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2.1 Your Notes Functions and Their Graphs Goal p Identify and graph functions. Vocabulary Relation A mapping, or pairing, of input values with output values Domain The set of input values of a relation Range The set of output values of a relation Function A relation for which each input has exactly one output Equation in two variables An equation that contains two variables Independent variable The input variable in an equation in two variables Dependent variable The output variable in an equation in two variables Example 1 Identify Functions Identify the domain and range. Then tell whether the relation is a function. Explain. a. Input Output 22 21 21 3 0 6 4 Solution b. Input Output 22 22 0 1 2 2 3 a. The domain consists of the inputs: 22, 21, 3 , and 6 . The range consists of the outputs: 21, 0 , and 4 . The relation is a function because each input is mapped onto exactly one output. b. The domain consists of the inputs: 22, 1 , and 2 . The range consists of the outputs: 22 , 0 , 2, and 3 . The relation is not a function because the input 2 is mapped onto both 0 and 3 . Copyright © McDougal Littell/Houghton Mifflin Company Lesson 2.1 • Algebra 2 C&S Notetaking Guide 25

2.1

Your Notes

Functions and Their Graphs

Goal p Identify and graph functions.

Vocabulary

Relation A mapping, or pairing, of input values

with output values

Domain The set of input values of a relation

Range The set of output values of a relation

Function A relation for which each input has

exactly one output

Equation in two variables An equation that contains

two variables

Independent variable The input variable in an

equation in two variables

Dependent variable The output variable in an

equation in two variables

Example 1 Identify Functions

Identify the domain and range. Then tell whether the

relation is a function. Explain.

a.

Input Output

22 21

21

3

0

6 4

Solution

b.

Input Output

22 22

0

1

2

2 3

a. The domain consists of the inputs: 22, 21, 3 ,

and 6 . The range consists of the outputs: 21, 0 ,

and 4 . The relation is a function because each

input is mapped onto exactly one output.

b. The domain consists of the inputs: 22, 1 , and 2 .

The range consists of the outputs: 22 , 0 , 2,

and 3 . The relation is not a function because the

input 2 is mapped onto both 0 and 3 .

Your Notes

VErTical linE TEST For FuncTionS

A relation is a function if and only if no vertical line

intersects the graph at more than one point .

Function Not a function

1

O

y

1

x

Example 2 Apply the Vertical Line Test

Graph the relations from Example 1. Use the vertical

line test to tell whether the relations are functions.

a. First, write the relation

as a set of ordered

pairs.

(–2, –1), (–1, 0) ,

(3, 0), (6, 4)

Then plot the points.

1

O

y

1

No vertical line

contains more

than one point so the

relation is a function .

x

1

O

y

1

b. First, write the relation

as a set of ordered

pairs.

(–2, –2) , (1, 0),

(2, 2) , (2, 3)

Then plot the points.

1

O

y

1

The vertical line at

x 5 2 contains two

different points in the

relation. So, the relation

is not a function.

x

Your Notes

Homework

GraphinG EquaTionS in Two VariablES

Step 1 Make a table of values. Write the ordered pairs .

Step 2 Plot enough solutions to recognize a pattern .

Step 3 Connect the points with a line or a curve .

Example 3 Graph an Equation

Graph y 5 22x 2 2.

Solution

1. Make a table of values . Write the ordered pairs .

Choose x 22 21 0 1 2

Evaluate y 2 0 22 24 26

(x, y): (22, 2 ), (21, 0 ), (0, 22 ), (1, 24 ), (2, 26 )

2. Plot the points. Notice that all

the points lie on a line .

3. Connect the points with a

line . Observe that the

graph represents a function .

Checkpoint Complete the following exercises.

1. Graph y 5 x 1 2. 2. Graph y 5 3x 2 1.

1

O

y

1

x

1

O

y

1

O

y

1

x

2.2

Your Notes

linear Functions and Function

notation

Goal p Identify, evaluate, and graph linear functions.

Vocabulary

Linear function A function that can be written in the

form y 5 mx 1 b where m and b are constants

Function notation Using f(x) to represent the

dependent variable of a function

Example 1 Identify a Linear Function

Tell whether the function is linear. Explain.

a. f(x) 5 2x3 1 6 2 x b. g(x) 5 9 1 4x

a. f is not a linear function because it has an

x 3-term .

b. g is a linear function because it can be rewritten as

g(x) 5 4x 1 9 , so m 5 4 and b 5 9 .

Checkpoint Tell whether the function is linear. Explain.

1. f(x) 5 6x 1 10

Yes; it is of the form

y 5 mx 1 b.

Example 2 Evaluate a Function

Evaluate the function when x 5 3.

2. g(x) 5 2x2 1 4x 2 1

No; it is not of the

form y 5 mx 1 b.

a. f(x) 5 22x 1 5 b. g(x) 5 x2 2 x 2 1

a. f (x) 5 22x 1 5 Write original function.

f ( 3 ) 5 22( 3 ) 1 5 Substitute 3 for x.

5 26 1 5 Multiply.

5 21 Simplify.

Your Notes

Homework

b. g (x) 5 x 2 2 x 2 1 Write original function.

g( 3 ) 5 ( 3 ) 2 2 3 2 1 Substitute 3 for x.

5 9 2 3 2 1 Multiply.

5 5 Simplify.

Checkpoint Evaluate the function when x 5 24.

3. f(x) 5 2x 2 3 4. g(x) 5 2x 2 1 3x

211 20

Example 3 Graph a Linear Function

Graph f(x) 5 2x 1 2.

Solution

Rewrite the function as y 5 2x 1 2. Find a point on

the graph by substituting a convenient value for x .

y 5 2x 1 2 Write equation.

y

y 5 2( 0 ) 1 2 Substitute 0 for x.

y 5 2 Simplify.

1

O 1

One point is (0, 2) . Find a second

point.

y 5 2(1) 1 2 Substitute 1 for x .

y 5 1 Simplify.

A second point is (1, 1) . Draw a line through the

two points.

Checkpoint Complete the following exercise.

5. Graph h(x) 5 22x 1 1.

1

O

y

1

x

2.3

Your Notes

When calculating

the slope, be sure

to subtract the

x‑coordinates and

the y‑coordinates

in the correct order.

A vertical line has

“undefined slope”

because for any

two points, the

slope formula’s

denominator

becomes 0, and

division by 0 is

undefined.

Slope and rate of change

Goal p Find and use the slope of a line.

Vocabulary

Slope For a nonvertical line, the ratio of the line’s

vertical change to its horizontal change

ThE SlopE oF a linE

The slope m of the nonvertical line passing through

the points (x1 , y1 ) and (x2 , y2 ) is given by the formula

m 5 y2 2 y1 } .

x2 2 x1 Example 1 Find the Slope of a Line

Find the slope of the line through (1, 3) and (6, 7).

Solution

Let (x 1 , y 1 ) 5 (1, 3) and (x 2 , y 2 ) 5 (6, 7) .

m 5 y2 2 y1 } 5

x2 2 x1 7 2 3

}

6 2 1

The slope of the line is

4

5 }

5

4

} .

5

claSSiFyinG linES by SlopE

Positive Negative Zero Undefined

O

y

x

O

y

Line rises from Line falls from Line is Line is

left to right. left to right . horizontal . vertical .

x

O

y

x

O

y

Your Notes

Example 2 Classify Lines Using Slope

Without graphing, tell whether the line through the

given points rises, falls, is horizontal, or is vertical.

a. (26, 22), (1, 3) b. (2, 21), (2, 2)

Solution

a. m 5

b. m 5

3 2 (22)

}

1 2 (26)

2 2 (21)

}

2 2 2

5 5

}

7 Because m > 0, the line rises .

5 3

}

0 Because m is undefined , the

line is vertical .

Checkpoint Complete the following exercises.

1. Find the slope of the line through (4, 2) and (7, 9).

7

}

3

2. Without graphing, tell whether the line through the

points (23, 2) and (1, 4) rises, falls, is horizontal, or

is vertical.

rises

Example 3 Compare Steepness of Lines

Tell which line is steeper.

Line 1: through (2, 3) and (21, 4)

Line 2: through (27, 22) and (24, 25)

Solution

Slope of line 1: m 5

Slope of line 2: m 5

4 2 3

}

21 2 2

25 2 (22)

}

24 2 (27)

5 1

}

23

or 2 1

}

3

5 23

}

3

5 21

Both lines have negative slope.

Because |m1 | > |m2 |, line 1 is steeper than

line 2 .

Your Notes

Homework

Checkpoint Tell which line is steeper.

3. Line 1: through (3, 5) and (6, 8)

Line 2: through (22, 4) and (10, 12)

line 1

Example 4 Slope as a Rate of Change

census The population of Smalltown was 10,000 in

2001 and 12,500 in 2006. Find the average rate of

change in the population.

Solution

Let (x1 , y1 ) 5 (2001, 10,000 ) and

(x2 , y2 ) 5 ( 2006 , 12,500 ).

Change in population

Average rate of change 5 }}

Change in time

12,500 2 10,000

5 }}

2006 2 2001

5 2500

}

5

5 500

The average rate of change is 500 people per year .

Checkpoint Complete the following exercise.

4. Census Suppose the population of Smalltown will be

20,000 in 2011. Use the information in Example 4

to find the average rate of change in the population

from 2001 to 2011.

1000 people per year

Your Notes

2.4 quick Graphs of linear

Equations

Goal p Use slope-intercept form and standard form to

graph equations.

Vocabulary

y-intercept The y-coordinate of a point where a

graph intersects the y-axis

Slope-intercept form A linear equation written in the

form y 5 mx 1 b where m is the slope and b is the

y-intercept of the equation’s graph

x-intercept The x-coordinate of a point where a

graph intersects the x-axis

Standard form of a linear equation A linear equation

written in the form Ax 1 By 5 C where A and B are

not both zero

GraphinG EquaTionS in SlopE-inTErcEpT Form

Step 1 Write the equation in slope-intercept form by

solving for y .

Step 2 Find the y-intercept and plot the

corresponding point.

Step 3 Find the slope and use it to plot a second

point on the line.

Step 4 Draw a line through the two points .

Your Notes

Example 1 Use Slope-Intercept Form to Graph a Line

Graph y 1 3 } x 5 1.

2

solution

1. Solve the equation for y by subtracting 3

} x from each

2

side:

y 5 1 2 3

} x or y 5 2

2 3

} x 1 1

2

2. Find the y-intercept. Comparing y 5 2 3

} x 1 1 to

2

y 5 mx 1 b, you can see that b 5 1 . Plot a point

at ( 0 , 1 ).

3. Find the slope. The slope is m 5 2 3

} , so

2 rise

}

run

From (0, 1), move down 3 units and right

2 units. Plot a second point at ( 2 , 22 ).

4. Draw a line through the two points.

1

O

y

1

x

GraphinG Equations in standard Form

Step 1 Write the equation in standard form .

Step 2 Find the x-intercept by letting y 5 0 and

solving for x . Plot the corresponding point.

Step 3 Find the y-intercept by letting x 5 0 and

solving for y . Plot the corresponding point.

Step 4 Draw a line through the two points .

5 23

}

2 .

Your Notes

Example 2 Draw Quick Graphs

Graph 4x 1 5y 5 20.

1. The equation is already in standard form .

2. Find the x-intercept. 4x 1 5( 0 ) 5 20 Let y 5 0.

SlopES oF parallEl and pErpEndicular linES

Consider two different

nonvertical lines l 1 and l 2

with slopes m 1 and m 2 .

Two lines are parallel if and only

if they have the same slope .

m 1 5 m 2

Two lines are perpendicular if and

only if their slopes are negative

reciprocals of each other.

m 1 5 2 1

}

m 2 or m 1 • m 2 5 –1

x 5 5 Solve for x.

The x-intercept is 5 , so plot the point (5, 0) .

3. Find the y-intercept. 4( 0 ) 1 5y 5 20 Let x 5 0 .

y 5 4 Solve for y.

The y-intercept is 4 , so plot the point (0, 4) .

4. Draw a line through

the intercepts.

l 1

1

O

l 2

y

1

O

1

O

y

1

l 2

x

Your Notes

Homework

Example 3 Graph Parallel and Perpendicular Lines

a. Draw the graph y 5 2x 2 3.

b. Graph the line that passes through (0, 2) and is

parallel to the graph of y 5 2x 2 3.

c. Graph the line that passes through (0, 2) and is

perpendicular to the graph of y 5 2x 2 3.

Solution

a. Draw the graph y 5 2x 2 3

using the y-intercept 23 and

slope 2 .

b. Graph the point (0, 2). Any

line parallel to the graph of

y 5 2x 2 3 has the same

slope, 2 . From (0, 2), move

up 2 units and right

1 unit. Plot a second point at (1, 4) . Draw a line

through the two points.

c. The slope of any line perpendicular to the

graph of y 5 2x 2 3 is the negative reciprocal

of 2, or 2 1

} . So, from (0, 2), move down 1 unit

2

and right 2 units. Plot a second point at

(2, 1) . Draw the line.

Checkpoint On the same grid, draw each graph.

1. y 5 2 1 }

3 x 1 1

2. The line parallel to

y 5 2 1 }

3 x 1 1 through

the point (0, 4)

3. The line perpendicular

to y 5 2 1 }

3 x 1 1 through

the point (0, 4)

c

1

O

y

1

O

Ex. 3

y

1

b

Ex. 1

Ex. 2

x

a

Your Notes

2.5 writing Equations of lines

Goal p Write linear equations.

Vocabulary

Point-slope form An equation of a line written in

the form y 2 y 1 5 m(x 2 x 1 ) where the line passes

through the point (x 1 , y 1 ) and has a slope of m

wriTinG an EquaTion oF a linE

Slope-Intercept Form Given the slope m and the

y-intercept b , use slope-intercept form: y 5 mx 1 b .

Point-Slope Form Given the slope m and a point

(x1 , y1 ) , use point-slope form: y 2 y1 5 m(x 2 x1 ) .

Example 1 Write an Equation Given Slope and y-Intercept

Write an equation of the

line shown.

y

Solution

1

From the graph, you can see

that the slope is m 5 2

O 2

3

}

2

.

Because the line intersects the

y-axis at ( 0, 1 ), the y-intercept

is b 5 1 .

Use the slope-intercept form to write an equation of

the line.

y 5 mx 1 b Use slope-intercept form.

y 5 2 3

}

2

x 1 1 Substitute 2 3

}

2

for m and 1 for b.

Your Notes

Example 2 Write an Equation Given Slope and a Point

Write an equation of the line that passes through (2, 1)

and has a slope of 2.

Solution

Because you know the slope and a point on the line,

use the point-slope form with (x1 , y1 ) 5 (2, 1) and

m 5 2 .

y 2 y1 5 m(x 2 x1 ) Use point-slope form.

y 2 1 5 2 (x 2 2 ) Substitute for m, x1 , and y1 .

Rewrite the equation in slope-intercept form.

y 2 1 5 2x 2 4 Distributive property

y 5 2x 2 3 Slope-intercept form

Checkpoint Write an equation of the line.

1. y

2. The line through (1, 5)

1

with a slope of 22

O

1

x

y 5 2

}

3 x 2 3 y 5 22x 1 7

Example 3 Write Equations of Parallel Lines

Write an equation of the line that passes through

(21, 1) and is parallel to the line y 5 22x 1 3.

Solution

The given line has a slope of 22 . Any line parallel

to this line must also have a slope of 22 . Use

point-slope form with (x1 , y1 ) 5 (21, 1) and

m 5 22 to write the equation of the line.

Your Notes

Homework

y 2 y 1 5 m(x 2 x 1 ) Use point-slope form.

y 2 1 5 22 [x 2 (21) ] Substitute 1 for y 1 , 22

for m, and 21 for x 1 .

y 2 1 5 22x 2 2 Distributive property

y 5 22x 2 1 Slope-intercept form

Example 4 Write Equations of Perpendicular Lines

Write an equation of the line that passes through

(23, 5) and is perpendicular to the line y 5 4x 2 1.

solution

The given line has a slope of 4 . The slope of any

line perpendicular to this line will be the negative

reciprocal of 4 , which is 2 1

} .

4

Use point-slope form with (x1 , y1 ) 5 (23, 5) and

m 5 2 1

} to write the equation of the line.

4

y 2 y 1 5 m(x 2 x 1 ) Use point-slope form.

y 2 5 5 2 1

} [x 2 (23) ]

4

Substitute 5 for y1 , 2 1

}

4

for m, and 23 for x1 .

1

y 2 5 5 2 } x 2

4 3

}

4

Distributive property

1

y 5 2 } x 1

4 17

}

4

Checkpoint Write an equation of the line.

3. The line through

(22, 3) parallel to

y 5 4x 2 6

y 5 4x 1 11

Slope-intercept form

4. The line through

(22, 3) perpendicular

to y 5 4x 2 6

y 5 2 1

} x 1

4 5

}

2

Your Notes

2.6 direct Variation

Goal p Write and graph direct variation equations.

Vocabulary

Direct variation Two variables x and y show direct

variation provided that y 5 kx where k is a nonzero

constant.

Constant of variation The nonzero constant k in a

direct variation equation y 5 kx

Example 1 Graph a Direct Variation Equation

Graph y 5 5

} x.

3

Solution

Plot a point at the origin .

Find a second point by substituting

a convenient value for x .

y 5 5 } x

3

Write equation.

y 5 5 } ( 3 )

3

Substitute 3 for x.

y 5 5 Simplify.

A second point is (3, 5) .

Plot the second point. Then draw a line through the

two points.

Checkpoint Graph the equation.

1. y 5 23x

1

O

y

1

x

1

O

y

1

Your Notes

The value of k is

called the spring

constant. This

constant is unique

for each spring.

Example 2 Write a Direct Variation Equation

physics According to Hooke’s law, the amount that a

spring is stretched varies directly with the force that is

applied to the spring. If 64 pounds of force F stretches

a spring a distance d of 8 inches, write an equation

so that F varies directly with d. How many pounds of

force are needed to stretch the spring to a distance

of 14 inches?

Solution

Find the constant of variation.

F 5 kd Write a direct variation equation.

64 5 k( 8 ) Substitute 64 for F and 8 for d.

8 5 k Simplify.

An equation is F 5 8d . To find how many pounds

of force is needed to stretch the spring 14 inches,

substitute 14 for d.

F 5 8 ( 14 ) 5 112

A force of 112 pounds will stretch the spring 14 inches.

Checkpoint Complete the following exercise.

2. Suppose that a force of 92 pounds is applied to

the spring discussed in Example 2. How far will

the spring be stretched?

11.5 inches

Your Notes

Homework

Example 3 Identify Direct Variation

The table gives sample cell phone bills, showing the

total monthly cost and the number of minutes used

per month. Tell whether total cost and the number of

minutes show direct variation. If so, write an equation

relating c and m.

Total cost, c (dollars) 35 45 80 15 28

Minutes used, m 100 129 229 43 80

solution

Find the ratio of c to m for each data pair.

35

}

100

15

}

43

5 0.35 45

}

129

28

< 0.35 } 5 0.35

30

Checkpoint Complete the following exercise.

3. Tell whether the data (21, 22), (1, 2), (3, 6),

(5, 10), (7, 14) show direct variation. If so, write an

expression relating x and y.

direct variation; y 5 2x

80

< 0.35 } < 0.35

229

The ratio of c to m in each pair is nearly the same , so

the data show direct variation . Substituting 0.35

for k in c 5 km gives the direct variation equation

c 5 0.35 m, which relates the total cost to number of

minutes used.

2.7

Your Notes

Scatter plots and correlation

Goal p See correlation in a scatter plot and find a

best-fitting line.

Vocabulary

Scatter plot A graph of a set of data pairs (x, y)

used to determine whether there is a relationship

between the variables x and y

Positive correlation A set of data pairs (x, y) for

which as x increases, y tends to increase

Negative correlation A set of data pairs (x, y) for

which as x increases, y tends to decrease

Relatively no correlation A set of data pairs (x, y) for

which there is no obvious pattern between x and y

Best-fitting line The line that most closely models

the data shown in a scatter plot

Example 1 Identify Correlation

Describe the correlation shown by each plot.

a. y

b.

1

O

1

x

Solution

a. The scatter plot shows a negative correlation: as

x increases, y decreases .

b. The scatter plot shows a positive correlation: as

x increases , y also increases .

1

O

y

1

Your Notes

Be sure that about

the same number

of points lie above

your best-fitting

line as lie below it.

approximatinG a BEst-FittinG LinE

Step 1 Draw a scatter plot of the data .

Step 2 Sketch a line that follows the trend of the

data points.

Step 3 Choose two points that appear to lie on the

line, and estimate their coordinates .

Step 4 Write an equation of the line that passes

through the two points from Step 3. This

equation gives a model for the data.

Example 2 Find a Best-Fitting Line

Football attendance The table gives the number of

people y who attended each of the first seven football

games x of the season. Approximate the best-fitting

line for the data.

x 1 2 3 4 5 6 7

y 722 763 772 826 815 857 897

solution

1. Draw a scatter plot.

950

y

2. Sketch the line that

900

best fits the data.

850

800

3. Choose two points on the

line. For this scatter plot you

750

700

650

might choose (1, 725) and

0

0 1 2 3 4 5 6 7 x

( 2 , 750).

Football game

4. Write an equation of the line. First find the slope using

the two points:

750 2 725 25

m 5 } 5 } 5 25

2 2 1 1

Now use point-slope form. Choose (x1 , y1 ) 5 (1, 725).

y 2 y 1 5 m(x 2 x 1 ) Point-slope form

y 2 725 5 25 (x 2 1 ) Substitute for y 1 , m, and x 1 .

Number of people

y 5 25x 1 700 Solve for y.

Your Notes

Homework

Example 3 Use a Best-Fitting Line

Use the equation of the best-fitting line from Example 2

to predict the number of people that will attend the

tenth football game.

Solution

Substitute 10 for x in the equation from Example 2.

y 5 25( 10 ) 1 700 5 250 1 700 5 950

You can predict that 950 people will attend the game.

Checkpoint Complete the following exercises.

For each scatter plot, tell whether the data show a

positive correlation, negative correlation, or relatively no

correlation.

1. y

2. y

1

O

1

positive correlation

x

1

O

1

relatively no correlation

3. School The table gives the average class score y on

each test for the first six chapters x of the textbook.

x 1 2 3 4 5 6

y 84 83 86 88 87 90

a. Approximate the best-fitting

line for the data.

y 5 2x 1 80

b. Use your equation to predict

the test score for the 9th test

that the class will take.

98

Average class score

90

88

86

84

82

y

0

0 1 2 3 4 5

Test

x

6

words to review

use your own words and/or an example to explain

the vocabulary word.

Relation

A mapping, or pairing,

of input values with

output values

Range

The set of output

values in a relation

Equation in two variables

y 5 2x 2 5

Dependent variable

The dependent variable

in the equation

y 5 2x 2 5 is y.

Function notation

f(x) 5 2x 1 4

y-intercept

The y-intercept of

y 5 2x 1 4 is 4.

x-intercept

The x-intercept of

y 5 2x 1 4 is 22.

Domain

The set of input values

in a relation

Function

A relation for which

each input has exactly

one output

Independent variable

The independent

variable in the equation

y 5 2x 2 5 is x.

Linear function

A function that can be

written in the form

y 5 mx 1 b where m

and b are constants

Slope

The slope of the line

passing through (2, 6)

and (5, 9) is

m 5 y 2 2 y 1

}

x 2 2 x 1

5

9 2 6

}

5 2 2

5 3

}

3 5 1.

Slope-intercept form

y 5 2x 1 4

Standard form of a linear

equation

2x 1 4y 5 7

Point-slope form

y 2 1 5 2(x 2 5)

Constant of variation

The constant of

variation in the direct

variation equation

y 5 6x is 6.

Positive correlation

This scatter plot shows

a positive correlation.

y

Relatively no correlation

This scatter plot

shows relatively no

correlation.

y

x

Direct variation

y 5 6x

Scatter plot

A graph of a set of

data pairs (x, y) used

to determine whether

there is a relationship

between the variables

x and y

Negative correlation

This scatter plot shows

a negative correlation.

y

Best-fitting line

The line that most

closely models the data

shown in a scatter plot

review your notes and chapter 2 by using the

chapter review on pages 115–118 of your textbook.

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