1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 3.1: The Cartesian Coordinate System

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o The components of the Cartesian coordinate system. o The graph of an equation. o The distance formula.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Components of the Cartesian Coordinate System o Many problems are naturally expressed with two or more variables. To solve these problems, we must determine all of the values of the variables that make the equation or inequality true. o For example, an equation with the variables and will have a solution consisting of a value for and a corresponding value for. A solution of the equation cannot consist of a value for only one of the variables. o Such equations are graphed on a two-dimensional coordinate system.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Cartesian Coordinate System The Cartesian coordinate system consists of two perpendicular real number lines (each of which is an axis). The point of intersection is called the origin of the system, and the four quarters defined by the two lines are called the quadrants of the plane.

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Cartesian Coordinate System Each point in the plane is identified by an ordered pair (a,b). In a given ordered pair the first coordinate indicates the horizontal displacement of the point from the origin, and the second coordinate indicates the vertical displacement.

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Cartesian Coordinate System Caution! Mathematics uses parentheses to denote ordered pairs as well as open intervals, which sometimes leads to confusion. You must rely on the context to determine the meaning of any parentheses you encounter. For instance, in the context of solving a one-variable inequality, the notation most likely refers to the open interval with endpoints at and where as in the context of solving an equation in two variables, probably refers to a point in the Cartesian plane.

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Graph of an Equation The horizontal number line is referred to as the x-axis, the vertical number line as the y-axis, and the two coordinates of the ordered pair as the x- coordinate and the y- coordinate.

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Graph of an Equation o The graph of an equation consists of a depiction in the Cartesian plane of all of those ordered pairs that make up the solution set of the equation. o We can find individual ordered pair solutions of a given equation by selecting numbers that seem appropriate for one of the variables and then solving the equation for the other variable.

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: The Graph of an Equation Sketch the graph of the following equation by plotting points.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: The Graph of an Equation Sketch a graph of the following equation by plotting points.

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Distance Formula Let and be the coordinates of two arbitrary points in the plane. By drawing the dotted lines parallel to the coordinate axes, we can form a right triangle. Note that we are able to determine the coordinates of the vertex at the right angle from the two vertices and.

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Distance Formula o The lengths of the two perpendicular sides of the triangle from the previous slide are easily calculated, as these lengths correspond to distances between numbers on the real number lines. o We can apply the Pythagorean Theorem to determine the distance, as labeled on the previous slide.

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Distance Formula Letting and represent two points on the Cartesian plane, the distance between these two points may be found using the following distance formula, derived from the Pythagorean Theorem:

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: The Distance Formula Determine the distance between and.

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Midpoint Formula Midpoint Formula The midpoint between two points and in the Cartesian plane has the following coordinates:

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: Using the Midpoint Formula Calculate the midpoint of the line segment connecting each pair of points. a.(5, 1) and (−1, 3) b.(3, 0) and (−6, 11) Solutions: In each case, we simply substitute the coordinates of each point into the midpoint formula. This has the effect of averaging both x-coordinates and both y-coordinates.

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: The Distance Formula Find the perimeter of the triangle whose vertices are the points (-1,-2), (2,-2), and (2,2). Is it a right triangle? Define isosceles and scalene.