A graph of function is a visual representation of elements of the function on the coordinate system. In this section, we will consider plotting elements (ordered pairs) of a special relation, called linear function.

1. Suppose a relation R is the set of ordered pairs of the points shown by the dots on the adjacent xy-coordinate system.

(a) List the elements of R by the roster (complete listing) method.
(b) Describe the elements of R by the set builder method.

2. Plot the following relations (set of ordered pairs) in the xy-coordinate plane.
( i.e., show the points corresponding to the ordered pairs by dots (•) or crosses(×).)

(a) R₁ = {(-1, -2), (0, 0), (1, 2), (2,4), (3, 6)}

(b) R₂ = {(x, y) | y =2x +1, x∈A} where A ={-2, -1, ..., 2}

3. Let R = A × B, where A = {x | x ∈ ℝ, 0 ≤ x ≤ 2 } and B ={2}. i.e., R = {(x, 2) | x ∈ A}.
(a) List as many elements of R as you can (at least five) and plot them on the xy- coordinate system.

(b) Observe that R has infinitely many elements. What do you get if you try to plot all elements of R?

4. Let R be a relation on A = {x | x∈ℝ, 0 ≤ x ≤ 2 } given by R = {(x, y) | y =x}.
(a) List as many elements of R as you can (at least five) and plot them on the Cartesian coordinate plane.

(b) What do you get if you try to plot all elements of R?

As you have seen in the above activity, the graph of a function (relation) on the xy- coordinate plane is the set of points on the plane whose coordinate pairs are the ordered pairs of the functions. In this section, we will focus on the graph a linear function defined below.

Definition

If a and b are fixed real numbers, a ≠ 0, then f(x) = ax + b for x∈ℝ is called a linear function.

Sometimes linear functions are written as y = ax + b because, as a set of ordered pairs, the linear function can be written as

• f = {(x, y) | y = ax + b, x∈ℝ}.
• If a = 0, f(x) = b is called a constant function.