Hyperbolas and Circles - navimath · 12/12/2013 · Hyperbola Characteristics The characteristics...
Transcript of Hyperbolas and Circles - navimath · 12/12/2013 · Hyperbola Characteristics The characteristics...
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 =1
𝑥
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Hyperbola Characteristics
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
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Locator Point
The locator point for this function is where the horizontal and vertical asymptotes intersect. Therefore we use the origin, (0,0).
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Standard Form
𝑓 𝑥 = −𝑎1
𝑥 − ℎ+ 𝑘
Reflects over x-axis when negative
Vertical Stretch or Compress Stretch: 𝑎 > 1
Compress: 0 < 𝑎 < 1
Horizontal Translation (opposite direction)
Vertical Translation
Impacts of h and k
Based on the graph at the right what inputs/outputs can our function never produce? This point is known as the hyperbolas ‘hole’
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Impacts of h and k
The coordinates of this hole are actually the values we cannot have in our domain and range. Domain: all real numbers for 𝑥 ≠ ℎ Range: all real numbers for 𝑦 ≠ 𝑘
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Impacts of h and k
This also means that our asymptotes can be identified as: Vertical Asymptote: x=h Horizontal Asymptote: y=k
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Example #1
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12
-10-9-8-7-6-5-4-3-2-1
123456789
1011
x
y
What is the equation for this graph?
𝑓 𝑥 =1
𝑥 − 3− 2
Example #2
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-4
-3
-2
-1
1
2
3
4
5
6
7
x
y
(-3,2)
You try:
𝑓 𝑥 =1
𝑥 + 4+ 1
Impacts of a
Our stretch/compression factor will once again change the shape of our function. The multiple of the factor will will determine how close our graph is to the ‘hole’ The larger the a value, the further away our graph will be. The smaller the a value , the closer our graph will be.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
Example #3
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11
-10-9-8-7-6-5-4-3-2-1
123456789
x
y
(3,3)
What is the equation for this function:
𝑓 𝑥 = 31
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.
Where r is the radius. The center of the circle, (0,0) is its
Locator Point.
𝑥2 + 𝑦2 = 𝑟2
Parent Function
Examples
1) x² + y² = 64
2) (x-3)² + y² = 49
3) x² + (y+4)² = 25
4) (x+2)² + (y-6)² = 16
State the coordinates of the center and the measure of radius for each.
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-4
-3
-2
-1
1
2
3
4
5
6
7
8
x
y
x² + (y-3)² = 4²
Now let’s find the equation given the graph:
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-4
-3
-2
-1
1
2
3
4
5
6
7
x
y
(-2,1)
(x-3)² + (y-1)² = 25
Now let’s find the equation given the graph:
Homework
Worksheet #6 GET IT DONE NOW!!! ENJOY YOUR BREAK!!!