1. SWBAT… define and evaluate functions Agenda 1. Warm-Up (10 min) 2. Review hw#3 & hw#4 (25 min) Warm-Up: 1.) How would the graph of y = |x| + 5 transform from the parent function graph, y = |x|? The graph of y = |x| + 5 would shift 5 units up from the parent function, y = |x|. 2.) How would the graph of y = |x + 1| transform from the parent function graph, y = |x|? (Hint: It does NOT shift up 1 unit, see HW#3, problem 2) The graph of y = |x + 1| would shift one unit to the left from the parent function, y = |x|. Review PPT4: Functions on the Infinity website Mon, 11/7

2. 3. Q: How would the graph of y = -|x + 1| transform from the parent function graph, y = |x|? A: The graph of y = -|x + 1| would shift 1 unit to the left and rotate around the x-axis from the parent function, y = |x|. (HW#3, Problem #5a)

3. y = -|x + 1| The vertex, or maximum point, is (-1, 0). Graph y = -|x + 1| by completing a table of values: x y -2 0 1 2 y =-|-2 +1| = -1 y =-|-1+ 1| = 0 y =-|0 + 1| = -1 y =-|1 +1| = -2 y =-|2 +1| = -3 y = -|x + 1| is shifted 1 unit to the left and rotated around the x-axis from the parent function, y = |x|

4. HW#3, Problem #6 5.Q: How would the graph of y = -|x + 1| + 3 transform from the parent function, y = |x|? A: The graph of y = -|x + 1| + 3 would shift 1 unit to the left shift, 3 units up, and rotate around the x- axis from the parent function graph, y = |x|.

5. When graphing 1. Since the ordered pairs are all positive, the origin can be at the bottom left of the graph 2. Include a title on the graph 3. Include an x-axis and y-axis label 4. The x-axis has to increase by the SAME amount for each box

6. SWBAT… define and evaluate functions Agenda 1. Warm-Up (5 min) 2. Notes on functions (25 min) 3. OYO problems (15 min) Warm-Up: 1.) Turn in HW#3 and HW#4 in the blue folder 2.) Set up Cornell notes. Topic is “Functions” HW#4: Function (page 1) Tues, 11/8

7. Objectives Today: 1. To define a function 2. To learn function notation 3. To evaluate functions Tomorrow: 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function 4. To write a function as an ordered pair

8. Ms. Sophia Papaefthimiou Infinity HS

9. What is a function? A function is like a machine: it has an input and an output.

10. Function Notation The most common name is "f", but you can have other names like "g" What goes into the function (input) is put inside parentheses after the name of the function Example: f(x) (pronounced “f of x”) The function is called "f“ "x" goes in (input) Question: 1.) A function is called “g” and “a” is the input. Write the function. 2.) What is the function name 3.) What is the input

11. Output Value Range Dependent Variable These are all equivalent names for the y. Input Value Domain Independent Variable These are all equivalent names for the x. Name of the function Function Notation: The Symbolic Form

12. Function Notation Function notation replaces the ___ in an equation with ___ Example: Given y = 3x + 2, write the equation in function notation f(x) = 3x + 2 Question: Write y = x 2 in function notation. y f(x)

13. Function Notation You used to say “y = 2x + 3; find the value of y when x = -1” y = 2x + 3 y = 2(-1) + 3 y = -2 + 3 y = 1 Now you say “f(x) = 2x + 3; find f(-1)” (pronounced as "f of x is two x plus three; find f of negative one") 1 f(-1) = 2x + 3

14. Ex: If f(x) = 2x + 3 find: 1. f(4x) 2. f(-1) + f(-9) 3. f(t – 5)

15. OYO (On Your Own) Problems Directions: If f(x) = 2x – 4 and g(x) = x² – 4x, find each value: 1. f(-3) 2. f(3x) 3. g(t) 4. f(q + 1) (Hint: f(q+1) = 2(q+1) – 4 5. f(2) + g(-2) 6. f(g(-2)) (Hint: Start from the inside out. Find g(-2) first) Then find f(12)

16. Revisit our objectives Today: 1. To define a function 2. To learn function notation 3. To evaluate functions

17. A function P is defined as follows: For x > 0, P(x) = x 5 + x 4 – 36x – 36 For x < 0, P(x) = -x 5 + x 4 + 36x – 36 What is the value of P(-1)? A. -70 B. -36 C. 0 D. 36

18. SWBAT… define and evaluate functions Agenda 1. Warm-Up (10 min) 2. Notes on functions (35 min) Warm-Up: If f(x) = -4x + 1 and g(x) = x², find: 1.) f(-10) 2.) g(-7) 3.) f(g(-7)) (Hint: Start from the inside out. Find g(-7) first) Then find f(49)) HW#3: Evaluating functions (page 2 & 3) Wed, 11/9

19. Objectives Yesterday 1. To define a function 2. To learn function notation 3. To evaluate functions Today 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function

20. SWBAT… define and evaluate functions Agenda 1. Warm-Up (10 min) 2. Notes on functions (10 min) 3. Work on hw Warm-Up: If f(x) = -3x + 5 and g(x) = x² – 6, find: 1.) f(-8) 2.) g(-4) 3.) f(g(-2)) (Hint: Start from the inside out. Find g(-2) first) Then find f(16)) HW#3: Evaluating functions (page 1 – 4) Thurs, 11/10

21. Function Mapping A set of points or equation where every input has exactly one output. In other words, the domain or x value can not be repeated

22. This is a function! There is only one arrow coming from each x. In other words, x can not be repeated This is a function! There is only one arrow coming from each x There is only one y for each x. It just so happens that it's always the same y for each x. Function Mapping (cont’d)

23. This one is not a function. There are two arrows coming from the number 1. The number 1 is associated with two different range elements. In order words, x is repeated. Function Mapping (cont’d)

24. Vertical Line Test No mater where we drop a vertical line, if the vertical line only hits the graph once, it is a function. (Note: I didn’t say anything about the x-axis) So, this graph is a function! What is this function called?  Quadratic function (2, 4) (-2, 4)

25. y x Q: Is the graph a function? Explain. A: Yes, this graph is a function because it passes the vertical line test; wherever you drop a vertical line, it will only hit the graph once. Q: What is the graph called? A: A line Q: What is the function called? A: A linear function Write the equation of the line in slope- intercept form, y = mx + b. Q: What is the slope, m? A: 3 Q: What is the y-intercept, b? A: -2 Vertical Line Test (cont’d) y = 3x – 2

26. Draw a graph, that would NOT pass the vertical line test. Exchange note books with the person next to you.  Write on their notebooks if their graph passes the vertical line test.  Give them feedback

27. Revisit our objectives Today: 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function WE MET OUR OBJECTIVES TODAY!!!!

28. Vertical Line Test (cont’d) Intersect at two points These graphs are not functions

29. SWBAT… find the domain and range of functions Agenda 1. Warm-Up (15 min) 2. 6 practice problems (20 min) 3. Exit slip (10 min) Warm-Up: 1. Write your HW in your planner for the week. 2. Graph the function f(x) = 1/2x 2 using a table of values. 3. Find the domain. Explain. 4. Find the range. Explain. HW#5: Evaluating functions (page 1 – 6) (Page 4 will be counted as a quiz grade) Mon, 11/14

30. x f(x) =1/2x 2 -4f(-4) = 8 -2f(-2) = 2 f(-1) = 0.5 0f(0) = 0 1f(1) = 0.5 2f(2) = 2 4f(4) = 8 The domain is all real numbers. Explanation: There are no restrictions on the domain, the x value. The range is y ≥ 0. Explanation: The graph will never be below 0.

31. Domain and Range Domain: The set of all x values in a function (width) Range: The set of all y values in a function (height) What can go into a function is called the Domain What comes out of a function is called the Range

32. Domain and Range (cont’d) f(x) = x 2 – 2 The domain is all real numbers. Explanation: There are no restrictions on the domain, the x value. The range is y ≥ -2. Explanation: The graph will never be below -2. (2, 4) (-2, 4) (0, -2)

33. Domain and Range (cont’d) Question: For the function f(x) = x 2, if the domain is {1, 2, 3}, what is the range?

34. Domain and Range (cont’d) Example: {(2, 4), (4, 5), (7, 3)} Question: What is the domain? What is the range? Is this a function? Explain. Example: {(2, 4), (2, 5), (7, 3)} Questions: What is the domain? What is the range? Is this a function? Explain.

35. Conclusion A function relates inputs to outputs The domain is the set of all x values in a function The range is the set of all y values in a function You can use a function and its given domain, the x-value, to find its range the y-value or f(x) value Parenthesis in function notation do not indicate multiplication.  “f(x)” means “plug in a value for x” An input, x, and its matching output, y or f(x), together are called an ordered pair Functions can be represented with a: 1. Table 2. Mapping 3. Equation 4. Graph

36. Question 1/6

37. Question 2/6 Suppose g(v) = v + 10. What is g(x – 1)? Answer: g(x – 1) = x – 1 + 10 g(x – 1) = x + 9

38. Question 3/6 What is the function name, input, and output of g(7)? Answer: g is the function name 7 is the input g(7) is the output

39. Question 4/6 Suppose g(x) = 3x + 2. Describe, in words, what the function g does. Answer: The function g takes an input x, multiplies by 3, and then adds 2.

40. Question 5/6 Write in function notation “the function g takes an input y adds 3, and then multiplies by 2.” (Hint: PEMDAS!!!!) Answer: g(y) = 2(y + 3)

41. What is the domain of (Hint: what numbers can you plug in for x to get an answer that makes sense) 1. All real numbers 2. All real numbers except for 0 3. All real numbers except for 3 4. All real numbers except for -3 Question 6/6

42. Exit slip: Do the problems on the back of last week’s agenda (collected in 10 min) HOMEWORK HW5: Functions – Page 1 - 6 1. All 4 pages 2. Page 4 will count as a quiz grade

43. Find the domain of each function:

44. Answers: 1.All real numbers except for 0 2.All real numbers except for -5 3.All real numbers 4.All real numbers greater than or equal to -9

45. Question 1/10 623623 125125 Is this a function? Write the below problem in your notebook

46. Question 2/10 If f(x) = -x 2 find a. f(3) b. f(-3)

47. Question 4/10 Given f(x) = 3x – 5 and the domain is {0, 2, -1} find the range

48. Question 3/10 Function f is defined by f(x) = -2x 2 + 6x – 3 Find f(-2) and write as an ordered pair