parabolas - mcclenahanand axis of a parabola and the equation of the parabola. suppose that you draw...
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Name __________________________________
Period __________
Date:
Topic: 9-3 Parabolas
Essential Question: What is the relationship among the
focus, directrix, and vertex of a parabola?
Standard: G-GPE.2 Derive the equation of a parabola given a focus and directrix.
Objective:
√( ) ( ) √( ) ( )
( ) ( )
( ) ( )
To learn the relationship between the focus, directrix, vertex,
and axis of a parabola and the equation of the parabola.
Suppose that you draw the line and plot the point
( ). Then plot several points P that appear to be the same
distance from the line as they are from the point F.
In the diagram, the distance from point P to the line is
measured along the perpendicular PD. To find an equation of
the path of P, you use the distance formula.
( )
( )
Summary
2
Parabola:
Example 1:
Solution
The last equation is of the form ( ) . In Lesson
7-5, you learned that the graph of such an equation is a
parabola with vertex ( ) and axis . Therefore, the
graph of the set of points P is a parabola with vertex ( ) and
axis . The following general definition of a parabola is
stated in terms of distance.
The important features of a parabola are shown in the diagram
below. Notice that the vertex is midway between the focus and
the directrix.
The vertex of a parabola is ( ) and the directrix is the line
. Find the focus of the parabola.
It is helpful to make a sketch. The
vertex is 3 units above the directrix.
Since the vertex is midway between
the focus and the directrix, the focus
is ( ). Answer
A parabola is the set of all points equidistant from a fixed
line, called the directrix, and a fixed point not on the line,
called the focus.
3
Exercise 1:
The vertex of a parabola is ( ) and the directrix is the line
. Find the focus of the parabola. Draw a sketch
The vertex of a parabola is ( ) and the focus is ( ). Find
the directrix of the parabola. Draw a sketch
4
Example 2:
Solution
Find an equation of the parabola having the point ( ) as
focus and the line as directrix. Draw the curve and label
the vertex V, the focus F, the directrix, and the axis of
symmetry.
From the definition, ( ) is on the parabola if and only if
, where PD is the perpendicular distance from P to
the directrix.
√( ) ( ) √( ) ( ( ))
( ) ( )
( )
(
) ( )
( )
To plot a few points, choose convenient values of y and
compute the corresponding values of x
x y
8
0 5
2
0 1
4
Notice that the parabola in Example 2 has a horizontal axis and
an equation of the form ( ) where the point
( ) is the vertex. This is similar to the equation of a
parabola that has a vertical axis, except that the roles of x and y
are reversed.
5
Exercise 2:
Find an equation of the parabola that has focus ( ) and
directrix . Then graph the parabola.
6
Exercise 2
continued:
Find an equation of the parabola that has focus ( ) and directrix
. Then graph the parabola.
7
Exercise 2
continued:
Find an equation of the parabola that has focus ( ) and
vertex ( ). Then graph the parabola.
8
If the distance between the vertex and the focus of a parabola
is | |, then it can be shown that
in the equation of the
parabola
The parabola whose equation is
( ) , where
,
opens upward if , downward if ; has
vertex ( ),
focus ( ),
directrix ,
and
axis of symmetry .
The parabola whose equation is
( ) , where
,
opens upward if , downward if ; has
vertex ( ),
focus ( ),
directrix ,
and
axis of symmetry .
9
Example 3:
Solution
Find the vertex, focus, directrix, and axis of symmetry of the
parabola . Then graph the parabola.
Complete the square using the terms in y:
( ) ( )
( )
Comparing this equation with ( ) , you can see that
Since,
,
Thus, the parabola opens to the right (since ).
The vertex is ( ), the focus is ( ), the directrix is
, and the axis of symmetry is . The graph is
shown above.
10
Exercise 3:
Find the vertex, focus, directrix, and axis of symmetry of the
parabola
. Then graph the parabola.
11
Exercise 3,
continued:
Find the vertex, focus, directrix, and axis of symmetry of the
parabola . Then graph the parabola.
12
Exercise 3,
continued:
Find the vertex, focus, directrix, and axis of symmetry of the
parabola . Then graph the parabola.
13
Example 4:
Solution
Exercise 4:
Find an equation of the parabola that has vertex (4, 2) and
directrix .
The distance from the vertex to the
directrix is 3, so | | Since the
directrix is above the vertex, the
parabola opens downward.
Therefore the squared term is the
term with x, and c is negative. If
,then
. Thus the
equation is
( ) .
Find an equation of the parabola that has vertex (3, 1) and
directrix . Graph the result.
14
Exercise 4,
continued:
Find an equation of the parabola that has vertex (3,-5) and
directrix . Graph the result.
Class work: p 415 Oral Exercises: 1-12
Homework: p 415 Written Exercises: 2-24 even
p 411 Written Exercises: 43-46
p 415 Written Exercises: 26-36 even
p 417 Mixed Review: 1-6