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

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

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

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

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’

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

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

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

Example #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.

Example #3 What is the equation for this function: 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 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.

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.

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

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

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

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