College Algebra Lecture Notes
Section 2.1
Page 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B
o P is the domain set
o 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.
o Examples:

A second way to express a relation is as a set of ordered pairs.
o Example: {(-2, -2), (-1, 1), (0, 2), (1, 1), (2, -1)}
o Example: {(-5, 2), (0, 2), (5, 6), (6, 5), (2, 0), (2, -5)}
o The first coordinate represents values of the independent variable.
o The second coordinate represents values of the dependent variable.
o The set of all first coordinates is called the domain.
o The set of all second coordinates is called the range.
Practice:
1. .
2. .
College Algebra Lecture Notes
Section 2.1
Page 2 of 6
B. The Graph of a Relation
 A third way to express a relation is with an equation.
o Example: y  2 x  1
o Example: x  y  1  2

A fourth way to express a relation is with a graph.
o Example: sometimes the graph is just given to us.
o Example: sometimes we make a graph of a relation specified by a set of ordered
pairs.
o Example: sometimes we make a graph of a relation specified by an equation.
Practice:
3. .
College Algebra Lecture Notes
Section 2.1
Page 3 of 6
C. The Equation of a Circle
The Midpoint Formula:
Given any line segment with endpoints P1   x1 , y1  and P2   x2 , y2  , the midpoint M is given by
 x  x y  y2 
M  1 2 , 1

2 
 2
The Distance Formula:
Given any two points P1   x1 , y1  and P2   x2 , y2  , the straight line distance between them is
d
 x2  x1    y2  y1 
2
2
The Equation of a Circle:
A circle of radius r with center at  h, k  has the equation
 x  h   y  k 
2
2
 r2
College Algebra Lecture Notes
Section 2.1
Page 4 of 6
Practice:
4. Compute the midpoint of the segment with endpoints at (-4, 7) and (3,-2).
5. Compute the distance between the points (-4, 7) and (3,-2).
College Algebra Lecture Notes
Section 2.1
Page 5 of 6
6. 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:
2
2
7. Graph the circle described by the equation  x  3   y  1  45 .
College Algebra Lecture Notes
Section 2.1
8. Graph the circle described by the equation x 2  y 2  2 x  4 y  4  0 .
Page 6 of 6