gps algebra unit 1: function families. function notation graphing basic functions function...

GPS Algebra

Unit 1: Function Families

Unit 1: Function Families• Function Notation• Graphing Basic Functions• Function Characteristics

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

Methods of Representing Relations

• Tables • Mappings

• Graphs • Function Notation

x y

-1 1

0 0

1 1

2 4

-1

0

1

2

0

1

4

f(x) = x2

Read: “f of x equals x squared”

Function notation is an efficient way to talk about functions

Is a Relation a Function?

Table/Mapping Check – repeats in domain (x)?

Graph Check – vertical line hits one or fewer points on the graph?

• 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)

x y

-1 1

0 0

1 1

2 4

x y

2 8

3 6

5 10

2 1

FUNCTION NOT A FUNCTION

FUNCTION

NOT

A

FUNCTI

ON

Domain and Range of a Function

• Domain– Inputs in a Table or Mapping– Independent Variable (x) on a Graph

• How far left and right my function goesx

f(x)• Range– Outputs in a Table or Mapping– Dependent Variable (y) on a Graph

• How far up and down my function goes

All Real #s

Domain and Range of a Function

• Domain: {-1, 0, 1, 2}• Range: {0, 1, 4}

x f(x)

-1 1

0 0

1 1

2 4

-1

0

1

2

0

1

4

• Range:

• Domain:

The farthest up/down the graph goes

The farthest left/right the graph goes

With continuous

functions, domain and

range are expressed in

interval notation.

0y

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) =

• The same substitution process can be used to complete a function table.

x f(x) = 3x + 5 f(x)

-2

-1

0

1

The inputs x arethe domain; theoutputs f(x) are the range

3(-2) – 1 = -6 – 1 = -7

3(-2) + 5 = -6 + 5

3(-1) + 5 = -3 + 5

3(0) + 5 = 0 + 5

3(1) + 5 = 3 + 5

-1

2

5

8

Graphing Basic Functions• Graphing parent functions for linear, absolute

value, quadratic, square root, cubic, and rational functions

• Introduction to transformations

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

Quadratic & Square Root Functions

• f(x) = x2 • f(x) = x


equal to 0

Domain: x is greater than or equal to 0

Range: y is greater than or equal to 0

Cubic & Rational Functions

• f(x) = x3 • f(x) =x

1


Domain: All Real #s except x cannot equal 0

Range: All Real #s except y cannot equal 0

Parent Graphs on the Move• Translation up


equal to -3


• Translation down

f(x) = x3 + 1

f(x) = |x| – 3

Vertical Stretch and Shrink• Coefficients determine the shape of a graph; a coefficient

outside the function results in a vertical stretch or shrink

f(x) = 3x2

Vertical stretch by 3(rises three

times as fast)

f(x) = ½x2

Vertical shrink by 2(rises half

as fast)

3f(x) is vertically stretched by 3 ½ f(x) is vertically shrunk by 2

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

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) = x

Domain: All Real #sRange: y is less than or

equal to 0

Domain: x is greater than or equal to 0Range: y is less than or equal to 0

Multiple Transformations• These transformations can occur together with changes to the

coefficient and what is added or subtracted to the function

f(x) = - ½ (x )2 + 3

Parent function: x2

(quadratic)

Reflect down

Vertical shrink by 2

Shift up 3

Domain: All Real #sRange: y is less than or equal to 3

Multiple Transformation Practice• Write the function for

the graph below• Graph the following

function

f(x) = -2(x )2 + 4f(x) = 3(x )2 – 3

Domain: All Real #sRange: y is greater than

or equal to 3

Domain: All Real #sRange: y is less than or

equal to 4

Function Characteristics• Analyzing graphs by determining domain, range,

zeros, intercepts, intervals of increase and decrease, maximums and minimums, and end behavior

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?!

Analyzing Functions:Domain and Range

• Domain: all real numbers in the input

• Range: all real numbers in the output

LINEAR


• Range: lowest point is -1; goes up forever y is greater than or equal to -1

ABSOLUTE VALUE

3( ) 1

2f x x 1

2

1)( xxf



• Range: lowest point is +1; goes up forever y is greater than or equal to 1

• 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

QUADRATIC SQUARE ROOT

( ) 2 3f x x

12)( 2 xxf



• 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

CUBIC RATIONAL

1( ) 1f x

x

3( ) 2( ) 1f x x

Analyzing Functions:Zeros and Intercepts

LINEAR ABSOLUTE VALUE

3( ) 1

2f x x 1

2

1)( xxf

• Zeros: one x-intercept: (-2, 0)• y-intercept: (0, -1)

• Zeros: two x-intercepts: (-1 , 0) and (1, 0)

• y-intercept: (0, -1)


• 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)


( ) 2 3f x x

12)( 2 xxf


• Zeros: (1, 0)• y-intercept: (0, -1)

CUBIC RATIONAL

1( ) 1f x

x

3( ) 2( ) 1f x x

• Zeros: (1, 0)• y-intercept: None

Analyzing Functions:Intervals of Decrease and Increase

• Intervals of decrease: x is less than and greater than 0

• Intervals of increase: none

3( ) 1

2f x x 1

2

1)( xxf


• Intervals of decrease: x is less than 0

• Intervals of increase: x is greater than 0


• 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


( ) 2 3f x x


• 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)

CUBIC RATIONAL

3( ) 2( ) 1f x x

1( ) 1f x

x

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.


• Maximums: none• Minimums: none

• Maximums: none• Minimums: (2, -1)

3( ) 1

2f x x 1

2

1)( xxf



• Maximums: none• Minimums: (0, 1)

• Maximums: none• Minimums: (0, -3)


( ) 2 3f x x

12)( 2 xxf




CUBIC RATIONAL

3( ) 2( ) 1f x x

1( ) 1f x

x

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)

Analyzing Functions:End Behavior

• End Behavior:– as x -, f(x)

• left arm up

– as x , f(x) -• right arm down

3( ) 1

2f x x 1

2

1)( xxf



• left arm up

– as x , f(x) • right arm up



• left arm up


• End Behavior:– as x , f(x)

• right arm up


( ) 2 3f x x

12)( 2 xxf


• End Behavior:– as x -, f(x) -

• left arm down


CUBIC

3( ) 2( ) 1f x x

f(x)

gps algebra unit 1: function families. function notation graphing basic functions function...

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