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