Section 1.1

Introduction to Graphing

Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives

Plot points. Determine whether an ordered pair is a solution of an equation. Find the x-and y-intercepts of an equation of the form Ax + By = C. Graph equations. Find the distance between two points in the plane and find the midpoint of a segment. Find an equation of a circle with a given center and radius, and given an equation of a circle in standard form, find the center and the radius. Graph equations of circles.

Cartesian Coordinate System

Example

To graph or plot a point, the first coordinate tells us to move left or right from the origin. The second coordinate tells us to move up or down.

Plot (3, 5).Move 3 units left.Next, we move 5 units up.Plot the point.

(–3, 5)

Solutions of Equations

Equations in two variables have solutions (x, y) that are ordered pairs.

•Example: 2x + 3y = 18

When an ordered pair is substituted into the equation, the result is a true equation. The ordered pair has to be a solution of the equation to receive a true statement.

Examples

a.Determine whether the ordered pair (5, 7) is a solution of 2x + 3y = 18.

2(5) + 3(7) ? 1810 + 21 ? 18 11 = 18 FALSE

(5, 7) is not a solution.

b.Determine whether the ordered pair (3, 4) is a solution of 2x + 3y = 18.

2(3) + 3(4) ? 18 6 + 12 ? 18 18 = 18 TRUE (3, 4) is a solution.

Graphs of Equations

To graph an equation is to make a drawing that represents the solutions of that equation.

x-Intercept

The point at which the graph crosses the x-axis.

An x-intercept is a point (a, 0). To find a, let y = 0 and solve for x.

Example

Find the x-intercept of 2x + 3y = 18.

2x + 3(0) = 18 2x = 18

x = 9

The x-intercept is (9, 0).

y-Intercept

The point at which the graph crosses the y-axis.

A y-intercept is a point (0, b). To find b, let x = 0 and solve for y.

Example

Find the y-intercept of 2x + 3y = 18.

2(0) + 3y = 18 3y = 18

y = 6

The y-intercept is (0, 6).

Example

Graph 2x + 3y = 18.

We already found the x-intercept: (9, 0)We already found the y-intercept: (0, 6)We find a third solution as a check. If x is replaced with 5, then

253y18103y18

3y8

y8

3Thus, is a solution.5,

8

3

Example (continued)

Graph: 2x + 3y = 18.x-intercept: (9, 0)y-intercept: (0, 6)Third point:

5,8

3

Example

Graph y = x2 – 9x – 12 .

(12, 24)2412

–2

32

32

26

12

–2

24

y

(10, –2)10

(5, 32)5

(4, 32)4

(2, 26)2

(0, 12)0

(1, –2)1(3, 24)3(x, y)x

Make a table of values.

The Distance Formula

The distance d between any two points (x1, y1) and (x2, y2) is given by

d (x2 x1)2 (y2 y1)2 .

Example

d (3 2 )2 ( 6 2)2

d 52 ( 8)2 2564

d 89 9.4

Find the distance between the points (–2, 2) and (3, 6).

Midpoint Formula

If the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are

x1 x2

2,y1 y2

2

.

Example

Find the midpoint of a segment whose endpoints are (4, 2) and (2, 5).

4 2

2,

25

2

2

2,

3

2

1, 3

2

Circles

A circle is the set of all points in a plane that are a fixed distance r from a center (h, k).

The equation of a circle with center (h, k) and radius r, in standard form, is

(x h)2 + (y k)2 = r2.

Example

Find an equation of a circle having radius 5 and center (3, 7).

Using the standard form, we have

(x h)2 + (y k)2 = r2

[x 3]2 + [y (7)]2 = 52

(x 3)2 + (y + 7)2 = 25.