1. Hyperbolas and Circles

2. 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

3. Hyperbola
A hyperbola is also known as a rational function and is expressed as Parent function and Graph: 𝑓 𝑥 = 1 𝑥

4. 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

5. Locator Point
The locator point for this function is where the horizontal and vertical asymptotes intersect.
Therefore we use the origin, (0,0).

6. Reflects over x-axis when negative
Standard Form
𝑓 𝑥 =−𝑎 1 𝑥−ℎ +𝑘
Reflects over x-axis when negative
Vertical Translation
Horizontal Translation
(opposite direction)
Vertical Stretch or Compress
Stretch: 𝑎>1
Compress: 0<𝑎<1

7. 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’

8. 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

9. Impacts of h and k
This also means that our asymptotes can be identified as:
Vertical Asymptote: x=h
Horizontal Asymptote: y=k

10. Example #1
What is the equation for this graph?
𝑓 𝑥 = 1 𝑥−3 −2

11. Example #2
You try:
𝑓 𝑥 = 1 𝑥+4 +1

12. 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.

13. Example #3
What is the equation for this function:
𝑓 𝑥 =3 1 𝑥 +2

14. Circle

15. The equation of a circle
What characterizes every point (x, y) on the circumference of a circle?

16. Every point (x, y) is the same distance r from the center
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.

17. The center of the circle, (0,0) is its Locator Point.
Parent Function
𝑥 2 + 𝑦 2 = 𝑟 2
Where r is the radius.
The center of the circle, (0,0) is its Locator Point.

18. x² + y² = 64 (x-3)² + y² = 49 x² + (y+4)² = 25 (x+2)² + (y-6)² = 16
Examples
State the coordinates of the center and the measure of radius for each.
x² + y² = 64
(x-3)² + y² = 49
x² + (y+4)² = 25
(x+2)² + (y-6)² = 16

19. Now let’s find the equation given the graph:
x² + (y-3)² = 4²

20. Now let’s find the equation given the graph:
(x-3)² + (y-1)² = 25

21. Homework
Worksheet #6
GET IT DONE NOW!!!
ENJOY YOUR BREAK!!!