### Day 1: Linear Functions and Graphs

#### by: BJ Jackson

Objectives:

1. to be able to find the domain and range of a relation

2. to be able to determine if a relation is a function.

3. to find ordered pairs that are in a function

4. to be able to write a linear function that models a given situation.

Definitions:

Relation: A pairing of two sets of numbers. Usually, this is thought of as any set of ordered pairs.

Domain: The first numbers in the set of ordered pairs or the set of all x-values in the given relation.

Range: The second numbers in the set of ordered pairs or the set of all y-values in the given relation.

Function: A pairing between two sets of numbers where the first element(x-value) of the set is paired with exactly one element of the second set(y-value).

Discussion

A relation is any set of ordered pairs which can be represented in many forms such as a table, a graph, a list of ordered pairs, etc.

Example 1: A table that pairs the age and weight of different people. So, for each person below we have two pieces of information. First, we are given their age which is first number in the set or the domain. Second, we are given their weight which is second number in the set or the range.

 Joe Fred Rita Jane Kelly Jose Edgar Age 14 20 18 24 19 30 27 Weight 150 180 110 125 142 210 197

Example 2: Another way of listing the data in the table above is as a set of ordered pairs with age used as the x-values and weight used as the y-values which would look like this.

(14,150), (20,180), (18,110), (24,125), (19, 142), (30,210), (27,197).

Example 3: Points could also be plotted on a graph and used to represent a relation which would look like the following.

Now, it should be easy for one to use the graph to find the coordinates of the points. Once we understand what a relation is, we can now find the domain and range of the relation and list them. This listing will have the following notation.

1) For domain: D={element, element,......}

2) For range: R={element, element,...}

Note, that we will list each element only once and usually in ascending order.

Example 4: List the domain and range of example 1.

D={14, 18, 19, 20, 24, 27, 30}

R={110, 125, 142, 150, 180, 197, 210}

On your own 1: List the relation in example 3 as, a set of points, in a table, and list the domain and range of the relation.

It is now time to understand when a relation is a function. A relation is a function when every element in the domain corresponds to one and only one value in the range. That is, every x has a specific y that it is associated with. It is often easier to determine when a relation is not a function then when it is a function. Example 5 will give some examples of relations that are not a function.

Example 5: Here is a relation that is not a function. See if you can look figure out why it is not a function before reading the explantion.

(1, 2), (3, 4), (5, 6), (1, 3)

To understand why it is not a function, start by looking at the domain. Are there any elements that repeat? Yes, the number 1 shows up twice in the domain. Now there is a potential problem because every element in the domain can have only 1 element in the range. Here, 1 has paired with two differnt elements, 2 and 3. So, this is not a function. Another way to look at would be as a graph.

Notice how the points (1,2) and (1,3) are directly above each other. This means that an element in the domain (x-values) has more then one element in the range (y-values).

On your Own 2: Are the relations in examples 1 - 3 functions?

1. (3, 7), (4, 11), (3, 7), (-8, 12)

2. (12.5, 9), (-11, -9), (8, -6.2), (-11, -10)

3. (1, 2), (2, 3), (3, 4), (3, 2 + 2)

The final concept of this section is linear function. A linear function is a function whose graph is a line. This means that there is the same slope(which will be covered in the next lesson) between every two points. In this section, we will take a function and write the equation given the be function in the form y = mx + b. The m (which stands for slope) is the difference between the the values and the b is the value when x = 0. If the x-values increase by something other than one, the difference in the y-values must be divided by the difference in the x-values. Once we have found the m and the b, we will substitute into the equation for m and b in order to have an equation that looks something like, y = 2x + 9. For all of these exercises, we will be given a function in the form of a table with the first row representing our x-value and the second row representing or y value.

Example 6:
 x 0 1 2 3 y 5 7 9 11

First, find the y-value when x = 0. In this case, y = 5 when x = 0. So, b = 5. Now, find the difference between the values which in this case is 2. So, m = 2. Now, write the equation that represents the linear function which is: y = 2x + 5.

Example 7:
 x 0 2 4 6 y 20 14 8 2

Here, b = 20 since that is the y-value when x = 0. The differnce in the y-values here is -6, but this is not the slope because the x-values increase by 2. So, in order to find m in this example one must divide the -6 by 2 to get -3. Now, substitute in the values to get the equation: y = -3x + 20.

On you Own 4: Try to write the equtions of the linear functions.

1.
 x 0 1 2 3 y 8 10 12 14

2.
 x 0 3 6 9 y 30 21 12 3

3.
 x 1 2 3 4 y 10 15 20 25