1. FUNCTIONS
Algebra I
C. Toliver

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

3. 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.

4. 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

5. 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

6. 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

7. 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

8. 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

9. How do you determine a function?
No x values are repeated.
Must pass the vertical line test

10. 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

11. Vertical Line Test .
. .
. . .

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

13. OBJECTIVE 2 The student will demonstrate an understanding of the properties and attributes of functions.

14. 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)

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

16. 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

17. 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?”

18. 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

19. Domain and Range

20. 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

21. Scatterplots Positive .. Correlation .. … .. . Negative .. Correlation
No Correlation
. Undefined Correlation .
………………

22. 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 │

23. 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

24. 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

25. 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.

26. 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.

27. 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.

28. 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?

29. 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

30. Representing Functions – Project Graduation
How many pizza slices must be sold to make a net profit of at least $250.00?

31. 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
≤ 1.50s
410 ≤ 1.50s
273.3 ≤s
274 slices must be sold