Algebra and Trigonometry 2e2-90

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TRY IT #6 Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a
diameter of 1600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker,” which is
placed 320 mm from the base.
ⓐ Find an equation that models a cross-section of the solar cooker. Assume that the vertex of
the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right
(i.e., has the x-axis as its axis of symmetry).
ⓑ Use the equation found in part ⓐ to find the depth of the cooker.
MEDIA
Access these online resources for additional instruction and practice with parabolas.
Conic Sections: The Parabola Part 1 of 2 (http://openstax.org/l/parabola1)
Conic Sections: The Parabola Part 2 of 2 (http://openstax.org/l/parabola2)
Parabola with Vertical Axis (http://openstax.org/l/parabolavertcal)
Parabola with Horizontal Axis (http://openstax.org/l/parabolahoriz)
12.3 SECTION EXERCISES
Verbal
1. Define a parabola in terms
of its focus and directrix.
2. If the equation of a parabola
is written in standard form
and is positive and the
directrix is a vertical line,
then what can we conclude
about its graph?
3. If the equation of a parabola
is written in standard form
and is negative and the
directrix is a horizontal line,
then what can we conclude
about its graph?
4. What is the effect on the
graph of a parabola if its
equation in standard form
has increasing values of
5. As the graph of a parabola
becomes wider, what will
happen to the distance
between the focus and
directrix?
Algebraic
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard
form.
6. 7. 8.
9. 10.
For the following exercises, rewrite the given equation in standard form, and then determine the vertex focus
and directrix of the parabola.
11. 12. 13.
14. 15. 16.
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17. 18. 19.
20. 21. 22.
23. 24. 25.
26. 27. 28.
29. 30.
Graphical
For the following exercises, graph the parabola, labeling the focus and the directrix.
31. 32. 33.
34. 35. 36.
37. 38. 39.
40. 41. 42.
43. 44.
For the following exercises, find the equation of the parabola given information about its graph.
45. Vertex is directrix is
focus is
46. Vertex is directrix is
focus is
47. Vertex is directrix is
focus is
48. Vertex is directrix
is focus is
49. Vertex is
directrix is focus
is
50. Vertex is directrix is
focus is
12.3 • The Parabola 1195
For the following exercises, determine the equation for the parabola from its graph.
51. 52. 53.
54. 55.
Extensions
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.
56. , Endpoints , 57. , Endpoints , 58. , Endpoints ,
59. , Endpoints
,
60. , Endpoints
,
Real-World Applications
61. The mirror in an
automobile headlight has a
parabolic cross-section
with the light bulb at the
focus. On a schematic, the
equation of the parabola is
given as At what
coordinates should you
place the light bulb?
62. If we want to construct the
mirror from the previous
exercise such that the
focus is located at
what should the
equation of the parabola
be?
63. A satellite dish is shaped
like a paraboloid of
revolution. This means that
it can be formed by
rotating a parabola around
its axis of symmetry. The
receiver is to be located at
the focus. If the dish is 12
feet across at its opening
and 4 feet deep at its
center, where should the
receiver be placed?
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64. Consider the satellite dish
from the previous exercise.
If the dish is 8 feet across
at the opening and 2 feet
deep, where should we
place the receiver?
65. The reflector in a
searchlight is shaped like a
paraboloid of revolution. A
light source is located 1
foot from the base along
the axis of symmetry. If the
opening of the searchlight
is 3 feet across, find the
depth.
66. If the reflector in the
searchlight from the
previous exercise has the
light source located 6
inches from the base along
the axis of symmetry and
the opening is 4 feet, find
the depth.
67. An arch is in the shape of a
parabola. It has a span of
100 feet and a maximum
height of 20 feet. Find the
equation of the parabola,
and determine the height
of the arch 40 feet from the
center.
68. If the arch from the
previous exercise has a
span of 160 feet and a
maximum height of 40
feet, find the equation of
the parabola, and
determine the distance
from the center at which
the height is 20 feet.
69. An object is projected so as
to follow a parabolic path
given by
where is the horizontal
distance traveled in feet
and is the height.
Determine the maximum
height the object reaches.
70. For the object from the
previous exercise, assume
the path followed is given
by
Determine how far along
the horizontal the object
traveled to reach
maximum height.
12.4 Rotation of Axes
Learning Objectives
In this section, you will:
Identify nondegenerate conic sections given their general form equations.
Use rotation of axes formulas.
Write equations of rotated conics in standard form.
Identify conics without rotating axes.
As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and
extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will
determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane
perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not
perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the
double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See Figure 1.
12.4 • Rotation of Axes 1197
Figure 1 The nondegenerate conic sections
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the
degenerate conic sections, which are shown in Figure 2. A degenerate conic results when a plane intersects the double
cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are
possible: a point, a line, or two intersecting lines.
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