1. Section 9.2 Graphing Simple Rational Functions

2. Basic Curve What does look like? y x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 -0.125 -0.25 -0.5 -2 2 1 0.5 0.25 0.125

3. Let’s look at some graphs! y x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 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 x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 Let’s look at more graphs! As the number on top becomes a larger negative, the branches in quadrants II and IV widen.

5. Let’s look at even more! y x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 Notice that the branches never touch the dotted line. The dotted line the branches never touch is called an ‘asymptote.’ x = 3x = -4 Notice that the number on the bottom will translate the graphs left or right (in the opposite direction). y = 0

6. How ‘bout a few more! y x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 y = -3 y = 4 Notice that the number on the right will translate the graphs up or down (in that direction). x = 0

7. 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.) x: The domain is all the possible values that are allowed to go into the x.

8. 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.) y: The range is all the possible values that are allowed to come out of a function.

9. Graph and state the domain and the range! y x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 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 = 5 x = -6

10. y x 10-8-426 -10 -6 -2 4 8 -10-64 10 -4 6 -108 2 -2 10 -8 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 = 2 x = 4 Graph and state the domain and the range!