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Section 9.2 Graphing Simple Rational Functions

Explore graphs of simple rational functions and learn about hyperbolas, branches, asymptotes, domain, and range.

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Section 9.2 Graphing Simple Rational Functions

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  1. Section 9.2 Graphing Simple Rational Functions

  2. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 Basic Curve • What does look like? -0.125 -0.25 -0.5 -1 -2 2 1 0.5 0.25 0.125

  3. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 Let’s look at some graphs! As the number on top becomes a larger positive, the branches in quadrants I and III widen. Each of the two pieces of the curve is called a ‘branch.’ The curve itself (both branches) is called a ‘hyperbola.’

  4. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 Let’s look at more graphs! As the number on top becomes a larger negative, the branches in quadrants II and IV widen.

  5. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 Let’s look at even more! Notice that the number on the bottom will translate the graphs left or right (in the opposite direction). y = 0 Notice that the branches never touch the dotted line. The dotted line the branches never touch is called an ‘asymptote.’ x = -4 x = 3

  6. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 How ‘bout a few more! y = 4 Notice that the number on the right will translate the graphs up or down (in that direction). y = -3 x = 0

  7. x: The domain is all the possible values that are allowed to go into the x. Let’s talk Domain! Recall: You can’t divide by 0! So what value of x would send in the math police? Therefore, x is allowed to be any real number, except zero. Notation (D: All Real Numbers, but x = 0.) So what value of x would make the denominator zero? Therefore, x is allowed to be any real number, except three. Notation (D: All Real Numbers, but x = 3.)

  8. y: The range is all the possible values that are allowed to come out of a function. Let’s talk Range! Since x cannot be 0, the expression 1/x cannot be 0. Therefore, y cannot be 0. Notation (R: All Real Numbers, but y = 0.) Since x cannot be 0, the expression 3/x cannot be 0. Therefore, y cannot be 0 – 3, which is –3. Notation (R: All Real Numbers, but y = -3.)

  9. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 Graph and state the domain and the range! y = 5 • We know we have a hyperbola. • How many units left/right? • How many units up/down? • What value can x not have? • What value can y not have? x = -6

  10. y Graph and state the domain and the range! 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 -10 y = 2 • We know we have a hyperbola. • How many units left/right? • How many units up/down? • What value can x not have? • What value can y not have? x = 4

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