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Download 8. Graphing Simple Rational Functions
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Transcript
Section 9.2 Graphing Simple Rational Functions Basic Curve 1 What does y look like? x x y -8 -4 -2 -1 -0.5 0.5 1 2 4 8 -0.125 -0.25 -0.5 -1 -2 2 1 0.5 0.25 0.125 y 10 8 6 4 2 x -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 Let’s look at some graphs! 1 y x 3 y x 5 y x As the number on top becomes a larger positive, the branches in quadrants I and III widen. y 10 8 6 4 2 x -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 Each of the two pieces of the curve is called a ‘branch.’ -8 -10 The curve itself (both branches) is called a ‘hyperbola.’ Let’s look at more graphs! 1 3 5 y y y x x x As the number on top becomes a larger negative, the branches in quadrants II and IV widen. y 10 8 6 4 2 x -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 Let’s look at even more! 3 y x3 3 y x4 y 10 8 6 Notice that the number on the bottom will translate the graphs y = 0 left or right (in the opposite direction). Notice that the branches never touch the dotted line. The dotted x = -4 line the branches never touch is called an ‘asymptote.’ -10 -8 -6 -4 -2 4 2 -2 x 2 4 -4 -6 -8 -10 x=3 6 8 10 How ‘bout a few more! 3 y 3 x 3 y 4 x y 10 8 6 Notice that the number on the right will translate the graphs up or down (in that direction). y=4 4 2 x -10 y = -3 -8 -6 -4 -2 -2 -4 -6 -8 -10 x=0 2 4 6 8 10 x: The domain is all the possible values that are allowed to go into the x. Let’s talk Domain! 1 y x 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.) 3 y x3 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.) y: The range is all the possible values that are allowed to come out of a function. Let’s talk Range! 1 y x 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.) 3 y 3 x 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.) Graph and state the domain and the range! 3 y 5 x6 y 10 8 y=5 6 4 •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? 2 x -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 x = -6 2 4 6 8 10 Graph and state the domain and the range! 3 y 2 x4 •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? y 10 8 6 4 y=2 2 x -10 -8 -6 -4 -2 -2 2 -4 -6 -8 -10 x=4 4 6 8 10