CHAPTER 7 STUDY GUIDE
7.1.
PARABOLA
Vertical parabola – x is squared but not y.
y  a( x  h) 2  k
1.
2.
3.
4.
5.
Vertex (h, k)
If a > 0 Opens UP
If a < 0 Opens DOWN
a > 1 then parabola is SKINNY ( or a < - 1)
If - 1 < a < 1 parabola is FAT
For every vertical parabola find the following:
a)
b)
c)
d)
Find the vertex
Find the y and x intercepts
Graph the parabola
Find the axis of symmetry for parabola
Practice:
1.
y  ( x  4) 2
2.
y  x2  2
3.
y  ( x  5) 2  4
4.
y  0.5( x  1) 2  3
5.
y  2x2  4x  5
- Use completing the square method
y / 2  ( 2 x 2  4 x  5) / 2
y
1
 x2  2x  2
2
2
y
1
 2  x2  2x
2
2
y
1
 2  1  x2  2x  1
2
2
y 3
  ( x  1) 2
2 2
y
3
 ( x  1) 2 
2
2
2
y  2( x  1)  3
6.
y  3x 2  24 x  46
- Use completing the square method
Horizontal parabola – y is squared but not x.
x  a( y  k ) 2  h
1.
2.
3.
4.
5.
Vertex (h, k)
If a > 0 Opens to the right
If a < 0 Opens to the left
If a  1 parabola is SKINNY
If - 1 < a < 1 parabola is FAT
For every horizontal parabola find the following:
a)
e)
f)
g)
Find the vertex
Find the y and x intercept
Graph the parabola
Find the axis of symmetry for parabola
1. x  y 2  2
2. x  ( y  1) 2
3. x  3( y  5) 2  2
4. x  0.5( y  3) 2  3
5. x  2 y 2  2 y  3
- Use completing the square method
x  ( 2 y 2  2 y  3) /( 2)
x
3
 y2  y 
2
2
x
3
  y2  y
2 2
x
3 1
1
   y2  y 
2 2 4
4
x
5
1 2
  (y  )
2 4
2
x
1 2 5
 (y  ) 
2
2
4
1 2
1
x  2 ( y  )  2
2
2
Application of Parabola
1. If an object is thrown upward with an initial velocity of 32 ft/s, then its height after
t seconds is: h  32t  16t 2
a) Find the maximum height attained by the object.
b) Find the total time in air.
2. A toy rocket is launched from the top of the building 50 feet toll at an initial
velocity of 200 ft/s, then its height after t seconds is: h  50  200t  16t 2
a) Determine the time at which the rocket reaches it’s maximum height
b) After how many seconds will it hit the ground?
3. Question 55 on page 535.
4. Question 56 on page 535.
7.2
ELLIPSE
Ellipse with the center at the origin of the system:
x2 y2

1
a 2 b2
a b
(If a = b - circle)
(x and y are squared – both have different
positive coefficients.)
a) center at (0,0)
b) a > b major axis is x – axis
c) a < b major axis is y - axis
Ellipse with the center outside of the origin of the system:
( x  h) 2 ( y  k ) 2

1
a2
b2
d) center at (h,k)
e) a > b major axis is x – axis
f) a < b major axis is y - axis
For every ellipse:
 Find the center of ellipse
 Find x and y intercepts (four vertices)
 Graph the ellipse
Graph ellipse by finding four vertices and the center.
1. 9 x 2  y 2  81
2. 4 x 2  9 y 2  36
3. 4 x 2  25 y 2  100
- Rewrite in standard form
4.
( x  2) 2 ( y  1) 2

1
14
9
5.
( x  1) 2 ( y  3) 2

1
9
25
6. Question 21 on Page 544
7. Question 47 on Page 546
7.3.
HYPERBOLLA
Vertical hyperbola with the center at the origin of the system
x2 y2

1
a 2 b2
(x and y are squared, but one has positive and other have
negative coefficients)
Horizontal hyperbola with the center at the origin of the system
y2 x2

1
b2 a 2
(x and y are squared, but one has positive and other have
negative coefficients)
Vertical hyperbola with the center at (h, k)
( x  h) 2 ( y  k ) 2

1
a2
b2
The four corner points of the rectangle: (a, b) (a, -b) (-a, b) (-a, -b)
b
Asymptotes (diagonals of the rectangle) are: y   x
a
For every hyperbola:
 Find the center
 Find the fundamental rectangle (four vertices)
 Find two asymptotes
 Graph hyperbola
Graphing both hyperbolas using FUNDAMENTAL RECTANGLE:
Practice:
1.
x2 y2

1
52 7 2
a = 5 and b = 7
Hyperbola centered at (0,0)
The four corner points of the rectangle: (5, 7) (5, -7) (-5, 7) (-5, -7)
7
Asymptotes (diagonals of the rectangle) are: y   x
5
Rewrite the equation in standard form and graph the hyperbola by finding a and b.
2.
x2 – y2 = 9
Example of hyperbola with the center outside the origin of the system
3.
( x  3) 2 ( y  2) 2

1
16
9
a = 4, b = 3, h = -3, k = 2
Center at (-3, 2)
7.4.
CONIC SECTION
The graphs of circle, parabola, ellipse and hyperbola are called conic sections
The standard equation for all conic section is:
Ax 2  Cy 2  Dx  Ey  F  0
Summary of the graphs:
1. Parabola with a vertical axis:




y  k  a( x  h) 2
x – squared but not y
If a > 0 open upward
If a < 0 open downwards
Vertex at (h, k)
2. Parabola with a horizontal axis




y – squared but not x
If a > 0 open to the right
If a < 0 open to the left
Vertex at (h, k)
3. Circle



( x  h) 2  ( y  k ) 2  r 2
Center at (h, k)
Radius is r
x and y are both squared and both have equal positive coefficients
4. Ellipse at the center




x  h  a( y  k ) 2
x2 y2

1
a 2 b2
(a > b)
Center at (0, 0)
x – intercepts are a and – a
y – intercepts are b and - b
x and y are both squared and both have different positive coefficients
5. Ellipse




( x  h) 2 ( y  k ) 2

1
a2
b2
x2 y2

1
a 2 b2
Center at (0, 0)
x – intercepts are a and – a
vertices of a fundamental rectangle at: (a, b) (-a, b) (a, -b) (-a, -b)
x is squared and has a positive coefficient
y is squared and has a negative coefficient
8. Vertical Hyperbola centered at (0,0)





(a > b)
Center at (h, k)
x – intercepts are (h + a) and (h – a)
x – intercepts are (k + b) and (k – b)
x and y are both squared and both have different positive coefficients
7. Horizontal Hyperbola centered at (0,0)





(a > b)
Center at (0, 0)
x – intercepts are b and – b
y – intercepts are a and - a
x and y are both squared and both have different positive coefficients
6. Ellipse




x2 y2

1
b2 a 2
y2 x2

1
a 2 b2
Center at (0, 0)
y – intercepts are a and – a
vertices of a fundamental rectangle at: (b, a) (-b, a) (b, -a) (-b, -a)
x is squared and has a negative coefficient
y is squared and has a positive coefficient
( x  h) 2 ( y  k ) 2

1
a2
b2
9. Horizontal hyperbola centered at (h, k)





Center at (h, k)
x – intercepts are (h + a) and (h – a)
vertices of a fundamental rectangle at: [(h + a), (k + b)] [(h – a),(k + b)] [(h + a),(k b)] [(h – a),(k - b)]
x is squared and has a positive coefficient
y is squared and has a negative coefficient
Based on the numerical coefficients of standard equation
Ax 2  Cy 2  Dx  Ey  F  0
we can recognize type of the conic section.




If either A = 0 or C = 0, but not both, then the equation is parabola
If A  C  0
Circle

If A  C and A  C  0 
Ellipse
If A  C  0
Hyperbola

Practice:
Identify equation:
1.
2.
3.
4.
5.
( x  2) 2  ( y  3) 2  52
x  25  y
2
2
2 x 2  8 x  2 y 2  30 y  512
x 2  6x  y  0
4 x 2  8 x  9 y 2  54 y  5  84
x2
y2
 1
4
9
2
7. x  4 y  8
8. y 2  4 y  x  4
9. 3x 2  12 x  3 y 2  11
6.
[Circle]
[Hyperbola]
[Circle]
[Parabola]
[Ellipse]