1. Unit 1: Function Families
GPS Algebra
Unit 1: Function Families

2. Unit 1: Function Families
Function Notation
Graphing Basic Functions
Function Characteristics

3. Function Notation Use and Purpose of Function Notation
Presentation of Functions in Tables, Mappings, Graphs, and Algebraic Function Notation
Determining Whether a Relation is a Function
Introduction of Domain and Range

4. Methods of Representing Relations
Tables
Mappings x y -1 1 2 4 -1 1 2 1 4
Graphs
Function Notation
f(x) = x2
Read: “f of x equals x squared”
Function notation is an efficient way to talk about functions

5. Is a Relation a Function?
A relation is a function if there is only one output for any of its inputs (i.e. an x input can only lead to one y output)
Table/Mapping Check – repeats in domain (x)? x y -1 1 2 4 x y 2 8 3 6 5 10 1
NOT A FUNCTION
FUNCTION
Graph Check – vertical line hits one or fewer points on the graph?
NOT A FUNCTION
FUNCTION

6. Domain and Range of a Function
Inputs in a Table or Mapping
Independent Variable (x) on a Graph
How far left and right my function goes x
Range
Outputs in a Table or Mapping
Dependent Variable (y) on a Graph
How far up and down my function goes
f(x)

7. Domain and Range of a Functionx
f(x) -1 1 2 4 -1 1 2
Domain: {-1, 0, 1, 2}
Range: {0, 1, 4} 1 4
With continuous functions, domain and range are expressed in interval notation.
Domain:
All Real #s
The farthest left/right the graph goes
Range:
The farthest up/down the graph goes

8. Using Function Notation
Function notation is used to represent relations which are functions. Finding f(-2), for example, is the same as evaluating an expression for x = -2.
Example: f(x) = 3x – 1; find f(-2)
Solution: f(-2) =
3(-2) – 1 =
-6 – 1 = -7
The same substitution process can be used to complete a function table. x
f(x) = 3x + 5
f(x) -2 -1 1
3(-2) + 5 = -1
3(-1) + 5 = 2
The inputs x are
the domain; the
outputs f(x) are the range
3(0) + 5 = 0 + 5 5
3(1) + 5 = 3 + 5 8

9. Graphing Basic Functions
Graphing parent functions for linear, absolute value, quadratic, square root, cubic, and rational functions
Introduction to transformations

10. Linear & Absolute Value Functions
f(x) = x
f(x) = |x|
Domain: All Real #s
Range: All Real #s
Domain: All Real #s
Range: y is greater than or equal to 0

11. Quadratic & Square Root Functions
f(x) = x2
f(x) =
Domain: All Real #s
Range: y is greater than or equal to 0
Domain: x is greater than or equal to 0
Range: y is greater than or equal to 0

12. Cubic & Rational Functions
f(x) = x3
f(x) =
Domain: All Real #s
Range: All Real #s
Domain: All Real #s except x cannot equal 0
Range: All Real #s except y cannot equal 0

13. Parent Graphs on the Move
Translation up
Translation down
f(x) = x3 + 1
f(x) = |x| – 3
Domain: All Real #s
Range: All Real #s
Domain: All Real #s
Range: y is greater than or equal to -3

14. Vertical Stretch and Shrink
Coefficients determine the shape of a graph; a coefficient outside the function results in a vertical stretch or shrink
3f(x) is vertically stretched by 3
½ f(x) is vertically shrunk by 2
f(x) = 3x2
Vertical stretch by 3
(rises three
times as fast)
f(x) = ½x2
Vertical shrink by 2
(rises half
as fast)
Domain: All Real #s; Range: y is greater than or equal to 0
Domain: All Real #s; Range: y is greater than or equal to 0

15. Reflections of a Function
The sign of a coefficient indicates whether it is reflected across the x-axis or y-axis
-f(x) is reflected across the x-axis
f(-x) is reflected across the y-axis
f(x) = -|x|
f(x) =
Domain: x is greater than or equal to 0
Range: y is less than or equal to 0
Domain: All Real #s
Range: y is less than or equal to 0

16. Multiple Transformations
These transformations can occur together with changes to the coefficient and what is added or subtracted to the function
Parent function: x2 (quadratic)
f(x) = - ½ (x )2 + 3
Reflect down
Shift up 3
Vertical shrink by 2
Domain: All Real #s
Range: y is less than or equal to 3

17. Multiple Transformation Practice
Write the function for the graph below
Graph the following function
f(x) = 3(x )2 – 3
f(x) = -2(x )2 + 4
Domain: All Real #s
Range: y is greater than or equal to 3
Domain: All Real #s
Range: y is less than or equal to 4

18. Function Characteristics
Analyzing graphs by determining domain, range, zeros, intercepts, intervals of increase and decrease, maximums and minimums, and end behavior

19. Does this function EVER stop?!
Analyzing Functions
Domain: how far left and right?
Range: how far up and down?
Zeros: x-intercepts – where does the function intersect the x-axis?
Intercepts: zeros and y-intercepts
Intervals of decrease and increase: where does the function go up and where does it go down?
Maximums and Minimums: what’s the highest and/or lowest the function goes?
End Behavior: as inputs approach infinity, what happens to the function?
Does this function EVER stop?!

20. Analyzing Functions: Domain and Range
LINEAR
ABSOLUTE VALUE
Domain: all real numbers in the input
Range: all real numbers in the output
Domain: all real numbers in the input
Range: lowest point is -1; goes up forever  y is greater than or equal to -1

21. Analyzing Functions: Domain and Range
SQUARE ROOT
QUADRATIC
Domain: furthest left is 0; goes right forever  x is greater than or equal to 0
Range: lowest point is -3; goes up forever  y is greater than or equal to -3
Domain: all real numbers in the input
Range: lowest point is +1; goes up forever  y is greater than or equal to 1

22. Analyzing Functions: Domain and Range
CUBIC
RATIONAL
Domain: all real numbers in the input
Range: all real numbers in the output
Domain: left and right forever, but skips over 2  All real's except 0
Range: up and down forever, but skips over -1  All real’s except -1

23. Analyzing Functions: Zeros and Intercepts
LINEAR
ABSOLUTE VALUE
Zeros: one x-intercept: (-2, 0)
y-intercept: (0, -1)
Zeros: two x-intercepts: (-1 , 0) and (1, 0)
y-intercept: (0, -1)

24. Analyzing Functions: Zeros and Intercepts
QUADRATIC
SQUARE ROOT
Zeros: no real zeros (the graph never intersects the x-axis)
y-intercept: (0, 1)
Zeros: one x-intercept: (2, 0)
y-intercept: (0, -3)

25. Analyzing Functions: Zeros and Intercepts
CUBIC
RATIONAL
Zeros: (1, 0)
y-intercept: (0, -1)
Zeros: (1, 0)
y-intercept: None

26. Analyzing Functions: Intervals of Decrease and Increase
LINEAR
ABSOLUTE VALUE
Intervals of decrease: x is less than and greater than 0
Intervals of increase: none
Intervals of decrease: x is less than 0
Intervals of increase: x is greater than 0

27. Analyzing Functions: Intervals of Decrease and Increase
QUADRATIC
SQUARE ROOT
Intervals of decrease: the left half of the function, x is less than 0
Intervals of increase: the right half of the function, x is greater than 0
Intervals of decrease: none (the entire function is uphill)
Intervals of increase: the entire domain of the function x is greater than 0

28. Analyzing Functions: Intervals of Decrease and Increase
CUBIC
RATIONAL
Intervals of decrease: none (the entire function is uphill)
Intervals of increase: the entire domain of the function x is less than and greater than 0
Intervals of decrease: the entire domain of the function, x is less than and greater than 0
Intervals of increase: none (the entire function is downhill)

29. Analyzing Functions: Maximums and Minimums
Maximum: the highest point of the graph
For example, a cannon is shot into the air. The maximum is where it changes from going up to going down (i.e. the highest it goes)
Minimum: the lowest point of the graph
For example, a stock broker is watching the market searching for a good time to buy Alpha-Bit stocks. She looks for a stock that appears to have reached a low cost and is about to begin to increase in value.

30. Analyzing Functions: Maximums and Minimums
LINEAR
ABSOLUTE VALUE
Maximums: none
Minimums: none
Maximums: none
Minimums: (2, -1)

31. Analyzing Functions: Maximums and Minimums
QUADRATIC
SQUARE ROOT
Maximums: none
Minimums: (0, 1)
Maximums: none
Minimums: (0, -3)

32. Analyzing Functions: Maximums and Minimums
CUBIC
RATIONAL
Maximums: none
Minimums: none
Maximums: none
Minimums: none

33. Analyzing Functions: End Behavior
As x approaches - (forever left): does the function approach - (forever down) or  (forever up)?
As x approaches  (forever right): does the function approach - (forever down) or  (forever up)?
Sample notation:
as x  -, f(x)  

34. Analyzing Functions: End Behavior
LINEAR
ABSOLUTE VALUE
End Behavior:
as x  -, f(x)  
left arm up
as x  , f(x)  -
right arm down
End Behavior:
as x  -, f(x)  
left arm up
as x  , f(x)  
right arm up

35. Analyzing Functions: End Behavior
QUADRATIC
SQUARE ROOT
End Behavior:
as x  -, f(x)  
left arm up
as x  , f(x)  
right arm up
End Behavior:
as x  , f(x)  
right arm up

36. Analyzing Functions: End Behavior
CUBIC
f(x)
End Behavior:
as x  -, f(x)  -
left arm down
as x  , f(x)  
right arm up