2-1: 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.

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 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.

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.

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

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

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

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

Value of a Function If g(s) = 2s + 3, find g(-2). g(-2) = 2(-2) + 3 =-4 + 3 = -1 g(-2) = -1

Value of a Function If h(x) = x2 - x + 7, find h(2c). h(2c) = (2c)2 – (2c) + 7 = 4c2 - 2c + 7

Value of a Function If f(k) = k2 - 3, find f(a - 1) f(a - 1)=(a - 1)2 - 3 (Remember FOIL?!) =(a-1)(a-1) - 3 = a2 - a - a + 1 - 3 = a2 - 2a - 2