1. Determining the Key Features of Function Graphs
10 February 2011

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

3. Domain Reminder: Domain is the set of all possible input or x-values
When we 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. Determining Domain Start at the origin
Move along the x-axis until you find the lowest possible x-value. This is your lower bound.
Return to the origin
Move along the x-axis until you find your highest possible x-value. This is your upper bound.

6. Examples
Domain:
Domain:

7. Example
Domain:

8. Determining Domain - Infinity

9. Examples
Domain:
Domain:

10. Your Turn:
In the purple Precalculus textbooks, complete problems 3, 7, and find the domain of 9 and 10 on pg. 160

11. Range The set of all possible output or y-values
When we 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. Determining Range Start at the origin
Move along the y-axis until you find the lowest possible y-value. This is your lower bound.
Return to the origin
Move along the y-axis until you find your highest possible y-value. This is your upper bound.

13. Examples
Range:
Range:

14. Examples
Range:
Range:

15. Your Turn:
In the purple Precalculus textbooks, complete problems 4, 8, and find the domain of 9 and 10 on pg. 160

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!!!

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

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

35. “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

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

37. 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)

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

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

40. Examples

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

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

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

44. Examples

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

46. 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!

47. 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) =

48. 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(–2) =
f(2) =

49. Example
Find f(1)
Find f(–0.5)
Find f(0)
Find f(–5)

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