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Part Two 407
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and whose inverse f
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is a true function, so that f 1( y) = x
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for all values of x in the domain of f, and for all values of y in the range of f Based on this information, what can we conclude about the nature of the mapping that f represents between the elements of its domain and the elements of its range
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Answer 12-10
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Every element x in the domain maps to exactly one element y in the range, and every element y in the range is mapped from exactly one element x in the domain Therefore, within the specified domain and range, the mapping that f represents is a one-to-one correspondence, technically known as a bijection
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13
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Question 13-1
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Answer 13-1
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The conic sections are geometric curves representing the intersection of a plane with a double cone There are four types: the circle, the ellipse, the parabola, and the hyperbola Figure 20-4 shows generic graphs of each type of conic section in the Cartesian plane
Question 13-2
How are the conic sections generated in 3D geometry
Answer 13-2
When the plane is perpendicular to the axis of the double cone, we get a circle, as shown in Fig 20-5A When the plane is not perpendicular to the axis of the cone but the intersection curve is closed, we get an ellipse ( Fig 20-5B) When we tilt the plane just enough to open up the curve, we get a parabola ( Fig 20-5C) When we tilt the plane still more, we get a hyperbola ( Fig 20-5D)
Question 13-3
What is meant by the term eccentricity with respect to a conic section How do the eccentricity values compare for a circle, an ellipse, a parabola, and a hyperbola
Answer 13-3
Eccentricity (symbolized e) is a nonnegative real number that defines the extent to which a conic section differs from a circle Here s how the eccentricity values compare for the four types of conic section: A circle has e = 0 An ellipse has 0 < e < 1
Review Questions and Answers
Circle
Ellipse
Parabola
Hyperbola
Figure 20-4
Illustration for Question and Answer 13-1
A parabola has e = 1 A hyperbola has e > 1
Question 13-4
What s the focus of a parabola What s the directrix of a parabola How are they related
Answer 13-4
The focus of a parabola is a point in the same plane as the parabola, and the directrix is a line in that plane that does not pass through the focus On a parabola, every point is equidistant from a specific focus and a specific directrix, as shown in Fig 20-6 For any particular focus and directrix in geometric space, there exists exactly one parabola Conversely, for any particular parabola in space, there exists exactly one focus, and exactly one directrix
Question 13-5
What s the focal length of a parabola
Part Two 409
Ellipse Circle
Parabola
Hyperbola
Figure 20-5
Illustration for Question and Answer 13-2
For any point on the parabola, these distances are equal Parabola Focus
Directrix
Figure 20-6
Illustration for Question and Answer 13-4
Review Questions and Answers Answer 13-5
The focal length of a parabola is the distance between the focus and the point on the parabola closest to the focus The focal length is also equal to half the distance between the focus and the point on the directrix closest to the focus
Question 13-6
What s the standard-form general equation for a circle in the Cartesian xy plane
Answer 13-6
The standard-form general equation is (x x0)2 + (y y0)2 = r2 where x0 and y0 are real-number constants that tell us the coordinates (x0,y0) of the center of the circle, and r is a positive real-number constant that tells us the radius of the circle
Question 13-7
What s the standard-form general equation for an ellipse in a Cartesian xy plane where the x axis is horizontal and the y axis is vertical
Answer 13-7
The standard-form general equation is (x x0)2/a2 + (y y0)2/b2 = 1 where x0 and y0 are real-number constants representing the coordinates (x0,y0) of the center of the ellipse, a is a positive real-number constant that tells us the length of the horizontal semiaxis, and b is a positive real-number constant that tells us the length of the vertical semi-axis
Question 13-8
What s the standard-form general equation for a parabola that opens upward or downward in a Cartesian xy plane where the x axis is horizontal and the y axis is vertical
Answer 13-8
The standard-form general equation is y = ax2 + bx + c where a, b, and c are real-number constants, and a 0 If a > 0, the parabola opens upward If a < 0, the parabola opens downward
Question 13-9
How can we locate the coordinates (x0,y0) of the vertex point on a parabola that opens upward or downward in a Cartesian xy plane where the x axis is horizontal and the y axis is vertical
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