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Chapter 2.1 The Rectangular Coordinate System
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by Example  Series

2.1 The Rectangular Coordinate System

Key Concept: Understand the rectangular coordinate system with respect to point within it, graphing functions and
finding relative distance between points.

Skills to Learn:

1. Know how to interpret points with the rectangular coordinate system

(rect system)

2. Know how to graphs functions using a pair-wise table of values

3. Know how to graph horizontal and vertical graphs

4. Know how to use the distance formula to find the distance between points

5. Know how to find the midpoint of any two point with the rect. system.

Basic Properties of the Rectangular Coordinate System:

The Rectangular Coordinate System

4 Points on the Rectangular Coordinate

Examples of Graphs

Examples 1. Graph the equation 4 + 2y = x

First write y = f(x):

2y = x - 4, y = ½ x -2

Construct Table of Points

x y = ½ x -2 Points (x, y)
-4 ½ (-4) -2 =-4 (-4, -4)
0 ½ (0) - 2

= -2

 (0, -2)
4 ½ (4) -2

=0

 (4, 0)

The y-intercept, when x = 0 is: y = ½ (0) -2

= -2 or (0, -2)

The x-intercept, when y = 0 is: 0 = ½ (x) - 2

So ½ x = 2, x = 4 or (4, 0)

Graphs of function y = ½ x -2

Examples 2. Graph the equation 3y =-6x + 3

First write y = f(x):

y = -2x + 1 by dividing by 3

Construct Table of Points

x y = -2x + 1 Points (x, y)
-1 -2 (-1) + 1

= 3

 (-1, 3)
0 -2 (0) + 1

= 1

 (0, 1)
1 -2 (1) + 1

= -1

 (1, -1)

The y-intercept, when x = 0 is: y = -2(0) +1

= 3 or (0, 3)

The x-intercept, when y = 0 is: 0 = -2x + 1

So -2x = -1, x = ½ or (½, 0)

Graphs of function y = -2x + 1

Examples 3. Graph these 2 functions and tell graphically where the intersect (1) y =-3x + 3 and (2) y = 4x +2

Construct Table of Points for both functions

x Fct

(1)

 Fct (2) Points

(1)

 Points (2)
-1 6 -2 (-1, 6) (-1, -2)
0 3 2 (0, 3) (0, 2)
1 0 6 (1, 0) (1, 6)

Intersection (Graphically), from graph both lines appear to intersect at coordinates

(0.15, 2.5)

Intersection (Algebraically),

let -3x + 3 = 4x + 2

So 3-2 = 4x + 3x, 7x = 1, x = 1/7 = 0.1429

Therefore y = 4(1/7)+2 = 2.5714

So coordinates where both intersect are:

(0.1429, 2.5714)

 Graphs of function Point of Intersection of two functions

Examples 4. Horizontal Lines, y = a

Example 5 - Vertical lines, x = a

Examples: Distance and Midpoint Formulas

Examples 6. Find the distance between these 2 points on a rectangular coordinate system; (-2, 4) and (2, -1)

Use the distance formula:

Let

Then

Note: is the point with the smallest value of x

 Distance Formula Illustrated

Examples 7. Find the distance between these 2 points on a rectangular coordinate system; (-3, -3) and (3, 4)

Use the distance formula:

Let

Then

Note: the similarity between the distance formula and Pythagorean theorem

 Distance Formula Illustrated

Examples 8. Find the midpoint of the line segment joining P(-3, 8) and Q(4, -4)

Use the midpoint formula:

Let

Then midpoint is:

Note: Pythagorean theorem ( )

 Distance & Midpoint Formulas Illustrated

Examples 9. Find the midpoint of the line segment joining P(-5, -4) and Q(3, 7)

Use the midpoint formula:

Let

Then midpoint is:

Note: Pythagorean theorem ( )

 Distance & Midpoint Formulas Illustrated