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 Y OR F(X) X OUTPUT INPUT DEPENDENT INDEPENDENT RANGE DOMAIN VERTICAL AXIS RISE DEPENDS ON IS A FUNCTION OF X INPUT INDEPENDENT DOMAIN HORIZONTAL AXIS RUN DETERMINES

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 Variables Underline 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 -4 x y 1 -2 4 3 9 16 x y -1 3 2 5 x y 2 1 3 5 4

Vertical Line Test . . . . . .

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? Now, you Try It, pg 42 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

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 .. … .. . Negative .. Correlation . . No Correlation . . .. . . . . . .. . Undefined 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=2x y=6 y=2x-7 3x+2y=9 y=2x2 y=3x2 +x-3 y=-2x2 +x+1 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=2x y=6 y=2x-7 3x+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-3 y=-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 Graduation 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 = $160 Sales = $1.50 X s slices of pizza = 1.50s Net 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 -160 250+160 ≤ 1.50s 410 ≤ 1.50s 1.50 1.50 273.3 ≤s 274 slices must be sold