Section 10.4 - Conics and Calculus
Parabolas—A parabola is the set of all points (x, y) in the plane that are equidistant from a fixed line (directrix) and a fixed
point (focus).
The Standard Form of the equation of a parabola with vertex (h, k) and directrix y  k  p is ( x  h) 2  4 p( y  k )
For this type, the axis is vertical (i.e. the parabola opens up or down)
For a directrix x  h  p , the equation is ( y  k ) 2  4 p( x  h) . Here the axis is horizontal, or the parabola opens to the side
Note: For parabolas, the vertex is always p units from both the focus and the directrix. If you know two pieces of
information from among vertex, focus, directrix, p, you can find everything there is to know about a parabola.
Also note that general form for a parabola is y  ax 2  bx  c (opens up or down) or x  ay 2  by  c (opens sideways).
When given in this form a 
1
and you can find the vertex using either College Algebra or Calc I techniques
4p
For each of the examples below, find the vertex, focus, and directrix of the parabola, and sketch the graph.
#1
#2
(𝑥 − 1)2 + 8(𝑦 + 2) = 0
𝑦2 + 𝑥 + 𝑦 = 0
Ellipse—An ellipse is the set of all points (x,y) the sum of whose distances from two distinct fixed points (foci) is constant.
The standard form for the equation of an ellipse with center (h,k) looks like
x  h 2   y  k 2  1
(horizontal major axis)
a2
b2
 x  h 2   y  k 2
b2
a2
1
(vertical major axis)
Facts:
c 2  a 2  b 2 (for an ellipse a is always bigger than b except in the degenerate case where they are equal and you
have a circle. So, to determine whether you have a vertical or horizontal major axis determine whether the bigger denominator
is under the x 2 term (horizontal major) or the y 2 term (vertical major)
c measures the distance from the center to a focus
a measures the distance from the center to a vertex (endpoint of the major axis)
b measures the distance from the center to an endpoint of the minor axis.
c
Eccentricity e  is a measure of the roundness or elongation of the ellipse. If e is close to 0, the ellipse is more circular in
a
shape. If e is close to 1 it is more elongated.
#3
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch the graph.
9𝑥 2 + 4𝑦 2 + 36𝑥 − 24𝑦 + 36 = 0
#4
Write the equation of an ellipse with vertices at (3,1) and (3,9) and that has a minor axis of length 6.
Hyperbolas—The set of all points in a plane, the difference of whose distances from two fixed points in the plane (the foci)
is a positive constant.
Standard forms for the equation of a hyperbola with center (h,k) look like
Or
 y  k 2  x  h 2
a2
b2
x  h 2   y  k 2
a2
b2
 1 (horizontal transverse axis)
 1 (vertical transverse axis)
Facts: c 2  a 2  b 2 where c is the distance from center to a focus, a is the distance from center to a vertex (endpoint of
transverse axis) and b is the distance from c to an endpoint of the conjugate axis.
b
If you have a horizontal transverse axis the asymptotes are y  k   x  h  . If the transverse axis is vertical the asymptotes
a
a
are y  k   x  h 
b
Examples:
#5 Find the center, foci, asymptotes, and vertices of the hyperbola, and sketch the graph.
𝑥 2 𝑦2
−
=1
25 9
#6
#7
Find an equation of the hyperbola with vertices at (2, ±3) and foci at (2, ±5).
Find all important features and sketch the graph:
3(𝑦 − 1)2 = 6 + 2(𝑥 + 1)2