Graphing Linear Relations and Functions 8.5G Identify functions using sets of ordered pairs, tables, mapping, and graphs

Vocabulary Relation Domain Range Function Input Output Independent variable Dependent variable Mapping Ordered pairs Linear function Continuous Discrete Vertical line test

Guiding Questions What are synonyms for the x-values? What are synonyms for y-values? How do you determine if a relation is a function? Why is it okay if the y-values repeat in a function? Explain how a why the vertical line test works. What is the difference between continuous and discrete? What is the difference between the independent variable and the dependent variable?

Relation: a set of ordered pairs Domain: the set of x-coordinates Relations & Functions Relation: a set of ordered pairs Domain: the set of x-coordinates Range: the set of y-coordinates When writing the domain and range, do not repeat values.

Relations and Functions Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: {0,1, 2, 3} State the range: R: {-6, 0, 4}

Relations and Functions Relations can be written in several ways: ordered pairs, table, graph, or mapping. We have already seen relations represented as ordered pairs.

Table {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}

Mapping Create two ovals with the domain on the left and the range on the right. Elements are not repeated. Connect elements of the domain with the corresponding elements in the range by drawing an arrow.

Mapping {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} 2 1 3 -6 4

Functions A function is a relation in which the members of the domain (x-values) DO NOT repeat. So, for every input (x-value) there is only one output (y-value) that corresponds to it. y-values can be repeated.

Is a relation a function? Focus on the x-coordinates, when given a relation If the set of ordered pairs have different x-coordinates, it IS A function If the set of ordered pairs have same x-coordinates, it is NOT a function Y-coordinates have no bearing in determining functions

A good example that you can “relate” to is students in our math class this year is the domain. The grade they earn out of the class is the range. Each student must be assigned a grade and can only be assigned ONE grade, but more than one student can get the same grade (we hope so---we want lots of A’s). That’s okay.

Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated.

Which mapping represents a function? Choice One Choice Two 2 –1 3 –2 3 1 –1 2 Choice 1

Which mapping represents a function? A. B. B

Which situation represents a function? a. The items in a store to their prices on a certain date b. Types of fruits to their colors There is only one price for each different item on a certain date. The relation from items to price makes it a function. A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function.

Graphs of a Function Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point. AKA: “The Pencil Test” Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function

Vertical Line Test Would this graph be a function? YES

Vertical Line Test Would this graph be a function? NO

Functions Discrete functions consist of points that are not connected. Continuous functions can be graphed with a line or smooth curve and contain an infinite number of points.

Is the following function discrete or continuous. What is the Domain Is the following function discrete or continuous? What is the Domain? What is the Range? Discrete

Is the following function discrete or continuous. What is the Domain Is the following function discrete or continuous? What is the Domain? What is the Range? continuous

Is the following function discrete or continuous. What is the Domain Is the following function discrete or continuous? What is the Domain? What is the Range? continuous

Is the following function discrete or continuous. What is the Domain Is the following function discrete or continuous? What is the Domain? What is the Range? discrete

Independent/Dependent Variables Independent variables (x-values) do not depend on anything. You can change them as you like, and see the results. It stands alone and isn’t changed by other variables you are trying to measure. Dependent variables (y-values) represent the results. It depends on other factors

Domain and Range in Real Life The number of shoes in x pairs of shoes can be expressed by the equation y = 2x. What is the independent variable? The # of pairs of shoes. What is the dependent variable? The total # of shoes.

Domain and Range in Real Life Mr. Landry is driving to his hometown. It takes four hours to get there. The distance he travels at any time, t, is represented by the function d = 55t (his average speed is 55mph. What is the independent variable? The time that he travels. What is the dependent variable? The total distance traveled.

Domain and Range in Real Life Johnny bought at most 10 tickets to a concert for him and his friends. The cost of each ticket was $12.50. What is the independent variable? The # of tickets bought. What is the dependent variable? The total cost of the tickets.

Function Notation When we know that a relation is a function, the “y” in the equation can be replaced with f(x). f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. The ‘f’ names the function, the ‘x’ tells the variable that is being used.

Value of a Function Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2. Find f(4): f(4) = 4 - 2 f(4) = 2

Guiding Questions (you should be able to answer) What are synonyms for the x-values? What are synonyms for y-values? How do you determine if a relation is a function? Why is it okay if the y-values repeat in a function? Explain how a why the vertical line test works. What is the difference between continuous and discrete? What is the difference between the independent variable and the dependent variable?