1. Applications A population P after t years is modeled by: a)What does P(0) represent ? b)What is the horizontal asymptote? c)What does the horizontal asymptote represent ?

2. Warm-up Determine all asymptotes for each function. 1. 2.

3. Two key things to REMEMBER… 1.A graph will only cross the x-axis at x-intercepts (zeros)! 2.A graph can not cross a vertical asymptote! Two key things to REMEMBER… 1.A graph will only cross the x-axis at x-intercepts (zeros)! 2.A graph can not cross a vertical asymptote! 3.4 Graph of a Rational Function

4. A. Sketch the graph of: a)Domain b)Reduce to lowest terms c) y- intercept: Evaluate f(0) d) x- intercepts : Zeros of numerator EvenOdd Multiplicity: Even – touches ; Odd – crosses e)VA : zeros of denominator (use reduced function) Even Multiplicity: Even (y values stay on same side of x-axis) Odd Odd (y values switch sides) f)Holes: terms which cancelled. g)HA/OA: End Behavior h)Test points each side of x=intercepts and VAs. i)Check: Does your graph pass the vertical line test? a)Domain b)Reduce to lowest terms c) y- intercept: Evaluate f(0) d) x- intercepts : Zeros of numerator EvenOdd Multiplicity: Even – touches ; Odd – crosses e)VA : zeros of denominator (use reduced function) Even Multiplicity: Even (y values stay on same side of x-axis) Odd Odd (y values switch sides) f)Holes: terms which cancelled. g)HA/OA: End Behavior h)Test points each side of x=intercepts and VAs. i)Check: Does your graph pass the vertical line test?

5. Analyzing Rational Graphs a)Domain b)Reduce to lowest terms c) y- intercept: d) x- intercepts : (and Multiplicity ) e) VA : (and Multiplicity ) f) Holes: g) HA/OA: h)Test points. i)Check:

6. Handout: 3.4 Graphing Rational Functions

7. Analyzing Rational Graphs x-int: (-1,0) y-int: (0,2/3) No symmetry Hole: (3,4/3) VA: x= -3 HA: y = 2

8. Analyzing Rational Graphs x-int: (-1,0) y-int: (0,2/3) No symmetry Hole: (3,4/3) VA: x= -3 HA: y = 2

9. 3.4 Handout

10. Analyzing Rational Graphs Analyze the graph of: x-int: None y-int: (0,-3/4) y-Axis Symmetry D: {x | x -2, 2 }

11. 3.4 Handout

12. Analyzing Rational Graphs Analyze the graph of: x-int: (-2,0) (-1,0) y-int: (0,-2) No Symmetry D: {x | x 1 }

13. 3.4 Handout

14. B. Build a rational function from information Recall for Polynomials… zerowhen c is a zero factor(x-c) is a factor Build a Rational Function with the following properties 1.VA at x= -2 and x= 2, HA at y = 2, x-intercepts at -3, 3 2.VA at x= 3, HA at y = 0, no x-intercepts

15. B. Find the equation from the graph Definition: Multiplicity of Vertical Asymptotes. ODD multiplicity of VA y-values change sign on each side of VA y-values do not change sign on each side of VA EVEN multiplicity of VA

16. Handout 3.4 #5)

17. Handout 3.4 #6

18. Handout 3.4 #7

19. 1. Direct Variation y varies directly with x is modeled by: k is the constant of proportionality 2. Inverse Variation y varies inversely as x, if there is a constant k such that:

20. 3. Joint Variation Problems y varies jointly with if there is a constant k such that: 4. Solving Variation Problems 1)Set up equation 2)Use known information to find k 3)Plug k back into 1) 4)Answer question using equation in 3)

21. Solving inequalities – true/false quiz True or False. 1.The solution set of is 2.The inequality can be solved by multiplying both sides by, resulting in the equivalent inequality True or False. 1.The solution set of is 2.The inequality can be solved by multiplying both sides by, resulting in the equivalent inequality