2.1 - 1 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc

2.1 - 1Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

2Graphs and Functions

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2

2.1Rectangular Coordinates and Graphs

• Ordered Pairs• The Rectangular Coordinate System• The Distance Formula• The Midpoint formula• Graphing Equations in Two Variables


Ordered Pairs

An ordered pair consists of two components, written inside parentheses.


Example 1 WRITING ORDERED PAIRS

CategoryAmount Spent

food $6443

housing $17,109

transportation $8604

health care $2976

apparel and services

$1801

entertainment $2835

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



Solution

(a) housing

Use the data in the second row: (housing, $17,109).


food $6443

housing $17,109


health care $2976


$1801

entertainment $2835



Solution

(b) entertainment

Use the data in the last row: (entertainment, $2835).


food $6443

housing $17,109


health care $2976


$1801

entertainment $2835


The Rectangular Coordinate System

Quadrant IQuadrant II

Quadrant III Quadrant IV

0x-axis

y-axis

a

bP(a, b)


The Distance Formula

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

P(– 4, 3) Q(8, 3)

R(8, – 2)8( , ) ( ) 12.4d P Q

Horizontal side of the triangle has length

Definition of distance

x

y



P(– 4, 3) Q(8, 3)

R(8, – 2)

( , ) ( ) 53 2 .d P Q

Vertical side of the triangle has length

x

y



P(– 4, 3) Q(8, 3)

R(8, – 2)

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

2 212 5 144 25 169 13.

x

y



P(– 4, 3) Q(8, 3)

R(8, – 2)

So the distance between (– 4, 3) and (8, – 2) is 13. x

y


The Distance FormulaTo 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.

2 22 1 2 1( , ) ( ) ( )d P R x x y y

P(x1, y1)Q(x2, y1)

R(x2, y2)

x

y


Distance Formula

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

2 22 1 2 1( , ) ( ) ( )d P R x x y y


Example 2 USING THE DISTANCE FORMULA

Solution

Find the distance between P(– 8, 4) and Q(3, – 2.)

2 2( ) 28 43 ( )

2 211 ( 6) Be careful when subtracting a

negative number.121 36

157

2 22 1 2 1( , ) ( ) ( )d P Q x x y y


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 APPLYING THE DISTANCE FORMULA

Are points M(– 2, 5), N(12, 3), and Q(10, – 11) the vertices of a right triangle?

Solution A triangle with the three given points as vertices 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 3

2 212( , ) ( 2 ( )3 5)d M N

200

196 4

APPLYING THE DISTANCE FORMULA


Example 3

2 210 2( , ) ( 5( 11) )d M Q

400

144 256

20



Example 3

2 210 12( , ) ( ) 1( 3)1d N Q

200

4 196



Example 3

The longest side, of length 20 units, is chosen as the possible hypotenuse. Since

2 22200 200 400 20

is true, the triangle is a right triangle with hypotenuse joining M and Q.



Collinear Points

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.



Are the points P(– 1, 5), Q(2, – 4), and R(4, – 10) collinear?Solution 22, ( 1 2) 5 ( 4) 9 81

3 0

90

1

d P Q

22, (2 4) 4 ( 10)

2 1

3 0

0

4 6 4d Q R

22, ( 1 4) 5 ( 10) 25 225

25 5 100

d P R



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

Solution

Because is true,

the three points are

5

c

10

ol

3 1

li

0 2 10

near.


Midpoint Formula

The midpoint M of the line segment with endpoints P(x1, y1) and Q(x2, y2) has the following coordinates.

1 2 1 2,2 2

x x y y


Example 5 USING THE MIDPOINT FORMULA

Use the midpoint formula to do each of the following.

Solution

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

( ) 3,

48 61,

2 21

2

Substitute in the midpoint formula.



Use the midpoint formula to do each of the following.

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

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

Solution



( 6)8

2x

6 16x

22x

x-value of P x-value of M

Substitute carefully.

x-value of P y-value of M

122

2y

12 4y

16y

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


Example 6 APPLYING THE MIDPOINT FORMULA

The following figure depicts how a graph might indicate the increase in the revenue generated by fast-food restaurants in the United States from $69.8 billion in 1990 to $164.8 billion in 2010. Use the midpoint formula and the two given points to estimate the revenue from fast-food restaurants in 2000, and compare it to the actual figure of $107.1 billion.





Solution The year 2000 lies halfway between 1990 and 2010, so we must find the coordinates of the midpoint of the segment that has endpoints(1990, 69.8) and (2010, 164.8).

69.8 164., (2000, 117.3)

2 21990 82010

Thus, our estimate is $117.3 billion, which is greater than the actual figure of $107.1 billion.


Example 7 FINDING ORDERED-PAIR SOLUTIONS OF EQUATIONS

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

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

(a) 4 1y x



Solution

(a) 4 1y x

4 1y x 4( ) 12y Let x = – 2.

8 1y Multiply.

9y

4 1y x 3 4 1x Let y = 3.

4 4x Add 1.

1 x Divide by 4.

This gives the ordered pairs (– 2, – 9) and (1, 3).Verify that the ordered pair (0, –1) is also a solution.



Solution

(b) 1x y Given equation

Let x = 1.

Square each side.

One ordered pair is (1, 2). Verify that the ordered pairs (0, 1) and (2, 5) are also solutions of the equation.

1x y

1 1y

1 1y

2 y Add 1.



Solution

(c) A table provides an organized method for determining ordered pairs.

2 4y x

Five ordered pairs are (–2, 0), (–1, –3), (0, –4), (1, –3), and (2, 0).

x y

– 2 0

– 1 – 3

0 – 4

1 – 3

2 0


Graphing an Equation by Point PlottingStep 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 Join the points from Step 3 with a smooth line or curve.


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

Example 8 GRAPHING EQUATIONS

(a) Graph the equation Solution

4 1.y x

4 1y x 0 4 1x 1 4x14

x

4 1y x 4( ) 10y 0 1y

1y

1,0 and 0, 1 .

4 These intercepts lead to the ordered pairs


Step 2 We find some other ordered pairs (also found in Example 7(a)).



4 1.y x

4 1y x 4( ) 12y Let x = – 2.

8 1y Multiply.

9y

4 1y x 3 4 1x Let y = 3.

4 4x Add 1.

1 x Divide by 4.

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


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



4 1.y x

Step 4 Join the points with a straight line.



(b) Graph the equation Solution

1.x y

Plot the ordered pairs found in Example 7b, and then join the points with a smooth curve. To confirm the direction the curve will take as x increases, we find another solution, (3, 10).



(c) Graph the equation Solution Plot the points and join them with a smooth curve.

2 4.y x

x y

– 2 0

– 1 – 3

0 – 4

1 – 3

2 0This curve is called a parabola.

Documents

2.1 - 1 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc