College Algebra Lecture Notes Section 2.1Page 1 of 6

Section 2.1: Rectangular Coordinates; Graphing Circles and Relations

Big Idea:Relationships between two quantities can be visualized on a graph.

Big Skill:You should be able to graph relations given by equations using a table of values, and graph the equation of a circle using the clues from the form of the equation.

A. Relations. Mapping Notation, and Ordered Pairs

  • A relation is a correspondence between two sets.
  • Mapping notation is used to show that one set has corresponding elements in another set:
  • P is the domain set
  • B is the range set
  • One way to express a relation is as a mapping that shows the correspondence from the elements in one set to the other set.
  • Examples:

  • A second way to express a relation is as a set of ordered pairs.
  • Example: {(-2, -2), (-1, 1), (0, 2), (1, 1), (2, -1)}
  • Example: {(-5, 2), (0, 2), (5, 6), (6, 5), (2, 0), (2, -5)}
  • The first coordinate represents values of the independent variable.
  • The second coordinate represents values of the dependent variable.
  • The set of all first coordinates is called the domain.
  • The set of all second coordinates is called the range.

Practice:

  1. .
  2. .

B. The Graph of a Relation

  • A third way to express a relation is with an equation.
  • Example:
  • Example:
  • A fourth way to express a relation is with a graph.
  • Example: sometimes the graph is just given to us.
  • Example: sometimes we make a graph of a relation specified by a set of ordered pairs.
  • Example: sometimes we make a graph of a relation specified by an equation.

Practice:

  1. .

C. The Equation of a Circle

The Midpoint Formula:

Given any line segment with endpoints and , the midpoint M is given by

The Distance Formula:

Given any two pointsand , the straight line distance between them is

The Equation of a Circle:

A circle of radius rwith center at has the equation

Practice:

  1. Compute the midpoint of the segment with endpoints at (-4, 7) and (3,-2).
  1. Compute the distance between the points (-4, 7) and (3,-2).

  1. Find the equation for a circle with a center at (-2, -1) and a radius of 5.

D. The Graph of a Circle

To quickly sketch a circle given its equation:

  • Manipulate the equation until it is in the standard form for the equation of a circle.
  • You may have to complete the square to do this.
  • Compare the numbers in the equation to the standard form of the equation to identify h, k, and r,
  • Plot the center at (h, k).
  • Plot points that are r units above, below, to the right, and to the left of the center.
  • Connect those four points with a circular curve.

Practice:

  1. Graph the circle described by the equation .
  1. Graph the circle described by the equation .