Chapter 1 Functions and Graphs

1. Chapter 1 Functions and Graphs

2. Quick Talk • What can you tell me about functions based on your peers’ lessons? • What can you tell me about graphs based on your peers’ lessons?

3. Review about factoring • Factor these following:

4. Challenge: Could you factor these? You should feel a sudden rush of urge that these can be factored…but how... • 1) • 2)

5. Quick Talk: What does factoring allow you to find?

6. Potential Answers • Rational Zeros when y=0

7. A mathematical modelis a mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. • There are 3 types of model: • Numerical Models • Algebraic model • Graphical models

8. The most basic type: Numerical Model • Numerical Model: use of numbers or data are analyzed to gain insights into phenomena

9. Group work: Numerical Model Example #1 • The chart shows growth in the number of engineers from 1980-2000. Is the proportion of female engineers over the years increasing? Explain

10. Answer • You have to be careful with the last example, the word “proportion” means what? • Once we converted all the data into ratios, we see that there is an increase from 1980-1995, but there is a drop from 1990-1995, then from 1995-2000 it increased again. From the data, the peak is at 1990, 32.3%

11. Algebraic Model • Algebraic model uses formulas or equations to relate variable quantities associated with the phenomena being studied.

12. Group Work: Algebraic Model Example #1 Note: • Think about what are you comparing. • What “formula” can you use? Or do you have to create one?

13. Answer

14. Group Work: Algebraic Model Example #2 • You went to Gamestop, it just happens every game you buy is discounted 15%( ) off the marked price. The discount is taken at the sales counter, and then a state sales tax of 7.5% and local tax of 1% is added on. • Questions: • 1) If you have $20, could you buy a game marked at $24.99? • 2) If you are determined to spend no more than $175, what’s the maximum total value of your marked purchases be?

15. Answer • 1) Identify your variables (ex: let m=market price, d=discounted price, k=constant, t=taxes, s=total sale price) • Your general equation should be: • d=km • s=d+td • When you substitute: s=km+t(km) • Now determine the values for each variable: • k=.85 m=? d=? t=.085 s=? Your actual equation (with numbers) should be: • d=.85m • s=.85m+.085(m) • 1) m=24.99, s=.85(24.99)+.085(24.99), s=21.24+2.12 , s= 23.36 Since you only have $20, 20 < 23.36, therefore, you can not buy the game • 2) s=175, so 175=.85(m)+.085(m), m=187.17, so the maximum total value of the market price should be at most $187.17

16. Graphing Model • Graphing model is a visible representation of a numerical model or an algebraic model that gives insight into the relationships between variable quantities.

17. Graphical model Example #1 • Graph this. What does it look like? Can you find an algebraic model that fits?

18. Answer • It is parabolic • Because (1,0.75) must satisfy the equation, after substitution

19. Graphical Model example #2 Create a graphic model, that fits the best with these data sets.

20. Answer • When you plot them, it looks like it’s linear, so let’s use a linear model y=mx+b • m=slope, b=y-intercept • , b=3.8 • y=.0145x+3.8 In statistics, this is known as “best fit line”

21. Group work Situation: • What methods are there to solve for the variable? Use one and find the solution

22. Answer • Potential ways: • Algebraic • Graphing • Factoring • Quadratic Formula • I used quadratic formula to solve it. You should have a solution that

23. Group Work Solve for this equation • What method did you choose and why?

24. Answer • X=0 or x=5/2 or x=-2/3

25. Group Work Solve this • Describe this graph and find the solutions

26. Group Work Solve this • Describe this graph and find the solutions

27. Homework Practice • Pgs 81-84 #1-9odd, 11-18, 29, 31, 35,43, 45, 48, 50

28. Basic parent Functions

29. Quick Talk: • What makes a function? What does it consists of?

30. Answer • One to one / passes vertical line test • Y and x • Equation • Domain • Range • Note: y is aka f(x)

31. Quick Talk: From what you just learned, which of these is not a function?

32. Quick Talk: What are parent functions?

33. Parent Function is the simplest function with characteristics.It is without any “transformations” or “shifting”

34. 12 parent functions: • As you are graphing it, please draw an arrow at the end.

35. The Identity Function

36. The Squaring Function

37. The Cubing Function

38. The Reciprocal Function

39. The Square Root Function

40. The Exponential Function

41. The Natural Logarithm Function

42. The Sine Function

43. The Cosine Function

44. The Absolute Value Function

45. The Greatest Integer Function

46. The Logistic Function

47. Important!!! Read this!!! • I neeeeeeedddddddd you guys to recognize, understand and know the parent functions!!!!! • Why? Because it will make finding domain and range easier.

48. What “shifts” can you possibly have on a parent function? • You can shift a function left or right (horizontal shifts) • You can shift a function up or down (vertical shifts) • Function can be steeper or flatter (linear) • Function can be wider or narrower (horizontal shrink/vertical stretch or horizontal stretch/vertical shrink) • Function can flip across x or y axis

49. Group Activity: graphing! Each group will select a member to put it on the whiteboard. With t chart. X:[-5,5] • Group 1 Group 6 • Group 2 Group 7 • Group 3 Group 8 • Group 4 Group 10 • Group 5

50. Shifts review • Parent function • Shift right 2 vertical stretch of 3, horizontal shrink of • Shift left 8 vertical shrink of , horizontal stretch of 2 • Shift up 10 Flip across the x-axis • Shift down 7 Flip across y-axis