Chapter 8, Section 1
Now we’ll really switch gears and go back to parabolas.
y = ax 2  bx  c
Long before algebra, parabolas were studied as geometric objects. It turns out that there
is a very nice truth about constructing a parabola.
If you have a line and a point not on that line and you draw in the points that are
equidistant from the point (called the focus) and the line (called the directrix), you will
draw a parabola
There are many positions for parabolas, not just cupped up or down. Each of these has a
focus and a directrix:
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The shortest distance from the focus to the directrix is the perpendicular segment between
them. This is defined to have length 2p. Note that the vertex of the parabola is midway
between the focus and the end of the perpendicular segment – it is p away from the focus
and p away from the directrix…so it is equidistant between the two objects.
Let’s first look at parabolas that are cupped up and have a vertex of ( 0, 0).
focus
vertex
directrix
These will have a focus of ( 0, p) and a directrix of y =  p.
In algebra, these all have the formula y = ax 2 .
We will use the new formula
for parabolas with the vertex at (0, 0).
x 2  4py
Note the algebra:
y = ax 2
and
x 2  4py
Solve for p:
What’s the shape for parabolas with the formula x 2  4py ?
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Illustration:
Domain:
y=
1 2
x .
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Range:
Vertex:
Graph shifting instructions:
Use x 2  4py to help find the following facts:
What is p for this parabola?
Where is the focus?
Where is the directrix?
What is the focal width and where are the endpoints of the focal chord?
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Now let’s look at cupped up parabolas with the vertex at (h, k)
Our new geometric equation becomes ( x  h ) 2  4p( y  k ) .
How does this compare to y  a(x  h)2  k ?
Let’s do some algebra with both of them:
Let’s take a nice parabola with it’s equation in standard form and convert it to geometric
form:
y  3(x  1)2  12
Domain?
Range?
Vertex?
x intercepts?
y intercept?
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Sketch the graph:
Put the focus and the directrix on the graph.
Well, let’s get p  the distance from the focus to the vertex and from the vertex to the
directrix 
Where is the focus? the directrix?
What if the formula is:
(x  h)2  4p(y  k) ?
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Another problem:
What are the focus and directrix of the cupped up parabola with x intercepts 1 and 5?
Write the following equation in geometric standard form and sketch the graph.
x 2  25  16y  10x
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There are also parabolas that open right, left, and at angles.
The ones that open right or left have the geometric formula:
(y  k)2  4p(x  h)
Note that the squared variable is the y term.
If it opens left, there’s a negative sign in front of the 4.
Graph the parabola
(y  2)2  8(x  1)
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A tangent line to a parabolas is a line that shares one point with the parabola. Note that
the (x, y) point is on BOTH graphs. When graphs intersect the point of intersection is on
both graphs.
The slope of the tangent is something you’ll learn how to calculate in Calculus I; for now,
let’s look at the other details:
Given:
y  ax 2  bx  c
the slope of the tangent line at point (e, f(e)) is 2ae + b, where a and b are from the
standard form of the parabola.
Write an equation of the line tangent to y  3x 2  5x  27 at x = 1.
Give the point or points of intersection between the given parabola and the line:
f (x)  x 2  10x  10
g(x)  11x  12
note on #60: use line g(x)= 8x + 14
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Ellipses and Hyperbolas
An ellipse is another conic section – a circle is a special case of an ellipse. As with
parabolas, these were studied hundreds of years before Christ and were well known to
mathematicians.
An ellipse is the set of all points in the plane, the sum of whose distances from two fixed
points is a positive constant.
Right away: notice that the formula will have an “add” in it. (“sum”)
Illustrations of types – major axis parallel x, major axis parallel y, major axis slanted
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Standard form equations :
note with a > b
(x  h) 2 ( y  k) 2

1
a2
b2
major axis parallel x
(x  h) 2 ( y  k) 2

b2
a2
major axis parallel y
Parts of an ellipse:
one center at (h, k)
c:
c 2  a 2  b 2 , c is the distance from the center to a focus point
(addition in formula, subtraction here)
Two foci located interior to the ellipse on the major axis:
(h + c, k) and (h  c, k) if the major axis is parallel to the x axis
(h, k + c) and (h, k  c) if the major axis is parallel to the y axis
a Major axis – length 2a, x = h or y = k … depends on who gets “a”
a Minor axis – length 2b, x = h or y = k … depends on who gets “b”
vertices:
ellipse points on the major axis
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An example:
x 2 y2

1
25 49
center:
major axis and length:
minor axis and length:
vertices:
c=
foci:
Sketch:
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Another example:
Describe everything and graph:
hint: 5  2.24
4x 2  9y 2  8x  36y  4  0
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Key points to remember:
There’s a plus sign in between the fractions on the left of the equation.
The major axis is parallel to the variable (axis) for the fraction with the biggest
denominator.
c = bigger  smaller …the opposite sign from the equation.
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Chapter 8, section 3: hyperbolas
A hyperbola is the set of all points in the plane, the difference between whose distances
from two fixed points is a positive constant.
Right away: note that the equation will have a “subtract” in it!
Illustrations: transverse axis parallel to x, transverse axis parallel to y, transverse axis
slanted.
nb: the transverse axis connects the branches of the hyperbola
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Standard form equations:
Transverse axis parallel to x
Transverse axis parallel to y
(x  h) 2 ( y  k) 2

1
a2
b2
( y  k) 2 (x  h) 2

1
a2
b2
note which variable has the plus sign! A t for transverse is very like a plus sign.
Parts of a hyperbola
Center
( h, k )
Vertices:
(h + a, k) and (h  a, k)
parallel to x
(h, k + a) and (h, k  a)
parallel to y
Foci:
c2  a 2  b2
(h + c, k) and (h  c, k)
(h, k + c) and (h, k  c)
parallel to x
parallel to y
Note: subract in equation, add here
Asymptotes:
replace 1 with 0 and solve for y.
Transverse axis
joins the vertices and has length 2a
x = h or y = k
Conjugate axis
is perpendicular to the transverse axis and goes through the center
has length 2b
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Let’s look at this one:
x 2 y2

1
16 9
center
vertices
transverse axis
conjugate axis
c=
foci are
asymptote lines are
graph
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Now let’s look at this one:
y2 x 2

1
9
4
center
transverse axis
conjugate axis
c
foci
vertices
asymptote lines
graph
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What is
4x 2  9y 2  8x  36y  4  0
What is
4x 2  9 y 2  16x  54y  29  0
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