Chapter Summary. How does Chapter 10 fit into the BIGGER PICTURE of algebra?

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1 Page of 5 0 Chapter Summar WHAT did ou learn? Find the distance between two points. (0.) Find the midpoint of the line segment connecting two points. (0.) Use distance and midpoint formulas in real-life situations. (0.) Graph and write equations of conics. parabolas (0., 0.6) circles (0.3, 0.6) ellipses (0.4, 0.6) hperbolas (0.5, 0.6) Classif a conic using its equation. (0.6) Solve sstems of quadratic equations. (0.7) Use conics to solve real-life problems. (0. 0.7) WHY did ou learn it? Find the distance a medical helicopter must travel. (p. 593) Find the diameter of a broken dish. (p. 59) Design a cit park. (p. 593) Model a solar energ collector. (p. 597) Model the region lit b a lighthouse. (p. 603) Model the shape of an Australian football field. (p. 64) Model the curved sides of a sculpture. (p. 67) Classif mirrors in a Cassegrain telescope. (p. 67) Find the epicenter of an earthquake. (p. 634) Find the area of The Ellipse at the White House. (p. 6) How does Chapter 0 fit into the BIGGER PICTURE of algebra? In Chapter 5 ou studied parabolas as graphs of quadratic functions, and in Chapter 9 ou studied hperbolas as graphs of rational functions. In a previous course ou studied circles, and possibl ellipses, in the contet of geometr. In Chapter 0 ou studied all four conic sections (parabolas, hperbolas, circles, and ellipses) as graphs of equations of the form A + B + C + D + E + F = 0. The conic sections are an important part of our stud of algebra and geometr because the have man different real-life applications. STUDY STRATEGY How did ou make and use a dictionar of graphs? Here is an eample of one entr for our dictionar of graphs, following the Stud Strateg on page 588. Dictionar of Graphs Circle with center at (h, k) and radius r (, ) (h, k) r Equation: ( º h) + ( º k) = r 64

2 Page of 5 0 Chapter Review VOCABULARY distance formula, p. 589 midpoint formula, p. 590 focus, p. 595, 609, 65 directri, p. 595 circle, p. 60 center, p. 60, 609, 65 radius, p. 60 equation of a circle, p. 60 ellipse, p. 609 verte, p. 609, 65 major ais, p. 609 co-verte, p. 609 minor ais, p. 609 equation of an ellipse, p. 609 hperbola, p. 65 transverse ais, p. 65 equation of a hperbola, p. 65 conic sections, p. 63 general second-degree equation, p. 66 discriminant, p THE DISTANCE AND MIDPOINT FORMULAS Eamples on pp EXAMPLES Let A = (º, 4) and B = (, º3). Distance between A and B = ( º ) + ( º ) = ( º ( º )) + ( º 3 º 4 ) Midpoint of AB Æ = M +, + = = = (º) +, 4 + (º3) = 0, A M 0, B Find the distance between the two points. Then find the midpoint of the line segment connecting the two points.. (º, º3), (4, ). (º5, 4), (0, º3) 3. (0, 0), (º4, 4) 4. (º, 0), (0, º8) 0. PARABOLAS Eamples on pp EXAMPLES The parabola with equation = 8 has verte (0, 0) and a horizontal ais of smmetr. It opens to the right. Note that = 4p = 8, so p =. The focus is ( p, 0)= (, 0), and the directri is =ºp =º. The parabola with equation = º8 has verte (0, 0) and a vertical ais of smmetr. It opens down. Note that = 4p = º8, so p = º. The focus is (0, p) = (0, º), and the directri is = ºp =. (, 0) 4 6 (0, ) Identif the focus and directri of the parabola. Then draw the parabola. 5. = 4 6. = º = 0 8. º = 0 Write the equation of the parabola with the given characteristic and verte (0, 0). 9. focus: (4, 0) 0. focus: (0, º3). directri: = º. directri: = 64 Chapter 0 Quadratic Relations and Conic Sections

3 Page 3 of CIRCLES Eamples on pp EXAMPLE The circle with equation + = 9 has center at (0, 0) and radius r = 9 = 3. Four points on the circle are (3, 0), (0, 3), (º3, 0), and (0, º3) = = = = 363 Write the standard form of the equation of the circle that has the given radius or passes through the given point and whose center is the origin. 7. radius: 5 8. radius: 0 9. point: (º, 3) 0. point: (, 8) 0.4 ELLIPSES Eamples on pp EXAMPLE The ellipse with equation + = 9 4 has a horizontal major ais because 9 > 4. Since 9 = 3, the vertices are at (º3, 0) and (3, 0). Since 4 =, the co-vertices are at (0, º) and (0, ). Since 9 º 4 = 5, the foci are at (º 5, 0) and ( 5, 0) = 34. º9 º 4 = º = 764 Write an equation of the ellipse with the given characteristics and center at (0, 0). 4. Verte: (0, 5), Co-verte: (, 0) 5. Verte: (4, 0), Focus: (º3, 0) 0.5 HYPERBOLAS Eamples on pp EXAMPLE The hperbola with equation º = has 4 9 a vertical transverse ais because the -term is positive. Since 4 =, vertices are (0, º) and (0, ). Since = 3, foci are (0, º 3 ) and (0, 3 ). Asmptotes are = 3 and = º 3. Chapter Review 643

4 Page 4 of continued Graph the hperbola. 6. º = º 9 = º 4 = 4 Write an equation of the hperbola with the given foci and vertices. 9. Foci: (0, º3), (0, 3) 30. Foci: (0, º4), (0, 4) 3. Foci: (º5, 0), (5, 0) Vertices: (0, º), (0, ) Vertices: (0, º), (0, ) Vertices: (º3, 0), (3, 0) 0.6 GRAPHING AND CLASSIFYING CONICS Eamples on pp EXAMPLE You can use the discriminant B º 4AC to classif a conic. For the equation + º = 0, the discriminant is B º 4AC = 0 º 4()() = º4. Because B º 4AC < 0, B = 0, and A = C, the equation represents a circle. To graph the circle, complete the square as follows. + º = 0 ( º 6 + 9) + ( + + ) = º ( º 3) + ( + ) = 4 The center of the circle is at (h, k) = (3, º) and r = 4 =. (3, ) Classif the conic section and write its equation in standard form. Then graph the equation º = º 0 + º 74 = º + 36 = º 4 º 8 º = SOLVING QUADRATIC SYSTEMS Eamples on pp EXAMPLE You can solve sstems of quadratic equations algebraicall. º º = 0 º + = 0 Solve the second equation for : = +. ( + ) º º 0( + ) + 3 = 0 Substitute into the first equation. º = 0, so = 3 or = 5. Simplif and solve. The points of intersection of the graphs of the sstem are (3, 5) and (5, 7). Find the points of intersection, if an, of the graphs in the sstem º = º 8 + = = º 6 = º 48 º + 9 = º = 0 + º º º 7 = º 35 = Chapter 0 Quadratic Relations and Conic Sections

5 Page 5 of 5 0 Chapter Test Find the distance between the two points. Then find the midpoint of the line segment connecting the two points.. (, 9), (5, 3). (º8, 3), (4, 7) 3. (º4, º), (3, 0) 4. (º, º5), (º3, 7) 5. (º, 6), (, 8) 6. (3, º), (4, 9) 7. + = = º 8 = = 5. ( º 4) = + 7. ( º 3) + ( + ) = 3. ( + 6) + ( º 7) = 4. ( º 4) º ( + 4) = 5. ( + ) º ( + ) = Write an equation for the conic section. 6. Parabola with verte at (0, 0) and directri = 5 7. Parabola with verte at (3, º6) and focus at (3, º4) 8. Circle with center at (0, 0) and passing through (4, 6) 9. Circle with center at (º8, 3) and radius 5 0. Ellipse with center at (0, 0), verte at (4, 0), and co-verte at (0, ). Ellipse with vertices at (3, º5) and (3, º) and foci at (3, º4) and (3, º). Hperbola with vertices at (º7, 0) and (7, 0) and foci at (º9, 0) and (9, 0) 3. Hperbola with verte at (4, ), focus at (4, 4), and center at (4, º) Classif the conic section and write its equation in standard form º º 3 = º + 4 = º 3 º 30 = º = 0 8. º 8 º = 0 9. º + º 6 º 6 º 4 = º = º = º = º º 3 = º = º = 0 Find the points of intersection, if an, of the graphs in the sstem = = = 8 º = º º = 0 = ARCHITECTURE The Roal Albert Hall in London is nearl elliptical in shape, about 30 feet long and 00 feet wide. Write an equation for the shape of the hall, assuming its center is at (0, 0). Then graph the equation. 40. SEARCH TEAM A search team of three members splits to search an area in the woods. Each member carries a famil service radio with a circular range of 3 miles. The agree to communicate from their bases ever hour. One member sets up base miles north of the first member. Where should the other member set up base to be as far east as possible but within range of communication? Chapter Test 645