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College Algebra & Trigonometry and Precalculus. 4 th EDITION. 2.1. Rectangular Coordinates and Graphs. Ordered Pairs The Rectangular Coordinate System The Distance Formula The Midpoint formula Graphing Equations. Ordered Pairs.

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4 th EDITION

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4 th edition

College Algebra & Trigonometry

and

Precalculus

4th EDITION


4 th edition

2.1

Rectangular Coordinates and Graphs

Ordered Pairs

The Rectangular Coordinate System

The Distance Formula

The Midpoint formula

Graphing Equations


Ordered pairs

Ordered Pairs

An ordered pair consists of two components, written inside parentheses, in which the order of the components is important.


Example 1

WRITING ORDERED PAIRS

Example 1

Use the table to write ordered pairs to express the relationship between each type of entertainment and the amount spent on it.


Example 11

WRITING ORDERED PAIRS

Example 1

a. DVD rental sales

Solution:

Use the data in the second row:

(DVD rental sales, $71).


Example 12

WRITING ORDERED PAIRS

Example 1

b. Movie tickets

Solution:

Use the data in the last row:

(Movie tickets, $30).


The rectangular coordinate system

The Rectangular Coordinate System

y-axis

Quadrant I

Quadrant II

P(a, b)

b

x-axis

a

0

Quadrant III

Quadrant IV


The distance formula

The Distance Formula

Using the coordinates of ordered pairs, we can find the distance between any two points in a plane.

y

Horizontal side of the triangle has length

Q(8, 3)

P(–4, 3)

x

R(8, –2)

Definition of distance.


The distance formula1

The Distance Formula

y

Vertical side of the triangle has length…

Q(8, 3)

P(–4, 3)

x

R(8, –2)


The distance formula2

The Distance Formula

y

By the Pythagorean theorem, the length of the remaining side of the triangle is

Q(8, 3)

P(–4, 3)

x

R(8, –2)


The distance formula3

The Distance Formula

y

So the distance between (–4, 3) and (8, –4) is 13.

Q(8, 3)

P(–4, 3)

x

R(8, –2)


The distance formula4

The Distance Formula

y

To obtain a general formula for the distance between two points in a coordinate plane, let P(x1, y1) and R(x2, y2) be any two distinct points in a plane. Complete a triangle by locating point Q with coordinates (x2, y1). The Pythagorean theorem gives the distance between P and R as

Q(x2, y1)

P(x1, y1)

x

R(x2, y2)


The distance formula5

The Distance Formula

Note Suppose that P(x1, y1) and R(x2, y2) are two points in a coordinate plane. Then the distance between P and R, written d(P, R) is given by


Example 2

USING THE DISTANCE FORMULA

Example 2

Find the distance between P(–8, 4) and

Q(3, –2.)

Solution:

Be careful when subtracting a negative number.


Using the distance formula

Using the Distance Formula

If the sides a, b, and c of a triangle satisfy a2 + b2 = c2, then the triangle is a right triangle with legs having lengths a and b and hypotenuse having length c?


Example 3

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3

Are points M(–2, 5), N(12, 3), and

Q(10, –11) the vertices of a right triangle.

Solution This triangle is a right triangle if the square of the length of the longest side equals the sum of the squares of the lengths of the other two sides.


Example 31

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3

y

N(12, 3)

M(–2, 5)

x

Q(10, –11)


Example 32

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3

y

N(12, 3)

M(–2, 5)

x

Q(10, –11)


Example 33

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3

y

N(12, 3)

M(–2, 5)

x

Q(10, –11)


Example 34

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3

y

N(12, 3)

M(–2, 5)

x

R(10, –11)


Colinear

Colinear

We can tell if three points are collinear, that is, if they lie on a straight line, using a similar procedure

Three points are collinear if the sum of the distances between two pairs of points is equal to the distance between the remaining pair of points.


Example 4

DETERMINING WHETHER THREE POINTS ARE COLLINEAR

Example 4

Are the points (–1, 5), (2,–4), and (4, –10) collinear?

Solution The distance between (–1, 5) and (2,–4) is


Example 41

DETERMINING WHETHER THREE POINTS ARE COLLINEAR

Example 4

Are the points (–1, 5), (2,–4), and (4, –10) collinear?

Solution The distance between (2,–4) and (4,–10) is


Example 42

DETERMINING WHETHER THREE POINTS ARE COLLINEAR

Example 4

Are the points (–1, 5), (2,–4), and (4, –10) collinear?

Solution The distance between the remaining pair of points (–1, 5) and (4,–10) is


4 th edition

Midpoint Formula

The midpoint of the line segment with endpoints (x1, y1) and (x2, y2) has coordinates


Example 5

USING THE MIDPOINT FORMULA

Example 5

Use the midpoint formula to determine the following.

a. Find the coordinates of M of the segment with endpoints (8, –4) and (–6,1).

Solution

Substitute in the midpoint formula.


Example 51

USING THE MIDPOINT FORMULA

Example 5

Use the midpoint formula to determine the following.

b. Find the coordinates of the other endpoint B of a segment with one endpoint A(–6, 12) and midpoint M(8, –2).

Solution

Let (x, y) represent the coordinates of B. Use the midpoint formula twice.


Example 52

USING THE MIDPOINT FORMULA

Example 5

Let (x, y) represent the coordinates of B. Use the midpoint formula twice.

x-value of A

x-value of M

x-value of A

y-value of M

Substitute carefully.

The coordinates of endpoint B are (22, –16).


Example 6

APPLYING THE MIDPOINT FORMULA TO DATA

Example 6

The graph indicates the increase of McDonald’s restaurants worldwide from 20,000 in 1996 to 31,000 in 2006.

Use the midpoint formula and the two given endpoints to estimate the number of restaurants in 2001, and compare it to the actual (rounded) figure of 30,000


Example 61

APPLYING THE MIDPOINT FORMULA TO DATA

Example 6

Number of McDonald’s Restaurants

Worldwide (in thousands)

35

30

25

20

2006

1996

Year


Example 62

APPLYING THE MIDPOINT FORMULA TO DATA

Example 6

Solution:2001 is halfway between 1996 and 2006. Find the coordinates of the midpoint of the segment that has endpoints (1996, 20) and (2006, 31).

Our estimate is 25,500, which is well below the actual figure of 30,000.


Example 7

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS

Example 7

For the equation, find at least three ordered pairs that are solutions

a.

Solution: Choose any real number for x or y and substitute in the equation to get the corresponding value of the other variable.


Example 71

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS

Example 7

a.

Solution:

Let x = –2.

Let y = 3.

Multiply.

Add 1.

Divide by 4.

This gives the ordered pairs (–2, –9) and (1, 3).

Find one more ordered pair that works!


Example 72

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS

Example 7

b.

Solution

Given equation

Let x = 1

Square both sides

One ordered pair is (1, 2).

Find two more ordered pair!


Example 73

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS

Example 7

c. A table provides an organization method for determining ordered pairs.

Solution:

Five ordered pairs are listed as solutions for this equation.


4 th edition

Graphing an Equation by Point Plotting

Step 1 Find the intercepts.

Step 2 Find as many additional ordered pairs as needed.

Step 3 Plot the ordered pairs from Steps 1 and 2.

Step 4 Connect the points from Step 3 with a smooth line or curve.


Example 8

GRAPHING EQUATIONS

Example 8

a. Graph the equation

Solution:

Step 1 Let y = 0 to find the x-intercept, and let x = 0 to find the y-intercept.


Example 81

GRAPHING EQUATIONS

Example 8

a. Graph the equation

Solution:

Step 2 We find some other ordered pairs.

Let y = 3.

Let x = –2.

Multiply.

Add 1.

Divide by 4.

This gives the ordered pairs (–2, –9) and (1, 3).


Example 82

GRAPHING EQUATIONS

Example 8

a. Graph the equation

Solution:

Step 3 Plot the four ordered pairs from Steps 1 and 2.

y

Step 4 Connect the points with a straight line.

x


Example 83

GRAPHING EQUATIONS

Example 8

b. Graph the equation

Solution:

y

Plot the ordered pairs found in example 7b, and develop a fourth point to confirm the direction the curve will take as x increases.

x


Example 84

GRAPHING EQUATIONS

Example 8

c. Graph the equation

Solution:

y

x

This curve is called a parabola.


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