Practice 6.1.2: Deriving the Equation of a Parabola
For problems 1–4, derive the standard equation of the parabola with the given focus and
directrix. Also, write the equation that shows how you applied the distance formula.
1. focus: (0, –3); directrix: y = 3
2. focus: (4, 0); directrix: x = –4
3. focus: (5, 0); directrix: x = 3
4. focus: (4, –8); directrix: y = 4
For problems 5 and 6, write the standard equation of the parabola with the given focus
and directrix.
5. focus: (–3, 2); directrix: x = –1
6. focus: (6.2, –1.8); directrix: y = –0.2
Use what you know about parabolas to solve problems 7–10.
7. Identify the vertex, focus, and directrix of the parabola whose equation is
1
2
y 2    x 4  .
2
continued
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Lesson 1: Deriving Equations
PRACTICE
UNIT 6 • MODELING GEOMETRY
Lesson 1: Deriving Equations
8. The diagram below shows a parabolic flashlight reflector. Light rays from the
center of the bulb at point F are reflected in parallel paths to form a beam of
light. A cross section of the reflector is a section of a parabola.
Cross-section view
F
The parabola is placed on a coordinate plane whose unit of distance is inches.
The focus F is (0.5, 0) and the directrix is x = –0.5. What is the standard equation
of the parabola?
9. The diagram below shows a railroad tunnel opening that is a parabolic curve.
Height
x-axis
P
Q
The diagram is placed on a coordinate plane so that points P and Q are on the
x-axis, the focus is (15, 10.5), and the directrix is y = 14.5. The unit of distance
on the grid is feet. What is the standard equation of the parabola? What is the
height of the opening? What is PQ, the width of the opening at ground level?
Sketch the parabola, showing the coordinates of P, Q, and the vertex.
continued
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Lesson 1: Deriving Equations
PRACTICE
UNIT 6 • MODELING GEOMETRY
Lesson 1: Deriving Equations
10. The diagram below shows a radio telescope dish. Incoming light rays reflect off of the
dish and toward the feed, located at point F. A cross section of the dish is a section of
a parabola. The feed is 48 inches above the vertex. The diameter of the dish at the top
is 10 feet.
F
Cross-section view
Depth
An astronomy student draws the parabola on a coordinate plane so that the vertex is
at the origin. What is the equation of the parabola on the plane? What is the depth of
the dish?
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Lesson 1: Deriving Equations