LESSON 47 (9.1- 9.3) PARABOLAS AS CONIC SECTIONS AND IDENTIFYING CONICS
You should learn to:
1. Write equations of parabolas in standard form for conics.
2. Classify conics from their general equations.
Terms to know: parabola, standard form for parabolas (as conics), axis of symmetry, general form equation for a
conic
Last semester in Lesson 13 you learned that the standard form equation for a quadratic function is y  a( x  h)2  k .
This is for parabolas which open upward or downward.
When parabolas are discussed as a type of conic sections, it is more convenient to write their equations in standard
form for parabolas (as conics).
For upward or downward curving parabolas, ( y  k )  a( x  h)2 .
opens upward for a  0
opens downward for a  0
For parabolas curving (opening) left or right, ( x  h)  a( y  k )2
opens right for a  0
opens left for a  0
Notice how similar the first of these equations is to y  a( x  h)2  k .
The value of a determines whether or not the parabola is stretched or squeezed. For a > 1, the parabola is
stretched (becomes thinner). For a < 1, the parabola is squeezed (becomes wider).
The axis of symmetry for the parabola is x  h for up or down parabolas and y  k for left or right parabolas.
Example 1: Find the vertex and axis of symmetry for each of the following conics. Then sketch the graph for each.
a. y  2  3( x  1)2
y  2  3( x  1) 2
vertex: (1, 2)
Axis of Symmetry: x  1
Opens down and it's
vertically stretched
b. x 
1
1
2
( y  1)2  4 x  ( y  1)  4
2
2
vertex: (4, 1)
Axis of Symmetry: y  1
Opens right and it's
horizontally shrunk
Example 2: Write the equation for the parabola given by x  2 y 2  4 y  1 in standard conic form. Then,
find the vertex and axis of symmetry for the parabola, and sketch its graph.
x  2 y 2  4 y  1
x  1  2 y 2  4 y
x  1  2  2( y 2  2 y  1)
x  3  2( y  1) 2
Vertex: (3,1)
Axis of Symmetry: y  1
Opens left and it's
horizontally stretched
Example 3: Write an equation for the parabola whose vertex is (0, 2) and which passes through the points
(2, 0) and (2, 4) .
( x  h)  a ( y  k ) 2
( x  h)  a ( y  k ) 2
vertex: (0, 2)
( x  0)  a ( y  2) 2
x  a ( y  2) 2
another point on the parabola: (2, 0)
2  a (0  2) 2
2  a (2) 2
2  4a
2 4a

4
4
1
a
2
Equation :
1
x   ( y  2) 2
2
Now that you have worked with all four types of conic sections (circles, ellipses, hyperbolas, and parabolas), you should
learn to classify conics from their equations in general form: Ax2  Bxy  Cy 2  Dx  Ey  F  0 .
Note: We will not deal with conics with xy terms in this course, so the equations you must classify will all be in the form
Ax2  Cy 2  Dx  Ey  F  0 . The following conditions produce the corresponding types of conic sections.
1.
2.
3.
4.
A  C (and A  0)
A and C have the same signs, but A  C (and A  0)
A and C have opposite signs (and A  0)
A  0 or C  0 (but not both) - only one squared term
Circles
Ellipses
Hyperbolas
Parabolas
Example 4: Classify each equation below as that of a circle, ellipse, hyperbola, or parabola.
a. 3x2  2 y 2  4 y  3  0
b. x 2  4 y 2  2 x  3  0
c. 2 y 2  3x  2  0
3x 2  2 y 2  4 y  3  0
A and C have opposite signs
(and A  0)
x2  4 y 2  2 x  3  0
A and C have the same signs,
but A  C (and A  0)
2 y 2  3x  2  0
A  0 or C  0 (but not both)
only one squared term
Hyperbola
Ellipse
parabola
d. x2  2 x  4 y  1  0
e. 3x2  6 x  3 y 2  9  0
x2  2x  4 y 1  0
A  0 or C  0 (but not both)
only one squared term
3 x 2  6 x  3 y 2  9  0
A  C (and A  0)
parabola
circle
LESSONS 45-47 (9.1-9.3) CONIC SECTION SUMMARY AND REVIEW
When graphing conic sections, remember to classify the conic, and then pick the form for the conic section which fits
the equation that you are given. You will likely have to perform algebraic operations (including completing the
square) to put the conic into standard form.
If you are given a graph, or data related to a graph, you can build an equation for a conic section by fitting the data into
the correct form of the appropriate conic. Find and use (h, k ) first. Then, use the rest of the data to find a and b or
r values.
Circles:
( x  h)2  ( y  k )2  r 2
Ellipses:
( x  h) 2 ( y  k ) 2

1
a2
b2
Hyperbolas:
( x  h) 2 ( y  k ) 2

1
a2
b2
( y  k ) 2 ( x  h) 2

1
b2
a2
Parabolas:
Center: (h, k )
radius: r
Center: (h, k )
For a  b, major axis is horizontal (length  2a )
and minor axis is vertical (length  2b)
For b  a, major axis is vertical (length  2b)
and minor axis is horizontal (length  2a)
Center: (h, k )
Vertices: (h  a, k )
b
Asymptotes: y  k   ( x  h)
a
Center: (h, k )
Vertices: (h, k  b)
b
Asymptotes: y  k   ( x  h)
a
y  k  a( x  h)2
Vertex: (h, k )
Axis of symmetry: x  h
x  h  a( y  k ) 2
Vertex: (h, k )
Axis of symmetry: y  k
ASSIGNMENT 47
Name _______________________ Per ____
(9.1-9.3) Page 668 (37-44, 53, 55-57 (don’t do the focus and directrix), 61, 65, 73, 74, 89) + Page 689 (49-58) +
Worksheet Problems 1-8 on the next page
Conics Review Worksheet
Graph the following conics. Write the center or vertex for each.
1.
( x  2)2  ( y  3)2  4
Center _________
2.
( x  1)2 y 2

1
9
16
y
Center ________
y
x
3.
x 2  4( y  2)2  16
Center __________
x
4.
1
x  2   ( y  3)2
2
y
Vertex ________
y
x
x
Write the equation for each conic, using the given information.
5.
3
A hyperbola with center at (-2, 1), vertices at (0, 1) and (-4, 1), and asymptotes y  1   ( x  2) .
2
6.
A parabola with vertex at (2, −1), that passes through the point (3, −3) and has a vertical axis.
Write each equation in standard form. Tell what type of conic section each equation represents.
7. 3x 2  y 2  6 x  4 y  11  0
8. x  5 y 2  30 y  47