Hyperbolas and Circles

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This content discusses the characteristics and transformations of hyperbolas and circles, including their equations, graphs, asymptotes, and locator points. It explains standard forms and the impacts of h and k values on the functions. Explore the concepts with examples and detailed explanations.

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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: ? ? =1 ? y 4 3 2 1 x -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4

Hyperbola Characteristics y 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 x -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4

Locator Point y The locator point for this function is where the horizontal and vertical asymptotes intersect. 4 3 2 1 x -4 -3 -2 -1 1 2 3 4 -1 -2 Therefore we use the origin, (0,0). -3 -4

Standard Form 1 ? ? = ? + ? ? Vertical Translation Reflects over x-axis when negative Horizontal Translation (opposite direction) Vertical Stretch or Compress Stretch: ? > 1 Compress: 0 < ? < 1

Impacts of h and k y Based on the graph at the right what inputs/outputs can our function never produce? 4 3 2 1 x -4 -3 -2 -1 1 2 3 4 -1 This point is known as the hyperbolas hole -2 -3 -4

Impacts of h and k The coordinates of this hole are actually the values we cannot have in our domain and range. y 4 3 2 1 x -4 -3 -2 -1 1 2 3 4 -1 Domain: all real numbers for ? Range: all real numbers for ? ? -2 -3 -4

Impacts of h and k y This also means that our asymptotes can be identified as: 4 3 2 1 x Vertical Asymptote: x=h -4 -3 -2 -1 1 2 3 4 -1 -2 Horizontal Asymptote: y=k -3 -4

Example #1 What is the equation for this graph? y 11 10 9 8 7 6 5 4 3 2 x 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 -1 -2 -3 -4 -5 1 -6 ? ? = 2 -7 ? 3 -8 -9 -10

Example #2 You try: y 7 6 5 4 3 2 (-3,2) 1 x -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 -1 1 ? ? = + 1 -2 ? + 4 -3 -4

Impacts of a Our stretch/compression factor will once again change the shape of our function. y 4 3 The multiple of the factor will will determine how close our graph is to the hole 2 1 x -4 -3 -2 -1 1 2 3 4 -1 The larger the a value, the further away our graph will be. -2 -3 -4 The smaller the a value , the closer our graph will be.

Example #3 What is the equation for this function: y 9 8 7 6 5 4 (3,3) 3 2 x 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 1 ?+2 -5 ? ? = 3 -6 -7 -8 -9 -10

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 ?2+ ?2= ?2 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)x + y = 64 2)(x-3) + y = 49 3)x + (y+4) = 25 4)(x+2) + (y-6) = 16