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Transcript
Answer Key
Name: ____________________________________
Date: __________________
THE LOCUS DEFINITION OF A PARABOLA
COMMON CORE ALGEBRA II
The circle had a relatively easy locus definition, i.e. the collection of all points equidistant from a given point.
Parabolas have a slightly more complex definition that we will explore in this lesson.
THE LOCUS DEFINITION OF A PARABOLA
A parabola is the collection of all points equidistant from a fixed point (known as its focus) and a fixed line
(known as its directrix).
1 2
x  1 is shown graphed below with selected points shown. For this parabola,
4
y
its focus is the point  0, 2  and its directrix is the x-axis.
Exercise #1: The parabola y 
(a) How far is the turning point  0, 1 from both the focus and
 6, 10 
directrix? How far is the point  2, 2  from both?
The point (0, 1) is one unit from both
directrix and focus and the point (2, 2) is
two units away from both.
 4, 5 
Focus
(b) Use the distance formula to verify that the point  4, 5  is
 2, 2 
the same distance away from the focus and directrix. Draw
line segments from the focus and directrix to this point to
visualize the distance. Repeat for the point  6, 10 
Clearly the point (4, 5) is 5 units away from the
directrix (the x-axis). To see how far it is from the
focus at (0, 2) we use the distance formula:
D
 4  0
2
Directrix
Clearly the point (6, 10) is 10 units away from the
directrix (the x-axis). To see how far it is from the
focus at (0, 2) we use the distance formula:
  5  2   16  9  25  5
2
D
 6  0
2
  10  2   36  64  100  10
2
(c) Use the distance formula to show that the equation of this parabola is y 
definition of a parabola.
We will pick a generic point on the
parabola and call it  x , y  . It is y-units
away from the directrix (the x-axis) and
we can set up its distance away from (0, 2)
as follows:
D
 x  0
2
  y  2
2
x
Due to the locus definition of a
parabola, we can now set these
two distances away to get the
equation:
y
 x  0
2
  y  2
2
1 2
x  1 based on the locus
4
And now some algebraic
manipulation:
y2   x  0   y  2
2
y2  x2  y2  4 y  4
0  x2  4 y  4
4 y  x2  4
y
x2  4 1 2
 x 1
4
4
COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA – LESSON #11
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
2
The algebra for finding the equation of a parabola based on its focus and directrix can be challenging, but you
know all of it from previous work you have done. Just be careful with each step.
Exercise #2: Consider a parabola whose focus is the point  0, 7  and whose directrix is the line y  3 .
(a) Sketch a diagram of the parabola below and
identify its turning point.
y
(b) Determine the equation of the parabola using
the locus definition.
Now with our generic point
 x  0   y  7
2
2
2
 y  3   x  0   y  7
2
y3
y3
2
y 2  6 y  9  x 2  y 2  14 y  49
2
 0, 5 
the
distance to the directrix is given by the
expression y  3 so:
 x, y 
 0, 7 
 x, y 
6 y  9  x 2  14 y  49
4
8 y  9  x 2  49
8 y  x 2  40
1
y  x2  5
8
y3
x
Any line and any point not on the line when used as the focus and directrix define a parabola. The most
challenging type of problem we will tackle in this course will be finding the equation of a parabola whose focus
point is not on one of the two axes. We will, however, stick with horizontal lines as our directrices.
Exercise #3: Determine the equation of the parabola whose focus is the point  4, 1 and whose directrix is the
horizontal line y  3 . First, draw a diagram that shows the parabola, then carefully use the distance formula to
derive its equation.
y
 x  4    y  1
2
2
2
 y  3    x  4    y  1
y3
 x, y 
2
2
y 2  6 y  9  x 2  8 x  16  y 2  2 y  1
6 y  9  x 2  8 x  2 y  17
8 y  9  x 2  8 x  17
 4, 1
y3
x
8 y  x2  8x  8
1
y  x2  x  1
8
y  3
COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA– LESSON #11
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
Answer Key
Name: ____________________________________
Date: __________________
THE LOCUS DEFINITION OF A PARABOLA
COMMON CORE ALGEBRA I HOMEWORK
FLUENCY
1. Fill in the following locus definition of a parabola with one of the words shown listed below. Words may be
used more than once.
point, line, equidistant, directrix, collection, focus
equidistant
collection
A parabola is the ____________________ of all points ______________________ from a fixed
line
point
____________________ and a fixed ______________________.
point
focus
line
directrix
The fixed _____________________ is known as the parabola's _______________________.
The fixed _____________________ is known as the parabola's _______________________.
2. The parabola whose equation is y 
1 2
x  2 is shown graphed on the grid below. Its directrix is the x-axis.
8
y
(a) Explain why the focus must be the point  0, 4  .
Label this point on the diagram.
The focus and directrix must be equidistant from the
turning point at (0, 2). The directrix is the x-axis so the
focus must be 2 units above the turning point at (0, 4).
(b) How far is the point  4, 4  from both the focus
and the directrix?
The point (4, 4) is four units from both the
focus and directrix.
(c) Show that the point  8, 10  is equidistant from
the focus and directrix.
The point (8, 10) is 10 units from the
directrix. To find its distance from the
focus, we have to use the distance
formula:
D
 8  0
2
  10  4 
x
Directrix
(d) Using the locus definition of a parabola, show
1
that the equation is y  x 2  2 .
8
 x  0   y  4
2
2
y2   x  0   y  4

y
y 2  x 2  y 2  8 y  16
2
 64  36  100  10
0  x 2  8 y  16
8 y  x 2  16
y
1 2
x 2
8
COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA – LESSON #11
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
2
2. Consider parabola that is the collection of all points equidistant from the point  0, 8  and the line y  2 .
(b) Draw a diagram of this parabola and label its
turning point on the diagram below.
(a) Give each of the following:
y2
Directrix: ___________________
y
 0, 8 
Focus: _____________________
 0, 8 
(c) Find the equation of this parabola using the locus definition.
 x, y 
y2
 0, 5 
 x  0   y  8
2
2
2
 y  2   x  0   y  8
y2
2
2
y2
x
y 2  4 y  4  x 2  y 2  16 y  64
4 y  4  x 2  16 y  64
12 y  4  x 2  64
12 y  x 2  60
1 2
y
x 5
12
3. Parabolas can be constructed using the classic geometric tools of a compass and a straightedge. The circles
below represent all the points equidistant from the focus  0, 4  . Given this focus point and a directrix of the
y
x-axis, do the following.
(a) Draw in the horizontal lines y  2, y  3, y  4, y  5, y  6 ,
y  7, and y  8 . These lines represent points that are a fixed
distance away from the x-axis (the directrix).
(b) Plot points at the intersections of the lines your drew in (a)
with the circles that are the same distance from the focus.
Connect with a smooth parabolic curve.
COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA– LESSON #11
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
x