Math 025 Section 7.1 Coordinate Systems

y-axis Quadrant II Quadrant I x-axis Origin Quadrant III Quadrant IV

Each point in the plane can be identified by an ordered pair (5, 7) The ordered pair tells the location of the point with reference to the origin Example: (5, 7) The numbers in the ordered pair are called the coordinates of the point 5 is the x-coordinate or abscissa 7 is the y-coordinate or ordinate The graph of a point is a dot placed at the location of the point

Graph the following ordered pairs: A(-2, -3) B(3, -2) C(0,2) D(-3,0) y C D x B A

Give the coordinates of A and B Give the abscissa of C y Give the abscissa of C B Give the ordinate of D C A D x Answers: Coordinates of A are (-4,2) Coordinates of B are (4, 4) Abscissa of C is -1 Ordinate of D is 1

Objective: To check solutions of an equation in two variables. Question: Is (-3, 7) a solution of y = -2x + 1 ? y = -2x + 1 Replace x with -3 7 = -2(-3) + 1 Replace y with 7 7 = 6 + 1 7 = 7 Both sides of the equation simplify to the same thing. Yes, (-3, 7) is a solution

Objective: To check solutions of an equation in two variables. Question: Is (3, -2) a solution of 3x – 4y = 15 ? 3x – 4y = 15 3(3) – 4(-2) = 15 Replace x with 3 Replace y with -2 9 + 8 = 15 17 = 15 Both sides of the equation are not the same. No, (3, -2) is not a solution

Problem: Graph the ordered-pair solutions of 2x – 3y = 6 Problem: Graph the ordered-pair solutions of 2x – 3y = 6 when x = -3, 0, 3 and 6 Solve 2x – 3y = 6 for y - 3y = -2x + 6 y = 2x – 2 3 x y 5 -3 2(-3) – 2 -4 3 2(0) – 2 -2 3 -5 3 2(3) – 2 3 6 2(6) – 2 2 3

Problem: Graph the ordered-pair solutions of y = 2x – 1 Problem: Graph the ordered-pair solutions of y = 2x – 1 when x = -2, 0, 1 and 3 y = 2x - 1 x y 3 -2 -5 -1 1 1 3 3 5

Objective: To determine if a set of ordered pairs is a function Definition of a relation A relation is any set of ordered pairs. Example: {(2, 3), (5, -4), (-7, 8), (-12, 8)} Definition of a function A function is a relation in which no two ordered pairs have the same first coordinate. Example: {(2, 3), (5, -4), (-7, 8), (-12, 8)} is a function

Objective: To determine if a set of ordered pairs is a function State whether each of the following relations is a function No {(5, 3), (5, -4), (-7, 12), (-5, 12)} Yes {(3, 3), (6, -5), (-7, 12), (-5, 12), (8, 3)} {(3, 3), (6, 3), (-7, 3), (-5, 3), (8, 3)} Yes {(2, 3), (2, -5), (2, 12), (2, 15), (2, 9)} No

Does the equation express y as a function of x ? y = 0.5x + 1 where x Î {-4, 0, 2} x y The relation for this domain is -4 0 2 -1 {(-4,-1), (0, 1), (2, 2)} 1 2 Yes, the equation is a function

Does the equation express y as a function of x ? |y| = x + 2 where x Î {-2, 0, 2} x y When x = 2, |y| = 0 so y = 0 -2 When x = 0, |y| = 2 so y = 2 or y = -2 2 When x = 2, |y| = 4 so y = 4 or y = - 4 -2 2 4 The relation for this domain is 2 -4 {(-2, 0), (0, 2), (0, -2), (2, 4), (2, -4)} No, the equation is not a function

Objective: To evaluate a function that is written in function notation. When an equation such as y = x2 + 3 defines y as a function of x, the following function notation is often used to emphasize that the relation is a function f(x) = x2 + 3 f(x) is read “the value of the function f at x” or “f of x” The expression f(4) means “the value of the function when x = 4” so f(4) = (4)2 + 3 = 16 + 3 = 19 This process is called evaluating the function

Objective: To evaluate a function that is written in function notation. Problem: Given f(x) = 5x + 1 find f(2) Solution: f(x) = 5x + 1 f(2) = 5(2) + 1 = 11 Problem: Given Q(r) = 4r2 – r – 3 find Q(3) Solution: Q(r) = 4r2 – r – 3 Q(3) = 4(3)2 – (3) – 3 = 36 – 3 – 3 = 30