1. Determining the Key Features of Function Graphs

2. The Key Features of Function Graphs - Preview
Domain and Range
x-intercepts and y-intercepts
Intervals of increasing, decreasing, and constant behavior
Parent Equations
Maxima and Minima

3. Domain Domain is the set of all possible input or x-values
To find the domain of the graph we look at the x-axis of the graph

4. Determining Domain - Symbols
Open Circle →
Exclusive
( )
Closed Circle →
Inclusive
[ ]

5. Identifying the Domain
Start at the far left of the graph.
Move along the x-axis until you find the lowest possible x-value of the graph. This is your lower bound. Note if you have a open circle or a closed circle.
Keep moving along the x-axis until you find your highest possible x-value of the graph. This is your upper bound. Note if you have a open circle or a closed circle.

6. Examples
Domain:
Domain:

7. Example
Domain:

8. Determining Domain - Infinity

9. Examples
Domain:
Domain:

10. Your Turn:
Complete problems 3, 7, and find the domain of 9 and 10 on pg. 160 from the Xeroxed sheets

11. Range The set of all possible output or y-values
To find the range of the graph we look at the y-axis of the graph
We also use open and closed circles for the range

12. Identifying the Range Start at the far left of the graph.
Move along the x-axis until you find the lowest possible x-value of the graph. This is your lower bound. Note if you have a open circle or a closed circle.
Keep moving along the x-axis until you find your highest possible x-value of the graph. This is your upper bound. Note if you have a open circle or a closed circle.

13. Examples
Range:
Range:

14. Examples
Range:
Range:

15. Your Turn:
Complete problems 4, 8, and find the range of 9 and 10 on pg. 160 from the Xeroxed sheets

16. X-Intercepts Where the graph crosses the x-axis Has many names:
Roots
Zeros

17. Examples
x-intercepts:
x-intercepts:

18. Y-Intercepts Where the graph crosses the y-axis y-intercepts:

19. Seek and Solve!!!

20. Types of Function Behavior
Increasing
Decreasing
Constant
When determining the type of behavior, we always move from left to right on the graph

21. Roller Coasters!!!
Fujiyama in Japan

22. Types of Behavior – Increasing
As x increases, y also increases
Direct Relationship

23. Types of Behavior – Constant
As x increases, y stays the same

24. Types of Behavior – Decreasing
As x increases, y decreases
Inverse Relationship

25. Identifying Intervals of Behavior
We use interval notation
The interval measures x-values. The type of behavior describes y-values.
Increasing: [0, 4)
The y-values are increasing
the x-values are between 0 inclusive and 4 exclusive
when

26. Identifying Intervals of Behavior
Increasing:
Constant:
Decreasing: x 1 1

27. Identifying Intervals of Behavior, cont.
Increasing:
Constant:
Decreasing: x -3 -1
Don’t get distracted by the arrows! Even though both of the arrows point “up”, the graph isn’t increasing at both ends of the graph!

28. Your Turn:
Complete problems 1 – 4 on The Key Features of Function Graphs – Part II handout.

29. 1. 2. 3. 4.

30. What do you think of when you hear the word parent?

31. Parent Function The most basic form of a type of function
Determines the general shape of the graph

32. Basic Types of Parent Functions
Linear
Absolute Value
Greatest Integer
Quadratic
Cubic
Square Root
Cube Root
Reciprocal

33. Parent Function Flipbook

34. Function Name: Linear Parent Function: f(x) = x “Baby” Functions: y 2

35. Greatest Integer Function
f(x) = [[x]]
f(x) = int(x)
Rounding function
Always round down

36. “Baby” Functions Look and behave similarly to their parent functions
To get a “baby” functions, add, subtract, multiply, and/or divide parent equations by (generally) constants
f(x) = x2 f(x) = 5x2 – 14
f(x) = f(x) =
f(x) = x3 f(x) = -2x3 + 4x2 – x + 2

37. “Baby” Functions, cont.
f(x) = |x|

38. Your Turn:
Create your own “baby” functions in your parent functions book.

39. Identifying Parent Functions
From Equations:
Identify the most important operation
Special Operation (absolute value, greatest integer)
Division by x
Highest Exponent (this includes square roots and cube roots)

40. Examples
f(x) = x3 + 4x – 3
f(x) = -2| x | + 11

41. Identifying Parent Equations
From Graphs:
Try to match graphs to the closest parent function graph

42. Examples

43. Your Turn:
Complete problems 5 – 12 on The Key Features of Function Graphs handout

44. Maximum (Maxima) and Minimum (Minima) Points
Peaks (or hills) are your maximum points
Valleys are your minimum points

45. Identifying Minimum and Maximum Points
Write the answers as points
You can have any combination of min and max points
Minimum:
Maximum:

46. Examples

47. Your Turn:
Complete problems 1 – 6 on The Key Features of Function Graphs – Part III handout.

49. Reminder: Find f(#) and Find f(x) = x
Find the value of f(x) when x equals #.
Solve for f(x) or y!
Find f(x) = #
Find the value of x when f(x) equals #.
Solve for x!

50. Evaluating Graphs of Functions – Find f(#)
Draw a (vertical) line at x = #
The intersection points are points where the graph = f(#)
f(1) =
f(–2) =

51. Evaluating Graphs of Functions – Find f(x) = #
Draw a (horizontal) line at y = #
The intersection points are points where the graph is f(x) = #
f(x) = –2
f(x) = 2

52. Example
Find f(1)
Find f(–0.5)
Find f(x) = 0
Find f(x) = –5

53. Your Turn:
Complete Parts A – D for problems 7 – 14 on The Key Features of Function Graphs – Part III handout.