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All Textbook Solutions for Algebra and Trigonometry (MindTap Course List)

61E 62E 63E 64E 65E 66E 6770. Using Cramers Rule to solve a system: Solve the system using Cramers Rule. {2x+7y=136x+16y=30 68E 69E 70E 71E 72E 73E 74E 1CT 2CT 3CT 4CT 5CT 6CT 7CT 8CT 9CT 10CT 11CT 12CT 13CT 14CT 15CT 16CT 17CT TEST Only one of the following matrix has an inverse. Find the determinant of each matrix, and use the determinant to identify the one that has an inverse. Then find the inverse. A=[141020101] B=[140020301]19CT 20CT The gray square in Table 1 has the following vertices: [00],[10],[11],[01] Apply each of the three transformations given in Table 1 to these vertices and sketch the result to verify that each transformation has the indicated effect. Use c=2 in the expansion matrix and c=1 in the shear matrix.Verify that multiplication by the given matrix has the indicated effect when applied to the gray square in the table. Use c=3 in the expansion matrix and c=1 in the shear matrix. T1=[1001] Reflection in yaxis T2=[100c] Expansion or contraction in ydirection T3=[10c1] Shear in ydirection Let T=[11.501] aWhat effect does T have on the gray square in the Table 1. bFind T1. cWhat effect does T1 have on the gray square? dWhat happens to the square if we first apply T, then T1?a Let T=[3001]. What effect does T have on the gray square in Table 1? b Let S=[1002]. What effect does S have on the gray square in Table 1? c Apply S to the vertices of the square, and then apply T to the result. What is the effect of the combined transformation? d Find the product matrix W=TS. e Apply the transformation W to the square. Compare to you final result in part c. What do you notice?The figure shows three outline versions of the letter F. The second one is obtained from the first by shrinking horizontally by a factor of 0.75, and the third is obtained from the first by shearing horizontally by a factor of 0.25. a Find a data matrix D for the first letter F. b Find a transformation matrix T that transforms the first F into the second. Calculate TD, and verify that this is a data matrix for the second F. c Find the transformation matrix S that transforms the first F into the third. Calculate SD, and verify that this is a data matrix for the third F.Here is a data matrix for a line drawing: D=[012100002440] aDraw the image represented by D. bLet T=[1101]. Calculate the matrix product TD, and draw the image represented by this product. What is the effect of the transformation T? cExpress T as a product of a shear matrix and a reflection matrix. See Problem 2. 2. Verify that multiplication by the given matrix has the indicated effect when applied to the gray square in the table. Use c=3 in the expansion matrix and c=1 in the shear matrix. T1=[1001] Reflection in yaxis T2=[100c] Expansion or contraction in ydirection T3=[10c1] Shear in ydirection A parabola is the set of all points in the plane that are equidistant from a fixed point called the ______ and a fixed line called the _______ of the parabola.2E 3E 4E 5E 6E 5-10 Graphs of Parabolas Match the equation with the graphs labeled I-IV. Give reasons for your answers. x2=6y 8E 5-10 Graphs of Parabolas Match the equation with the graphs labeled I-IV. Give reasons for your answers. y28x=0 10E 11E 12E 11-24 Graphing Parabolas An equation of a parabola is given. a Find the focus, directrix, and focal diameter of the parabola. b Sketch a graph of the parabola and its directrix. y2=24x 11-24 Graphing Parabolas An equation of a parabola is given. a Find the focus, directrix, and focal diameter of the parabola. b Sketch a graph of the parabola and its directrix. y2=16x 15E 11-24 Graphing Parabolas An equation of a parabola is given. a Find the focus, directrix, and focal diameter of the parabola. b Sketch a graph of the parabola and its directrix. x=2y2 17E 18E 11-24 Graphing Parabolas An equation of a parabola is given. a Find the focus, directrix, and focal diameter of the parabola. b Sketch a graph of the parabola and its directrix. 5y=x2 11-24 Graphing Parabolas An equation of a parabola is given. a Find the focus, directrix, and focal diameter of the parabola. b Sketch a graph of the parabola and its directrix. 9x=y2 21E 22E 23E 11-24 Graphing Parabolas An equation of a parabola is given. a Find the focus, directrix, and focal diameter of the parabola. b Sketch a graph of the parabola and its directrix. 8x2+12y=0 25-30 Graphing Parabolas Use a graphing device to graph the parabola. x2=16y 26E 27E 28E 29E 30E 31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Focus: F(0,6)32E 33E 31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Focus: F(5,0)35E 31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Focus: F(112,0)31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Directrix: x=4 38E 39E 40E 41E 42E 31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Focus on the positive x-axis, 2 units away from the directrix.31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Focus on the negative y-axis, 6 units away from the directrix.45E 31-48 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given conditions. Open upward with focus 5 units away from the vertex.47E 48E 49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.51E 52E 49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.54E 49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.49-58 Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown.SKILLS 59-60 Families of Parabolasa Find equations for the family of parabolas with the given description. b Draw the graphs. What do you conclude? The family of parabolas with vertex at the origin and with directrixes y=12,y=1,y=4,andy=8.SKILLS 59-60 Families of Parabolas a Find equations for the family of parabolas with the given description. b Draw the graphs. What do you conclude? The family of parabolas with vertex at the origin, focus on the positive y-axis and with focal diameters 1,2,4and8.APPLICATIONS Parabolic Reflector A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is 12 cm. aFind an equation of the parabola. bFind the diameter d(C,D) of the opening, 20 cm from the vertex.APPLICATIONS Satellite Dish A reflector for a satellite dish is parabolic in cross section, with the receiverat the focus F. The reflector is 1ft deep and 20ft wide from rim to rim see the figure. How far is the receiver from the vertex of the parabolic reflector?APPLICATIONS Suspension Bridge In a suspension bridge the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 600 m apart, and the lowest point of the suspension cables is 150 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex. Note: This equation is used to find the length of cable needed in the construction of the bridge.64E DISCUSS WRITE: Parabolas in the Real World Several examples of the use of parabolas are given in the text. Find other situation in real life in which parabolas occur. Consult a scientific encyclopedia in the reference section of your library, or search the internet.66E CONCEPTS An ellipse is the set of all points in the plane for which the __________ of the distances from two fixed points F1 and F2 is constant. The points F1 and F2 are called the ___________ of the ellipse.2E 3E 4E 5E 6E SKILLS 5-8Graphs of Ellipses Match the equation with the graphs labeled I-IV. Give reasons for your answers. 4x2+y2=4 8E 9E 10E 11E 12E SKILLS 9-28 Graphing Ellipse An equation of an ellipse is given. a Find the vertices, foci and eccentricity of the ellipse. b Determine the lengths of the major and minor axes. c Sketch a grapg of the ellipse. x249+y225=1 SKILLS 9-28 Graphing Ellipse An equation of an ellipse is given. a Find the vertices, foci and eccentricity of the ellipse. b Determine the lengths of the major and minor axes. c Sketch a grapg of the ellipse. x29+y264=1 15E SKILLS Graphing Ellipse An equation of an ellipse is given. a Find the vertices, foci and eccentricity of the ellipse. b Determine the lengths of the major and minor axes. c Sketch a grapg of the ellipse. 4x2+25y2=100 17E 18E SKILLS 9-28 Graphing Ellipses An equation of an ellipse is given. a Find the vertices, foci, and eccentricity of the ellipse. b Determine the lengths of the major and minor axes. c Sketch a graph of the ellipse. 16x2+25y2=1600 SKILLS 9-28 Graphing Ellipses An equation of an ellipse is given. a Find the vertices, foci, and eccentricity of the ellipse. b Determine the lengths of the major and minor axes. c Sketch a graph of the ellipse. 2x2+49y2=98 21E 22E 23E 24E 9-28. Graphing Ellipses An equation of ellipse is given a Find the vertices, foci and eccentricity of ellipse. b Find the length of major and minor axes. c Sketch the graph of an ellipse. x2+4y2=1 9-28. Graphing Ellipses An equation of ellipse is given a Find the vertices, foci and eccentricity of ellipse. b Find the length of major and minor axes. c Sketch the graph of an ellipse. 9x2+4y2=1 27E 28E 29E 30E 29-34 Finding the Equation of an Ellipse Find an equation for the ellipse whose graph is shown.29-34 Finding the Equation of an Ellipse Find an equation for the ellipse whose graph is shown.33E 34E 35E 36E 35-38 Graphing Ellipses Use a graphing device to graph the ellipse. 6x2+y2=36.38E 39E 39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 40. Foci: (0,3), vertices: (0,5).41E 42E 39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 43. Foci: (0,10), vertices: (0,7).39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 44. Foci: (15,0), vertices: (6,0).45E 46E 47E 39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 48. Foci: (5,0), length of major axis: 12.39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 49. Endpoints of major axis: (10,0), distance between foci: 6.50E 51E 52E 53E SKILLS 39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Eccentricity: 0.75, foci: (1.5,0)SKILLS 39-56 Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Eccentricity: 3/2, foci on y-axis, length of major axis: 4.56E 57E 58E 59E 60E SKILLS Plus Ancillary Circle The ancillary circle of an ellipse is the circle with radius equal to half the length of the minor axis and center the same as the ellipse see the figure. The ancillary circle is thus the largest circle that can fit within an ellipse. a Find an equation for the ancillary circle of the ellipse x2+4y2=16. b For the ellipse and ancillary circle of part a, show that if (s,t) is a point on the ancillary circle, then (2s,t) is a point on the ellipse.62E 63E How Wide Is an Ellipse at a Focus? A latus rectum for an ellipse is a line segment perpendicular to the major axis at a focus, with endpoints on the ellipse, as shown in the figure. Show that the length of a latus rectum is 2b2/a for the ellipse x2a2+y2b2=1a>b APPLICATIONS Perihelion and Aphelion The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbits at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earths distance from the sun is 147,000,000km at perihelion and 153,000,000km at aphelion. Find an equation for the earths orbit. Place the origin at the center of the orbits with the sun on the x-axis.66E APPLICATIONS Lunar Orbit For an object in an elliptical orbit around the moon, the points in the orbit that are closest to and farthest from the center of the moon are called perilune and apolune, respectively. These are the vertices of the orbit. The Apollo 11 spacecraft was placed in a lunar orbit with perilune at 68 mi and apolune at 195 mi above the surface of the moon. Assuming that the moon is a sphere of radius 1075 mi, find an equation for the orbit of Apollo 11. Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the x-axis.Plywood Ellipse A carpenter wishes to construct an elliptical table top from a 4 ft by 8 ft sheet of plywood. He will trace out the ellipse using the thumbtack and string method illustrated in Figure 2 and 3. What length of string should he use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?69E 70E 71E DISCUSS Is It an Ellipse? A piece of paper is wrapped around a cylindrical bottle, and them a compass is used to draw a circle on the paper, as shown in the figure. When the paper is laid flat, is the shape drawn on the paper in ellipse? You dont need to prove your answer, but you might want to do the experiment and see what you get.CONCEPTS A Hyperbola is the set of all points in the plane for which the of the distances from two fixed points F1 and F2 is constant. The point F1 and F2 are called the of the hyperbola.2E 3E 4E 5E 6E SKILLS Graphs of Hyperbolas Match the equation with the graphs labeled IIV. Give reasons for your answers. 16y2x2=144 8E 9E 10E 11E Graphing Hyperbolas An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. x29y264=1 Graphing Hyperbolas An equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. y2x225=1 14E 15E SKILLS 926Graphing HyperbolasAn equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. x216y212=1 17E SKILLS 926Graphing Hyperbolas An equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. 25y29x2=225 SKILLS 926Graphing Hyperbolas An equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. 4y29x2=144 20E 21E SKILLS 926Graphing Hyperbolas An equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. 3y2x29=0 23E SKILLS 926Graphing Hyperbolas An equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. x23y2+12=0 SKILLS 926Graphing Hyperbolas An equation of a hyperbola is given. a Find the vertices, foci, and asymptotes of the hyperbola. b Determine the length of the transverse axis. c Sketch a graph of the hyperbola. 4y2x2=1 26E 27E 28E 29E 30E 31E 27-32. Finding the equation of a Hyperbola Find the equation of the hyperbola whose graph is shown.33E 34E 35E 36E 37-50. Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. Foci: (5,0), vertices: (3,0)38E 39E 37-50 Finding the equation of a HyperbolaFind the equation for the hyperbola that satisfies the given conditions. Foci: (6,0), vertices: (2,0)41E 42E 37-50 Finding the equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. Vertices (0,6), hyperbola passes through (5,9)37-50 Finding the equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. Vertices (2,0), hyperbola passes through (3,30).45E 46E 47E 48E 37-50 Finding the equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. Foci (5,0), length of transverse axis:6 50E 51E 52E 53E 54E 55E 56E Comet Trajectories Some comets, such as Halleys Comet, are a permanent part of the solar system, traveling in the elliptical orbits around the sun. Other comets pass through the solar system only once, following a hyperbolic path with the sun at a focus. The figure below shows the paths of such a comet. Find an equation of the path, assuming that the closest the comet comes to the sun is 2109mi and that the path the comet was taking before it neared the solar system is at a right angle to the path it continues on after leaving the solar system.58E 59E 60E Suppose we want to graph an equation in x and y. a If we replace x by x3, the graph of the equation is shifted to the _____ by 3 units. If we replace x by x+3, the graph of the equation is shifted to the ____ by 3 units. b If we replace y by y1, the graph of the equation is shifted ______ by 1 unit. If we replace y by y+1, the graph of the equation is shifted _______ by 1 unit 2E 3E 4E 5E 6E SKILLS 5-12 Graphing Shifted Ellipses An equation of an ellipse is given.a Find the center, vertices, and foci of the ellipse. b Determine the lengths of the major and minor axes. c Sketch a graph of the ellipse. x29+(y+5)225=1 8E 9E 10E 11E 12E SKILLS 13-20 Graphing Shifted Parabolas An equation of a parabola is given.a Find the vertex, focus, and directrix of the parabola. b Sketch a graph showing the parabola and its directrix. (x3)2=8(y+1)14E 15E 16E 17E 18E SKILLS 13-20 Graphing Shifted Parabolas An equation of a parabola is given.a Find the vertex, focus, and directrix of the parabola. b Sketch a graph showing the parabola and its directrix. y26y12x+33=0 SKILLS 13-20 Graphing Shifted Parabolas An equation of a parabola is given.a Find the vertex, focus, and directrix of the parabola. b Sketch a graph showing the parabola and its directrix. x2+2x20y+41=0 21E 22E 23E 24E 21-28 Graphing Shifted HyperbolasAn equation of hyperbola is given. a Find the centre, vertices, foci, and asymptotes of the hyperbola. b Sketch a graph showing the hyperbola and its asymptotes. (x+1)29(y+1)24=1 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 35-46Finding the Equation of a Shifted Conic Find an equation for the conic section with the given properties. The hyperbola with center C(1,4), vertices V1(1,3) and V2(1,11), and foci F1(1,5) and F2(1,13)38E 35-46Finding the Equation of a Shifted Conic Find an equation for the conic section with the given properties. The parabola with vertex V(3,5) and directrix y=2 40E 41E 42E 43E 44E 45E 46E 47E 48E SKILLS 47-58 Graphing Shifted Conics Complete the square to deter-mine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyper-bola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. x25y22x+20y=44 50E 51E 52E 53E 54E 47-58 Graphing Shifted Conics Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and the lengths of the major and minor axes. If it is parabola, find the vertex, focus, and directrix. If it is hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. if the equation has no graph, explain why. 4x2+16=4(y2+2x).SKILLS 47-58 Graphing Shifted Conics Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and the lengths of the major and minor axes. If it is parabola, find the vertex, focus, and directrix. If it is hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. if the equation has no graph, explain why. x2y2=10(xy)+1.57E 58E 59E 60E 59-62. Graphing Shifting Conics Use a graphing device to graph the conic. 9x2+36=y2+36x+6y 62E

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