What are the coordinates for the y-intercept(s) of the parabola? f. In order to graph the parabola, more information is needed. What are the points on the parabola that have a y-coordinate of 3? g. Sketch the parabola. h. What are the coordinates of the vertex? 1006 Chapter Circles and Parabolas
Problem 4 Making Sense of the Constant p Consider the sketch of the parabola. Let p represent the distance from the vertex to the focus. y R Q( x, y) vertex focus x directrix 1. Label the vertex, the focus, and the directrix. 2. Label the vertex with its coordinates. 3. Label the axis of symmetry with the equation for its line. 4. Label the distance, p, on the graph. 5. Label the focus with its coordinates..4 Equation of a Parabola 1007
6. Label the directrix with the equation for its line. Explain your reasoning. 7. What is the relationship between the directrix and the axis of symmetry? 8. What is the distance from the focus to the directrix? Label this distance on the graph. 9. Write an equation for the parabola. a. Let d 1 represent the distance from point Q on the parabola to the focus. Write an equation using the Distance Formula to represent d 1. Simplify the equation. b. Line segment QR represents the perpendicular distance from point Q on the parabola to the directrix. Draw in line segment QR. What are the coordinates of point R? Label the coordinates on the graph. 1008 Chapter Circles and Parabolas
c. Let d 2 represent the distance from point Q to point R. Write an equation using the Distance Formula to represent d 2. Simplify the equation. d. What do you know about the relationship between d 1 and d 2? e. Write an equation for the parabola using parts (a) through (d). Simplify the equation so that one side of the equation is the squared term. f. What is the significance of the equation in Question 9, part (e)?.4 Equation of a Parabola 1009
Problem 5 Using the Constant p to Graph a Parabola 1. Consider the parabola represented by the equation y 2 5 20x. a. Identify the vertex of the parabola. b. Identify the axis of symmetry. What does the form of the equation tell you? What do you think the graph will look like? c. Determine the value of p. Show your work. d. Determine the coordinates of the focus. Justify your reasoning. e. Determine the equation of the directrix. Justify your reasoning. f. Graph the parabola using the information from parts (a) through (e). g. Describe the concavity of the parabola. Justify your reasoning. 1010 Chapter Circles and Parabolas
2. Consider the parabola represented by the equation x 2 5 212y. a. Identify the vertex of the parabola. b. Identify the axis of symmetry. How does this equation compare to the one in Question 1? c. Determine the value of p. Show your work. d. Determine the coordinates of the focus. Justify your reasoning. e. Determine the equation of the directrix. Justify your reasoning. f. Graph the parabola using the information from parts (a) through (e). g. Describe the concavity of the parabola. Justify your reasoning..4 Equation of a Parabola 1011
3. Sketch the parabola x 2 5 28y. Label the vertex, the axis of symmetry, the focus, and the directrix. Determine the value of p and the concavity of the parabola. What s your prediction for the shape of this parabola? 1012 Chapter Circles and Parabolas
4. Sketch the parabola y 2 5 210x. Label the vertex, the axis of symmetry, the focus, and the directrix. Determine the value of p and the concavity of the parabola. 5. Analyze each equation and its corresponding graph in Questions 1 through 4. What do you notice about the sign of the constant p and the concavity of the parabolas?.4 Equation of a Parabola 10
Problem 6 Applications of Parabolas 1. The main cables of a suspension bridge are parabolic. The parabolic shape allows the cables to bear the weight of the bridge evenly. The distance between the towers is 900 feet and the height of each tower is about 75 feet. 75 ft 900 ft Write an equation for the parabola that represents the cable between the two towers. 1014 Chapter Circles and Parabolas
2. The cross section of a satellite dish is a parabola. The satellite dish is 5 feet wide at its opening and 1 foot deep. The receiver of the satellite dish should be placed at the focus of the parabola. How far should the receiver be placed from the vertex of the satellite dish?.4 Equation of a Parabola 1015
3. Many carnivals and amusement parks have mirrors that are parabolic. When you look at your reflection in a parabolic mirror, your image appears distorted and makes you look taller or shorter depending on the shape of the mirror. The focal length of a mirror is the distance from the vertex to the focus of the mirror. Consider a mirror that is 72 inches tall with a vertex that is concave 6 inches from the top and bottom edges of the mirror. What is the focal length of the mirror? 6 inches 1016 Chapter Circles and Parabolas
Talk the Talk Determine the following for each parabola: The vertex The focus The axis of symmetry The value of p The directrix The concavity Then, graph the parabola. 1. x 2 5 18y 2. y 2 1 44x 5 0.4 Equation of a Parabola 1017
3. Complete the table. Parabola Centered at Origin y y Graph p p x p p x Equation of Parabola Orientation of Parabola Axis of Symmetry Coordinates of Vertex Concavity Coordinates of Focus Equation of Directrix Be prepared to share your solutions and methods. 1018 Chapter Circles and Parabolas
Simply Parabolic More with Parabolas.5 Learning Goal In this lesson, you will: Solve problems using characteristics of parabolas. Scientists explore deep space by using large antennas to listen for distant radio waves. Parabolic antennas amplify faint signals by using the properties of parabolas to focus them onto a receiver. The worldwide Deep Space Network also helps to keep track of exploratory spacecraft like the two Voyager spacecrafts, which have nearly left our Solar System! 1019
Problem 1 Going the Equidistance In this activity, we will use the distance formula to determine the equation of points that are equidistant from a given point (the focus) and a given line (the directrix) where the vertex is a point other than the origin. 1. Consider the graph shown. a. Determine an equation for all the points equidistant from the point (23, 25) and the line y 5 3. 10 8 6 y 4 y 3 2 d 2 12 10 8 6 4 ( 3, 5) 2 2 4 6 0 d 1 2 4 (x, y) 6 8 x d 1 d 2 8 10 1020 Chapter Circles and Parabolas
b. Complete the following table for the equation you determined in part (a). Then, graph the equation with the directrix in the grid below. Calculate the value of p, and identify and label the directrix, the focus, the vertex, and the axis of symmetry. x 27 23 4 y 2. Consider the graph shown. a. Determine the equation for all the points equidistant from the point (7, 5) and the line x 5 1. y 14 12 x 1 10 d 1 d 2 d 1 (x, y) 8 d 2 6 4 (7, 5) 2 8 6 4 2 2 4 0 2 4 6 8 10 12 x.5 More with Parabolas 1021
b. Complete the following table for the equation you determined in part (a). Then, graph the equation with the directrix in the grid. Calculate the value of p, and identify and label the directrix, the focus, the vertex, and the axis of symmetry. x 4 6 6 y The standard forms of parabolas with vertex at (h, k) are (x 2 h) 2 5 4p(y 2 k) and (y 2 k) 2 5 4p(x 2 h). 3. Rewrite the equations from Questions 1 and 2 in this form. How do these equations compare with the standard form equation of a circle: (x 2 h) 2 1 (y 2 k) 2 5 r 2? 1022 Chapter Circles and Parabolas
4. Transform each equation into standard form and: Calculate the value of p, the vertex, the focus, the equation of the directrix, and the equation of the axis of symmetry. Sketch its graph with the focus and directrix. Determine if the parabola is concave up or down, left or right. Graph and label the parabola. a. ( y 1 1) 2 5 12(x 2 3) b. y 2 1 8y 1 8x 1 16 5 0.5 More with Parabolas 1023
c. 4x 2 2 40x 1 48y 1 4 5 0 5. Write an equation in standard form for each parabola. Then, graph and label the parabola. a. A parabola with a vertex at (3, 2) and a focus at (3, 4). 1024 Chapter Circles and Parabolas
b. A parabola with a vertex at (4, 1) and a directrix at x 5 2..5 More with Parabolas 1025
6. Complete the table. Parabola y y Graph p p (h, k) (h, k) p p x x Equation of Parabola (x 2 h) 2 5 4p( y 2 k) ( y 2 k) 2 5 4p(x 2 h) Orientation of Parabola Axis of Symmetry Coordinates of Vertex Coordinates of Focus Equation of Directrix Concavity 1026 Chapter Circles and Parabolas
Problem 2 Archway The finish line of a 5K race is an archway of balloons. The archway is formed by two parabolas, one representing the top of the archway and one representing the bottom of the archway as shown. 9 ft The width of the archway on the ground is 60 feet. The height of the top of the archway is 18 feet. The height of the bottom of the archway is 9 feet. The framework of the archway consists of vertical posts 10 feet apart with posts connecting the tops and bottoms of adjacent vertical posts. Calculate the sum of the lengths of the posts by answering each question. 1. Graph the archway on the coordinate plane. Let the x-axis represent the ground. Let the vertex of each arch lie on the y-axis. 2. Determine the coordinates of each point described. Then, label each point on the coordinate plane. a. The vertex of the top of the archway. b. The vertex of the bottom of the archway. c. The points where the archway touches the ground..5 More with Parabolas 1027
3. Calculate an equation for the parabola representing the top of the archway. 4. Calculate an equation for the parabola representing the bottom of the archway. 1028 Chapter Circles and Parabolas
5. What are the x-coordinates of each vertical post? 6. Determine the coordinates representing the endpoints of the five vertical posts. Label each in the coordinate plane. 7. Calculate the length of each vertical post. 8. Determine the coordinates representing the endpoints of the four non-vertical posts..5 More with Parabolas 1029
9. Calculate the length of each non-vertical post. 10. What is the total length of all the posts in the archway? Be prepared to share your solutions and methods. 1030 Chapter Circles and Parabolas
Chapter Summary Key Terms locus of points (.4) parabola (.4) focus of a parabola (.4) directrix of a parabola (.4) general form of a parabola (.4) standard form of a parabola (.4) axis of symmetry (.4) vertex of a parabola (.4) concavity (.4).1 Classifying Polygons on the Coordinate Plane You can classify polygons inscribed in circles in the coordinate plane, using midpoints and theorems. Example nabc is inscribed in circle E with coordinates A (25, 0), B (5, 0), and C (3, 4). Classify nabc by sides and by angles. 10 8 y Slope of AC : 25 2 3 5 28 0 2 4 24 5 2 6 4 C Slope of AB : 25 2 5 5 210 0 2 0 2 5 undefined 0 A E Slope of BC : 5 2 3 0 2 4 5 2 24 5 2 1 10 8 6 4 2 0 2 2 Since the slope of AC and BC 4 are negative 6 reciprocals, the segments are perpendicular 8 and form a right angle. Therefore the triangle 10 is a right triangle. Length of AC : (25 2 3) 2 1 (0 2 4) 2 5 (28) 2 1 (24) 2 5 64 1 16 5 80 2 4 B 6 8 10 x Length of AB : (25 2 5) 2 1 (0 2 0) 2 5 (210) 2 5 100 5 10 Length of BC : (5 2 3) 2 1 (0 2 4) 2 5 2 2 1 (24) 2 5 4 1 16 5 20 Since all three sides have a different length, the triangle is scalene. 1031
.2 Using the Standard Form of the Equation of a Circle The standard form of the equation of a circle is (x 2 h) 2 1 (y 2 k) 2 5 r 2 where the center of the circle is the point (h, k) and the radius is of length r. Example A circle is expressed by the equation (x 2 2) 2 1 (y 1 3) 2 5 16. So, the center of the circle is the point (2, 23) and the circle has a radius of 4..2 Transforming the General Form of the Equation of a Circle into Standard Form The general form of the equation of a circle is Ax 2 1 Cy 2 1 Dx 1 Ey 1 F 5 0 where A, C, D, and E are constants, A 5 C, and x fi y. A method called complete the square can be used to transform the equation of a circle in general form into standard form. Example x 2 1 y 2 1 10x 1 4y 2 12 5 0 (x 2 1 10x) 1 (y 2 1 4y) 5 12 Rewrite the equation equal to the constant. x 2 1 10x 1 25 1 y 2 1 4y 1 4 5 12 1 25 1 4 Complete the square in each parentheses. (x 1 5) 2 1 (y 1 2) 2 5 41 Factor each trinomial..3 Determining if a Point Is on a Circle Given the coordinates of the center of a circle, and a point on the circle, the Pythagorean Theorem can be used to determine if another given point is on the circle. Example Given circle A with center point (25, 2), point B (22, 2) on the circle, and a point X (23, 0). Using the Pythagorean Theorem the length of AX is 8. Since AX is not equal to 3, point X is not on the circle. y 8 7 6 5 c 2 5 2 2 1 2 2 4 c 2 5 4 1 4 c 2 5 8 c 5 8 8 7 6 A 5 4 B 3 X 2 1 3 2 1 0 1 x 1032 Chapter Circles and Parabolas
.4 Determining the Equation of a Parabola A parabola is the set of all points in a plane equidistant from a fixed point, called the focus, and a fixed line, called the directrix. Example If a parabola has a focus at (2, 0) and a directrix of x 5 22, the equation of the parabola is y 2 5 8x. Distance from focus to a point on the parabola: (2 2 x) 2 1 (0 2 y) 2 Distance from directrix to the same point on the parabola: (22 2 x) 2 1 (y 2 y) 2 (2 2 x) 2 1 (0 2 y) 2 5 (22 2 x) 2 1 (y 2 y) 2 (2 2 x) 2 1 (2y) 2 5 (22 2 x) 2 (2 2 x) 2 1 (2y) 2 5 (22 2 x) 2 4 2 4x 1 x 2 1 y 2 5 4 1 4x 1 x 2 y 2 5 8x y 5 4 3 ( 2, y) (x, y) 2 1 (2, 0) 5 4 3 2 1 0 1 2 3 4 5 1 2 3 4 5 x.4 Determining Information About a Parabola with a Vertex at the Origin You can determine information about a parabola with a vertex at the origin from its equation and graph. Examples y When the equation of a parabola is in the form x 2 5 4py, then the parabola has the following characteristics: vertical and concave up or down axis of symmetry is y axis, or x 5 0 p p x vertex at (0, 0) has a focus at (0, p) directrix is at y 5 2p When the equation of a parabola is in the form y 2 5 4px, then the parabola has the following characteristics: horizontal and concave left or right axis of symmetry is x axis, or y 5 0 vertex at (0, 0) p y p x has a focus at (p, 0) directrix is at x 5 2p Chapter Summary 1033
.5 Determining Information About a Parabola with a Vertex Not at the Origin You can determine information about a parabola with a vertex not at the origin from its equation and graph. Examples When the equation of a parabola is in the form (x 2 h) 2 5 4p(y 2 k), then the parabola has the following characteristics: y vertical and concave up or down axis of symmetry is the line x 5 h vertex at (h, k) p p (h, k) has a focus at (h, k 1 p) directrix is at y 5 k 2 p x When the equation of a parabola is in the form (y 2 k) 2 5 4p(x 2 h), then the parabola has the following characteristics: y horizontal and concave left or right axis of symmetry is the line y 5 k vertex at (h, k) (h, k) p p has a focus at (h 1 p, k) directrix is at x 5 h 2 p x 1034 Chapter Circles and Parabolas