Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN: 9781305071742
Author: James Stewart, Lothar Redlin, Saleem Watson
Publisher: Cengage Learning
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Maximization
Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.
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Chapter 12.1, Problem 60E

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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 , 4   and   8 .

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Chapter 12 Solutions

Algebra and Trigonometry (MindTap Course List)

Ch. 12.1 - A parabola is the set of all points in the plane...Ch. 12.1 - Prob. 2ECh. 12.1 - Prob. 3ECh. 12.1 - Prob. 4ECh. 12.1 - Prob. 5ECh. 12.1 - Prob. 6ECh. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 8ECh. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 10E
Ch. 12.1 - Prob. 11ECh. 12.1 - Prob. 12ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - Prob. 15ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - Prob. 21ECh. 12.1 - Prob. 22ECh. 12.1 - Prob. 23ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - 25-30 Graphing Parabolas Use a graphing device to...Ch. 12.1 - Prob. 26ECh. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Prob. 29ECh. 12.1 - Prob. 30ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 32ECh. 12.1 - Prob. 33ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 35ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Prob. 40ECh. 12.1 - Prob. 41ECh. 12.1 - Prob. 42ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 45ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 47ECh. 12.1 - Prob. 48ECh. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 51ECh. 12.1 - Prob. 52ECh. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 54ECh. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - SKILLS 59-60 Families of Parabolasa Find equations...Ch. 12.1 - SKILLS 59-60 Families of Parabolas a Find...Ch. 12.1 - APPLICATIONS Parabolic Reflector A lamp with a...Ch. 12.1 - APPLICATIONS Satellite Dish A reflector for a...Ch. 12.1 - APPLICATIONS Suspension Bridge In a suspension...Ch. 12.1 - Prob. 64ECh. 12.1 - DISCUSS WRITE: Parabolas in the Real World Several...Ch. 12.1 - Prob. 66ECh. 12.2 - CONCEPTS An ellipse is the set of all points in...Ch. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - SKILLS 5-8Graphs of Ellipses Match the equation...Ch. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - SKILLS 9-28 Graphing Ellipse An equation of an...Ch. 12.2 - SKILLS 9-28 Graphing Ellipse An equation of an...Ch. 12.2 - Prob. 15ECh. 12.2 - SKILLS Graphing Ellipse An equation of an ellipse...Ch. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - SKILLS 9-28 Graphing Ellipses An equation of an...Ch. 12.2 - SKILLS 9-28 Graphing Ellipses An equation of an...Ch. 12.2 - Prob. 21ECh. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - 9-28. Graphing Ellipses An equation of ellipse is...Ch. 12.2 - 9-28. Graphing Ellipses An equation of ellipse is...Ch. 12.2 - Prob. 27ECh. 12.2 - Prob. 28ECh. 12.2 - Prob. 29ECh. 12.2 - Prob. 30ECh. 12.2 - 29-34 Finding the Equation of an Ellipse Find an...Ch. 12.2 - 29-34 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 33ECh. 12.2 - Prob. 34ECh. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - 35-38 Graphing Ellipses Use a graphing device to...Ch. 12.2 - Prob. 38ECh. 12.2 - Prob. 39ECh. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 41ECh. 12.2 - Prob. 42ECh. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 45ECh. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 50ECh. 12.2 - Prob. 51ECh. 12.2 - Prob. 52ECh. 12.2 - Prob. 53ECh. 12.2 - SKILLS 39-56 Finding the Equation of an Ellipse...Ch. 12.2 - SKILLS 39-56 Finding the Equation of an Ellipse...Ch. 12.2 - Prob. 56ECh. 12.2 - Prob. 57ECh. 12.2 - Prob. 58ECh. 12.2 - Prob. 59ECh. 12.2 - Prob. 60ECh. 12.2 - SKILLS Plus Ancillary Circle The ancillary circle...Ch. 12.2 - Prob. 62ECh. 12.2 - Prob. 63ECh. 12.2 - How Wide Is an Ellipse at a Focus? A latus rectum...Ch. 12.2 - APPLICATIONS Perihelion and Aphelion The planets...Ch. 12.2 - Prob. 66ECh. 12.2 - APPLICATIONS Lunar Orbit For an object in an...Ch. 12.2 - Plywood Ellipse A carpenter wishes to construct an...Ch. 12.2 - Prob. 69ECh. 12.2 - Prob. 70ECh. 12.2 - Prob. 71ECh. 12.2 - DISCUSS Is It an Ellipse? A piece of paper is...Ch. 12.3 - CONCEPTS A Hyperbola is the set of all points in...Ch. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - SKILLS Graphs of Hyperbolas Match the equation...Ch. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - Prob. 11ECh. 12.3 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 12.3 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - SKILLS 926Graphing HyperbolasAn equation of a...Ch. 12.3 - Prob. 17ECh. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - Prob. 20ECh. 12.3 - Prob. 21ECh. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - Prob. 23ECh. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - Prob. 31ECh. 12.3 - 27-32. Finding the equation of a Hyperbola Find...Ch. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Prob. 35ECh. 12.3 - Prob. 36ECh. 12.3 - 37-50. Finding the Equation of a Hyperbola Find an...Ch. 12.3 - Prob. 38ECh. 12.3 - Prob. 39ECh. 12.3 - 37-50 Finding the equation of a HyperbolaFind the...Ch. 12.3 - Prob. 41ECh. 12.3 - Prob. 42ECh. 12.3 - 37-50 Finding the equation of a Hyperbola Find an...Ch. 12.3 - 37-50 Finding the equation of a Hyperbola Find an...Ch. 12.3 - Prob. 45ECh. 12.3 - Prob. 46ECh. 12.3 - Prob. 47ECh. 12.3 - Prob. 48ECh. 12.3 - 37-50 Finding the equation of a Hyperbola Find an...Ch. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.3 - Prob. 53ECh. 12.3 - Prob. 54ECh. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Comet Trajectories Some comets, such as Halleys...Ch. 12.3 - Prob. 58ECh. 12.3 - Prob. 59ECh. 12.3 - Prob. 60ECh. 12.4 - Suppose we want to graph an equation in x and y. a...Ch. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - SKILLS 5-12 Graphing Shifted Ellipses An equation...Ch. 12.4 - Prob. 8ECh. 12.4 - Prob. 9ECh. 12.4 - Prob. 10ECh. 12.4 - Prob. 11ECh. 12.4 - Prob. 12ECh. 12.4 - SKILLS 13-20 Graphing Shifted Parabolas An...Ch. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - Prob. 18ECh. 12.4 - SKILLS 13-20 Graphing Shifted Parabolas An...Ch. 12.4 - SKILLS 13-20 Graphing Shifted Parabolas An...Ch. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - 21-28 Graphing Shifted HyperbolasAn equation of...Ch. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - Prob. 31ECh. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - 35-46Finding the Equation of a Shifted Conic Find...Ch. 12.4 - Prob. 38ECh. 12.4 - 35-46Finding the Equation of a Shifted Conic Find...Ch. 12.4 - Prob. 40ECh. 12.4 - Prob. 41ECh. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - Prob. 44ECh. 12.4 - Prob. 45ECh. 12.4 - Prob. 46ECh. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - SKILLS 47-58 Graphing Shifted Conics Complete the...Ch. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Prob. 53ECh. 12.4 - Prob. 54ECh. 12.4 - 47-58 Graphing Shifted Conics Complete the square...Ch. 12.4 - SKILLS 47-58 Graphing Shifted Conics Complete the...Ch. 12.4 - Prob. 57ECh. 12.4 - Prob. 58ECh. 12.4 - Prob. 59ECh. 12.4 - Prob. 60ECh. 12.4 - 59-62. Graphing Shifting Conics Use a graphing...Ch. 12.4 - Prob. 62ECh. 12.4 - Prob. 63ECh. 12.4 - Prob. 64ECh. 12.4 - Prob. 65ECh. 12.4 - Prob. 66ECh. 12.4 - Orbit of a satellite A satellite is in an...Ch. 12.4 - DISCUSS: A Family of Confocal Conics Conics that...Ch. 12.5 - Suppose the x- and y-axes are routed through an...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - Prob. 6ECh. 12.5 - Prob. 7ECh. 12.5 - 3-8 Rotation of Axes Determine the XY-coordinates...Ch. 12.5 - Prob. 9ECh. 12.5 - Prob. 10ECh. 12.5 - Prob. 11ECh. 12.5 - Prob. 12ECh. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - Prob. 15ECh. 12.5 - Prob. 16ECh. 12.5 - Prob. 17ECh. 12.5 - Prob. 18ECh. 12.5 - Prob. 19ECh. 12.5 - Prob. 20ECh. 12.5 - Prob. 21ECh. 12.5 - 15-28 Graphing a Rotated Conic a Use the...Ch. 12.5 - Prob. 23ECh. 12.5 - Prob. 24ECh. 12.5 - Prob. 25ECh. 12.5 - Prob. 26ECh. 12.5 - Prob. 27ECh. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Prob. 31ECh. 12.5 - Prob. 32ECh. 12.5 - Prob. 33ECh. 12.5 - Prob. 34ECh. 12.5 - Prob. 35ECh. 12.5 - Prob. 36ECh. 12.5 - Prob. 37ECh. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.6 - CONCEPTS All conics can be described geometrically...Ch. 12.6 - Prob. 2ECh. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - Prob. 5ECh. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - Prob. 9ECh. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - Prob. 14ECh. 12.6 - Prob. 15ECh. 12.6 - Prob. 16ECh. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - Prob. 20ECh. 12.6 - Prob. 21ECh. 12.6 - Prob. 22ECh. 12.6 - Prob. 23ECh. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - Prob. 26ECh. 12.6 - Prob. 27ECh. 12.6 - Prob. 28ECh. 12.6 - Prob. 29ECh. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.6 - Prob. 32ECh. 12.6 - Prob. 33ECh. 12.6 - Prob. 34ECh. 12.6 - Prob. 35ECh. 12.6 - Prob. 36ECh. 12.6 - Prob. 37ECh. 12.6 - Prob. 38ECh. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Prob. 43ECh. 12.6 - Prob. 44ECh. 12.6 - Prob. 45ECh. 12.6 - Prob. 46ECh. 12.6 - Prob. 47ECh. 12.CR - Prob. 1CCCh. 12.CR - Prob. 2CCCh. 12.CR - a Give the geometric definition of a hyperbola. b...Ch. 12.CR - Prob. 4CCCh. 12.CR - Prob. 5CCCh. 12.CR - Prob. 6CCCh. 12.CR - Prob. 7CCCh. 12.CR - Prob. 8CCCh. 12.CR - Prob. 1ECh. 12.CR - Prob. 2ECh. 12.CR - Prob. 3ECh. 12.CR - Prob. 4ECh. 12.CR - Prob. 5ECh. 12.CR - Prob. 6ECh. 12.CR - Prob. 7ECh. 12.CR - Prob. 8ECh. 12.CR - Prob. 9ECh. 12.CR - Prob. 10ECh. 12.CR - Prob. 11ECh. 12.CR - Prob. 12ECh. 12.CR - Prob. 13ECh. 12.CR - Prob. 14ECh. 12.CR - Prob. 15ECh. 12.CR - Prob. 16ECh. 12.CR - Prob. 17ECh. 12.CR - Prob. 18ECh. 12.CR - Prob. 19ECh. 12.CR - Prob. 20ECh. 12.CR - Prob. 21ECh. 12.CR - Prob. 22ECh. 12.CR - Prob. 23ECh. 12.CR - Prob. 24ECh. 12.CR - Prob. 25ECh. 12.CR - Prob. 26ECh. 12.CR - Prob. 27ECh. 12.CR - Prob. 28ECh. 12.CR - Prob. 29ECh. 12.CR - Prob. 30ECh. 12.CR - Prob. 31ECh. 12.CR - Prob. 32ECh. 12.CR - Prob. 33ECh. 12.CR - Prob. 34ECh. 12.CR - Prob. 35ECh. 12.CR - Prob. 36ECh. 12.CR - Prob. 37ECh. 12.CR - Prob. 38ECh. 12.CR - Prob. 39ECh. 12.CR - Prob. 40ECh. 12.CR - Prob. 41ECh. 12.CR - Prob. 42ECh. 12.CR - Prob. 43ECh. 12.CR - Prob. 44ECh. 12.CR - Prob. 45ECh. 12.CR - Prob. 46ECh. 12.CR - Prob. 47ECh. 12.CR - Prob. 48ECh. 12.CR - Prob. 49ECh. 12.CR - Prob. 50ECh. 12.CR - 43-54 Indentifying and Graphic a conicDetermine...Ch. 12.CR - Prob. 52ECh. 12.CR - Prob. 53ECh. 12.CR - Prob. 54ECh. 12.CR - Prob. 55ECh. 12.CR - Prob. 56ECh. 12.CR - Prob. 57ECh. 12.CR - Prob. 58ECh. 12.CR - Prob. 59ECh. 12.CR - Prob. 60ECh. 12.CR - Prob. 61ECh. 12.CR - Prob. 62ECh. 12.CR - Prob. 63ECh. 12.CR - Prob. 64ECh. 12.CR - Prob. 65ECh. 12.CR - Prob. 66ECh. 12.CR - Prob. 67ECh. 12.CR - Prob. 68ECh. 12.CR - Prob. 69ECh. 12.CR - Prob. 70ECh. 12.CR - Prob. 71ECh. 12.CR - Prob. 72ECh. 12.CR - Prob. 73ECh. 12.CR - Prob. 74ECh. 12.CR - Prob. 75ECh. 12.CR - Prob. 76ECh. 12.CR - Prob. 77ECh. 12.CR - Prob. 78ECh. 12.CR - Prob. 79ECh. 12.CR - Prob. 80ECh. 12.CT - Prob. 1CTCh. 12.CT - Prob. 2CTCh. 12.CT - Prob. 3CTCh. 12.CT - Prob. 4CTCh. 12.CT - Prob. 5CTCh. 12.CT - Find an equation for the hyperbola with foci (0,5)...Ch. 12.CT - 7-9 Find an equation for the conic whose graph is...Ch. 12.CT - Prob. 8CTCh. 12.CT - Prob. 9CTCh. 12.CT - Prob. 10CTCh. 12.CT - 10-12 Determine whether the equation represents...Ch. 12.CT - Prob. 12CTCh. 12.CT - Prob. 13CTCh. 12.CT - Find an equation for the parabola with focus (2,4)...Ch. 12.CT - A parabolic reflector for a car headlight forms a...Ch. 12.CT - aUse the discriminant to determine whether the...Ch. 12.CT - Prob. 17CTCh. 12.FOM - Prob. 1PCh. 12.FOM - Prob. 2PCh. 12.FOM - Prob. 3PCh. 12.FOM - Prob. 4PCh. 12.FOM - Prob. 5PCh. 12.FOM - Prob. 6P
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Finding The Focus and Directrix of a Parabola - Conic Sections; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=KYgmOTLbuqE;License: Standard YouTube License, CC-BY