FUNCTIONS Algebra I C. Toliver. Definition of a Relation What is a relation? Any set of ordered...

FUNCTIONS

Algebra I

C. Toliver

Definition of a Relation

What is a relation?Any set of ordered pairs (x,y).

Definition of a Function

What is a function?A set of ordered pairs (x,y). Each x- coordinate is paired with only one y-

coordinate.

Other Definitions

Domain: the set of input values; the set of first coordinates in an ordered pair

Range: the set of output values; the set of second coordinates in an ordered pair

Variable: a letter or symbol used to represent a value Independent: not subject to control by others; self-

governing Dependent: relying on or subject to something else for

support

Functions

XINPUT

INDEPENDENTDOMAIN

HORIZONTAL AXISRUN

DETERMINES

Y OR F(X)OUTPUT

DEPENDENTRANGE

VERTICAL AXISRISE

DEPENDS ONIS A FUNCTION OF

Dependent and Independent Variables A function is a set of ordered pairs (x,y)

where:x is the independent variable and

y is the dependent variable The value of y depends on the value of x

Dependent and Independent Variables

Underline the dependent variables below: The distance traveled; the time driven at

speed r The total cost of gasoline; the number of

gallons purchased The square feet of floor space; the number

of tiles needed to cover the floor

Dependent and Independent VariablesUnderline the dependent variables below: The distance, d traveled; the time, t driven

at 60 mph The total cost, c of gasoline; the number of

gallons, g purchased The square feet of floor space; the number

of tiles, t needed to cover the floor

How do you determine a function?

No x values are repeated. Must pass the vertical line test

Which table does not represent a function?

x y

1 2

4 1

-1 2

-4 1

x y

1 1

-2 4

3 9

1 16

x y

-1 -1

3 3

2 2

5 5

x y

2 0

1 1

3 5

4 0

Vertical Line Test

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. .

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Vertical Line Test – circle the graphs that are not functions.

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. .

. .

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OBJECTIVE 2

The student will demonstrate an understanding of the properties and attributes of functions.

Domain and Range pg 37

Domain – all the x coordinates in the ordered pairs (x,y)

Range – all the y coordinates in the ordered pairs (x,y)

Domain and Range

Identify the domain and range of the function below:{(3,9), (5,39), (9,23), (6,14)}

Domain and Range

Identify the domain and range of the function below:{(3,9), (5,39), (9,23), (6,14)}

The domain is all the x values {3,5,9,6}

The range is all the y values {9,39,23,14}

Now you Try It, pg 38

Domain and Range

You may be asked to identify the domain and/or range of a function

Ask yourself “What are the reasonable values for the domain of this function?”

Ask yourself “What are the reasonable values for the range of this function?”

Domain and Range

What’s reasonable?Can the value be zero?Can the value be positive?Can the value be negative?What is the lowest value possible?What is the highest value possible?Must all the values be whole numbers?

Now, you Try It, pg 42

Domain and Range

Irrational Numbers, 2, -2 , 43

Rational Numbers3/2, 4.1, -2/6, 5,.05

Integers…-1,0,1,…

Whole Numbers0,1,2,3,…

Natural Numbers1,2,3,….

Real Numbers

Domain and Range

Remember: A closed circle on a graph means the point is in the

solution An open circle on a graph means the point is not in

the solution A solid line on a graph means the line is in the

solution A dashed line on a graph means the line is not in the

solution

Scatterplots

Positive .. Correlation ..

…

..

. Undefined Correlation

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.

. ………………

. . No Correlation

. . ..

. . . .

. ..

. Negative

.. Correlation

..

. .

Parent Functions

What is a linear function? y=mx+b

What is a quadratic function? y=ax2+bx+c

What is an absolute value function?y=│x │

Parent Functions What is a parent function? These are examples of ________ functions. The parent function for

each of these ________functions is __________?y=2xy=6y=2x-73x+2y=9

These are examples of ________ functions. The parent function for each of these ________functions is __________? y=2x2

y=3x2 +x-3y=-2x2 +x+1

These are examples of ________ functions. The parent function for each of these ________functions is __________? y= 2│x│y= │x-3│y= -│x│+1

Parent Functions What is a parent function? These are examples of linear functions. The parent function for

each of these linear functions is y=x .y=2xy=6y=2x-73x+2y=9

These are examples of quadratic functions. The parent function for each of these quadratic functions is y=x2 .y=2x2

y=3x2 +x-3y=-2x2 +x+1

These are examples of absolute value functions. The parent function for each of these absolute value functions is y=│x│ .y= 2│x│y= │x-3│y= -│x│+1

Representing Functions

• Project Graduation sells pizza for $1.50 per slice. There are eight slices in each large pizza. Each week, Project Graduation buys 40 large pizzas at a cost of $4.00 per pizza. A local store donates the paper plates and napkins.

Representing Functions – Project Graduation Write an equation for the net profit from

pizza sales. Let p = net sales profit in dollars and s = the number of pizza slices sold.

Identify the dependent and independent variables.

Representing Functions –Project Graduation1. Project Graduation sells pizza for $1.50 per slice. There are eight

slices in each large pizza. Each week, Project Graduation buys 40 large pizzas at a cost of $4.00 per pizza. A local store donates the paper plates and napkins.

Write an equation for the net profit from pizza sales. Let p = net sales profit in dollars and s = the number of pizza slices sold.

Cost = 40 pizzas x $4.00 per pizza = $160Sales = $1.50 X s slices of pizza = 1.50sNet profit, p = sales – cost p=1.50s-160

Identify the dependent and independent variables. p, net profit is dependent; s pizza slices is independent.

Representing Functions – Project Graduation If all the pizza slices are sold, how much

profit will Project Graduation make? What is the domain of the function? What

is the range?

Representing Functions –Project Graduation If all the pizza slices are sold, how much

profit will Project Graduation make?s=40 pizzas x 8 slices per pizza = 320 slices

p=1.50x320 – 160 = $320 What is the domain of the function? What is

the range?Domain: 0≤s≤320, where s is a whole number

Range: -$160≤p≤$320

Representing Functions – Project Graduation How many pizza slices must be sold to

make a net profit of at least $250.00?

Representing Functions – Project Graduation How many pizza slices must be sold to make a

net profit of at least $250.00?250 ≤1.50s -160250+160 ≤ 1.50s410 ≤ 1.50s410 ≤ 1.50s1.50 1.50273.3 ≤s274 slices must be sold