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!!!