Understanding Functions and Graphing Lines: A Comprehensive Guide

1. 2-1 Functions

2. What is a Function? Definition: __________________________ ___________________________________ The x values of a function are called the ____________________ and all the y values are called the _________________ ___________________________________ X is called the “______________________” while y is the “______________________”

3. Well, what would a non function look like? • Equations that would not be functions: • _____________________________________________________ • _____________________________________________________

4. Domain? What was that? -the x values. The easiest way to define the domain ___________ _________________________________________ _________________________________________ ________________________________________ • _____________________________________ b) ____________________________________ Either is acceptable.

5. Examples: Find the Domain of each

6. Function Notation The algebraic expression is a function. There are LOTS of functions out there (any equation you can dream up where an x will produce only one y value is a function) but I am going to use this one for now. To show that something IS a function, it is written like this: Don’t worry! ____________________________ _______________________________________

7. OK – how do we use it? Lets use the sample from before. 1. Given find f(1), f(-2) and f(0). The function is simply an instruction of what to do to x. ___________________________________ Plug 1 in for all x’s and solve for y and put as a ordered pair (x,y) f(1)= f(-2)= f(0)=

8. Examples Find the domain of 3. 4. 5.

9. 2.2 Graphing Lines Going from an equation to a picture

11. What methods can I use to graph line? • ___________________________ ___________________________ ___________________________ ___________________________ ___________________________ Please graph 2x + 3y = 6

12. What method can I use to graph line? 2. _____________________________ _____________________________ Lets review slope for a minute

13. SLOPE • Slope = Please graph y = -3x +4

14. Parallel Lines Perpendicular Lines Horizontal Lines Vertical Lines Special Things

15. Other Review Items ____________ ____________ ____________

16. 2.3 Equations of Lines Going the other direction – from a picture to the equation

17. There are 3 standard forms of equations • Slope intercept form ______________ • Standard form ______________ ____________________________ 3. Point slope form

18. So, what do you need to have to find the equation of the line? Lets try one: Slope=2 and the y-int = 5

19. Convert y = 1.5 x – 6 to standard form. • ________________________________ • __________________________________ 2. Convert 10x – 2y = 3 to slope/intercept

20. Find the equation of the line that has Slope = 3, y intercept = 10 Slope = 3, x intercept = 10 Slope = 3, passes through (10, 10)

21. Parallel to -4x + 2y = 10 and passes through (-1, -1) 7. Parallel to x + 2y = 1 and passes through the point of intersection of the lines y = 3x – 2 and y = 2x + 1.

22. Triangle ABC has vertices A(-4,-2) L(2,8) G(6,2) • Write the equation of AL

23. Triangle ABC has vertices A(-4,-2) L(2,8) G(6,2) • Find the equation of the perpendicular bisector • of LG. • Steps: • ___________________ • ___________________ • ____________________ • 3. ________________ L G A

24. Triangle ABC has vertices A(-4,-2) L(2,8) G(6,2) • Find the equation of the altitude to AG • Steps: • 2. ________________ L G A

25. 2-4 A Variety of Graphs Piecewise Functions

26. What are Piecewise Functions? Piecewise functions are defined ___________________________________ ___________________________________ ___________________________________ ___________________________________

27. Graphing absolute Values

28. How will we graph? ______________________________ ______________________________ ______________________________ ______________________________ ______________________________

29. Graphing Absolute Value • _______________________________________ __________________________________________ __________________________________________ __________________________________________ • ______________________________________ _________________________________________ _________________________________________ _________________________________________

30. Examples 1. 2.

31. The next kind of piecewise function The form of this function is similar to this: This looks worse than it is. Essentially the function is split into multiple functions based on particular domains. ______________________________ ________________________________________

32. _____________________________________ • _____________________________________ • x y x y x y

33. 2-5 Systems of Equations Finding a solution that works for multiple equations

34. Warm Up Please graph on one set of axes the following:

35. Solutions for multiple equations? That is, where 2 lines intersect. How can 2 lines intersect?

36. What methods have you already learned for finding where 2 line intersect? • _______________________________________ • _______________________________________ __________________________________________ __________________________________________ • _______________________________________ __________________________________________ __________________________________________

37. What method do you have to use? Unless specified (i.e. follow directions) you may use ANY method you want.  I want you to be happy. Examples:

38. Steps to solve 3 Equations 3 Variables 1. __________________________________ 2. __________________________________ 3. ___________________________________ 4. ___________________________________ 5. ___________________________________

39. A golfer scored only 4’s and 5’s in a round of 18 holes. His score was 80. How many of each score did he have?

40. 2. Tuition plus Room/Board at a local college is $24,000. Room/Board is $400 more than one-third the tuition. Find the tuition.

41. 3. Mr. Tem bought 7 different shirts for the coaches of his baseball team. The blue long sleeved shirts cost $30 each and the white short sleeved shorts cost $20 each. If he paid a total of $160, how many of each shirt did he buy??

42. 4.. Rob invests money, some at 10% and some at 20% earning $20 in interest per year. Had the amounts invested been reversed, he would have received $25 in interest. How much has he invested all together?

43. 6. The sum of two numbers is 20. The larger is 5 less than twice the smaller. What are the numbers??

44. 2-6 Graphing Quadratic Functions

45. No more linear functions What happens graphically when an equation’s high power is 2? _____________________________ _____________________________

46. The Parabola (The Picture)

47. Looking at Trends 5 4 3 2 1 -4 -3 -2 -1 1 2 3 4

48. So, we see some trends We probably won’t use trends; much like absolute values, one easy way to graph parabolic functions is to plot the vertex and then plot 2 points on either side of the x coordinate of the vertex.

49. The Parabola (The Equation) From what we saw, these are the trends: Add/Subtract inside the squared quantity? ________________________ Add/Subtract outside the squared quantity? ________________________ Multiply/Divide inside or outside? ________________________

50. The Parabola (The Equation) a ____________________________ (h, k) _________________________ If a < 0, what will happen to the graph?