Transcript Section 9.3 - Shelton State Community College

Section 9.3
The Parabola
Finally, something familiar!
• The parabola is oft discussed in MTH 112, as it
is the graph of a quadratic function:
f ( x)  ax  bx  c
2
x
b
2a look familiar?
• Does
• Our discussion of the parabola will be consistent
with our discussion of the other conic sections.
The Parabola
• A parabola is the set of all points in the
plane that are the same distance from a
given point (focus) as they are from a
given line (directrix).
• It is important to note that the focus is not
a point on the directrix.
• While the directrix can be any line, we only
consider horizontal and vertical ones.
Parabolic Parts
• The axis of symmetry of a parabola is an
imaginary line perpendicular to the
directrix that passes through the focus.
• The axis of symmetry intersects the
parabola at a point called the vertex.
• Let p be the distance from the vertex to
the focus. It follows that p is also the
distance from the vertex to the directrix.
More Pictures
Equations
• The standard form of the equation of a parabola
with horizontal directrix is
x  h 
2
 4 p y  k 
• When p is positive, the parabola opens upward.
• When p is negative, the parabola opens
downward.
Equations (cont.)
• The standard form of the equation of a parabola
with vertical directrix is
y  k 
2
 4 px  h
• When p is positive, the parabola opens to the
right.
• When p is negative, the parabola opens to the
left.
Giggle, giggle
•
1.
2.
3.
•
The latus rectum is a line segment that:
passes through the focus;
Is parallel to the directrix;
Has its endpoints on the parabola.
The length of the latus rectum is 4p.
Pictures
Finally…
•
•
•
•
•
Draw the picture.
Draw the picture.
Draw the picture.
And, when all else fails….
Draw the picture.
Examples
• Find the vertex, focus and directrix, and sketch
the graph.
1. x2 = 24y
2. y2 = 40x
3. (x + 2)2 = -4(y – 1)
• Find the standard form of the equation the
parabola so described:
1. Focus is (12, 0); directrix is x = -12
2. Vertex is (3, -1); focus is (3, -2)
More Examples
• Convert to standard form by completing
the square on x:
x2 + 6x – 12y – 15 = 0
• Convert to standard form by completing
the square on y:
y2 – 12y + 16x + 36 = 0