Graphing Linear Relations and Functions Objectives: Understand, draw, and determine if a relation is a function. Graph & write linear equations, determine domain and range. Understand and calculate slope.

Domain & Range A relation is a set of ordered pairs. Domain: first components in the relation (independent) Range: second components in the relation (dependent, the value depends on what the domain value is) Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element.

EXAMPLE Consider the function: eye color (assume all people have only one color, and it is not changeable) It IS a function because when asked the eye color of each person, there is only one answer. i.e. {(Joe, brown), (Mo, blue), (Mary, green), (Ava, brown), (Natalie, blue)} NOTE: the range values are not necessarily unique.

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 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)}

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

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.

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.

Graphing a function Horizontal axis: x values Vertical axis: y values Plot points individually or use a graphing utility (calculator or computer algebra system) Example:

Graphs of functions

Graphs of a Function Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.

What is the domain & range of the function with this graph?

x y x y Does the graph represent a function? Name the domain and range. Yes D: all reals R: all reals Yes D: all reals R: y ≥ -6

x y x y Does the graph represent a function? Name the domain and range. No D: x ≥ 1/2 R: all reals No D: all reals R: all reals

Does the graph represent a function? Name the domain and range. Yes D: all reals R: y ≥ -6 No D: x = 2 R: all reals x y x y

Can you identify domain & range from the graph? Look horizontally. What all x-values are contained in the graph? That’s your domain! Look vertically. What all y-values are contained in the graph? That’s your range!

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) = f(4) = 2

Value of a Function If g(s) = 2s + 3, find g(-2). g(-2) = 2(-2) + 3 = = -1 g(-2) = -1

Value of a Function If h(x) = x 2 - x + 7, find h(2c). h(2c) = (2c) 2 – (2c) + 7 = 4c 2 - 2c + 7

HOMEWORK: Complete worksheet packet Worksheet packet due November 6 th Sit quietly AND work until the bell rings!!!!!