Section 1.1

Rectangular Coordinates;

Graphing Utilities;

Introduction to Graphing Equations

SECTION 1.1 OBJECTIVES

1. Use the Distance Formula2. Use the Midpoint Formula3. Graph Equations by Hand by Plotting Points4. Graph Equations Using a Graphing Utility5. Use a Graphing Utility to Create Tables6. Find Intercepts from a Graph7. Use a Graphing Utility to Approximate Intercepts

x axis

y axis

origin

Rectangular or Cartesian Coordinate System

Rectangular Coordinates

Let's plot the point (6,4)

(-3,-5)

(0,7)Let's plot the point (-6,0)

(6,4)

(-6,0)

Let's plot the point (-3,-5) Let's plot the point (0,7)

Rectangular Coordinates

Quadrant I x > 0, y > 0

Quadrant II x < 0, y > 0

Quadrant III x < 0, y < 0

Quadrant IVx > 0, y < 0

Coordinate Axis

Coordinate Axis on Graphing Utility

Coordinate Axis on Graphing Utility

Coordinate Axis on Graphing Utility

Point on Axis on Graphing Utility

Find the coordinates of the point shown. Assume the coordinates are integers.

(- 2, 2)

OBJECTIVE 1

Horizontal or Vertical Segments

The Distance between two points is the absolute value of their difference

Not every pair of points lies on a vertical or horizontal line so the distance formula must be used.

Find the distance d between the points (2, 5) and (4, 8)

1 1 1

2 2 2

( , ) (2,5)

( , ) (4,8)

P x y

P x y

2 2

1 2

2 2

, 4 2 8 5

2 3

13

d P P

Find the length of the line segment shown.

2 2

1 2

2 2

, 3 4 2 5

7 3

49 9

58

d P P

A = (– 4, – 1), B = (1, 11), and C = (1, – 1)

a.

b. Length of AB= 13 Length of BC= 12 Length of AC= 5

c. 2 2 2

?2 2 25 12 13

25 144 169

169 169

a b c

d.

1

21

5 12230

A bh

A

A

A = (– 4, – 1), B = (1, 11), and C = (1, – 1)

OBJECTIVE 2

Development of Midpoint Formula

Find the midpoint of a line segment from P1 = (3, -5) to P2 = (1, 7). Plot the points P1 and P2 and their midpoint.

3 1 5 7, 2,1

2 2

OBJECTIVE 3

Example of Data Plotted by Hand

Determine if the following points are on the graph of the equation - 3x +y = 6

(b) (2, 0)(a) (0, 4) (c) (-1, 3)

3 2 0 6

6 6

Not on the graph

3 0 4 6

4 6

Not on the graph

3 1 3 6

6 6

On the graph

Steps to Graph an Equation by Hand by Plotting Points

1. Find all points (x, y) that satisfy the equation. To determine these points, choose values of x and use the equation to find the corresponding values for y. Create a table of values.

2. Plot the points listed in the table. Now connect the points to obtain the graph of the equation.

X Y

-2 1

-1 3

0 5

1 7

2 9

3y x

X Y

-2 - 8

-1 -1

0 0

1 1

2 8

OBJECTIVE 4

Steps for Graphing an Equation Using a Graphing

Utility1. Solve the equation for y in terms of x

2. Enter the equation to be graphed into your graphing utility. (y= editor)

3. Choose an initial viewing window. Without any knowledge about the behavior of the graph, it is common to choose the standard viewing window as the initial viewing window.

4. Graph the equation

5. Adjust the viewing window until a complete graph is obtained

Solve for y: – 2x + 5y + 3 = – 1

5 1 2 3

5 2 4

2 4

5 5

y x

y x

y x

2Use a graphing utility to graph the equation 2 12x y 22 12y x

OBJECTIVE 5

2Create a table that displays the points on the graph of 2 12

for 3, 2, 2, 0,1, 2, 3

x y

x

OBJECTIVE 6

Intercepts of a Graph

.X-intercepts are: (- 3, 0 ), (4.5, 0), (3/2, 0)

Y-intercepts are (0, - 3.5), (0, - 4/3), (0, 3)

OBJECTIVE 7

Approximating Intercepts Using a Graphing Utility

1. Use 2nd Calc, Value on the TI-83/84 calculator to find the y-intercept by entering 0 for the x-value.

2. Use 2nd Calc, Zero on the TI-83/84 calculator to find the x-intercept. See the owners’ manual for specific instructions.

3. For other calculators, check your owners’ manual.

22 12x y