Hyperbolas and Circles

Learning Targets
 To recognize and describe the
characteristics of a hyperbola and
circle.
 To relate the transformations,
reflections and translations of a
hyperbola and circle to an equation or
graph
Hyperbola
A hyperbola is also known as a rational function and is expressed as
Parent function and Graph: 𝑓 𝑥 =
y
4
3
2
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
1
𝑥
Hyperbola Characteristics
y
4
3
2
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
The characteristics of a
hyperbola are:
• Has no vertical or
horizontal symmetry
• There are both horizontal
and vertical asymptotes
• The domain and range is
limited
Locator Point
y
4
3
2
1
x
-4
-3
-2
-1
1
-1
2
3
4
The locator point for
this function is where
the horizontal and
vertical asymptotes
intersect.
-2
-3
-4
Therefore we use the
origin, (0,0).
Standard Form
1
𝑓 𝑥 = −𝑎
+𝑘
𝑥−ℎ
Reflects over x-axis
when negative
Vertical Stretch or Compress
Stretch: 𝑎 > 1
Compress: 0 < 𝑎 < 1
Vertical Translation
Horizontal Translation
(opposite direction)
Impacts of h and k
y
4
3
2
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
Based on the graph at the
right what inputs/outputs
can our function never
produce?
4
This point is known as the
hyperbolas ‘hole’
Impacts of h and k
y
4
3
2
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
The coordinates of this
hole are actually the
values we cannot have in
our domain and range.
4
Domain: all real numbers
for 𝑥 ≠ ℎ
Range: all real numbers for
𝑦 ≠𝑘
Impacts of h and k
y
This also means that our
asymptotes can be
identified as:
4
3
2
1
x
-4
-3
-2
-1
1
-1
2
3
4
Vertical Asymptote: x=h
-2
-3
-4
Horizontal Asymptote: y=k
Example #1
What is the equation for this graph?
11 y
10
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
x
1 2 3 4 5 6 7 8 9 10 11 12
𝑓 𝑥 =
1
−2
𝑥−3
Example #2
You try:
y
7
6
5
4
3
(-3,2)
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
x
1
2
3
-1
-2
-3
-4
𝑓 𝑥 =
4
5
6
7
8
9
1
+1
𝑥+4
Impacts of a
Our stretch/compression
factor will once again change
the shape of our function.
y
4
3
2
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
The multiple of the factor will
will determine how close our
graph is to the ‘hole’
4
The larger the a value, the
further away our graph will
be.
The smaller the a value , the
closer our graph will be.
Example #3
What is the equation for this function:
9 y
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
(3,3)
x
1 2 3 4 5 6 7 8 9 10 11
𝑓 𝑥 =3
1
𝑥
+2
Circle
The equation of a circle
 What characterizes every point (x, y) on the
circumference of a circle?
Every point (x, y) is the
same distance r from the
center. Therefore,
according to the
Pythagorean distance
formula for the distance of
a point from the origin.
Parent Function
𝑥2 + 𝑦2 = 𝑟2
Where r is the radius.
The center of the circle, (0,0) is its
Locator Point.
Examples
State the coordinates of the center and
the measure of radius for each.
1) x² + y² = 64
2) (x-3)² + y² = 49
3) x² + (y+4)² = 25
4) (x+2)² + (y-6)² = 16
Now let’s find the equation given the graph:
8
y
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
x
1
2
3
4
5
6
7
8
9
-1
-2
-3
-4
x² + (y-3)² = 4²
Now let’s find the equation
given the graph:
(x-3)² + (y-1)² = 25
(-2,1)
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
y
x
1 2 3 4 5 6 7 8 9
Homework
Worksheet #6
GET IT DONE NOW!!!
ENJOY YOUR BREAK!!!