Week 3. Due for this week… Homework 3 (on MyMathLab – via the Materials Link) Monday night at...

Week 3

Due for this week…

Homework 3 (on MyMathLab – via the Materials Link) Monday night at 6pm.

Read Chapter 5 (The last of the new material for MTH 208)

Do the MyMathLab Self-Check for week 3. Learning team planning for week 5.

Slide 2Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Introduction to Graphing

The Rectangular Coordinate System

Scatterplots and Line Graphs

3.1


The Rectangular Coordinate System

One common way to graph data is to use the rectangular coordinate system, or xy-plane.In the xy-plane the horizontal axis is the x-axis, and the vertical axis is the y-axis.The axes intersect at the origin.The axes divide the xy-plane into four regions called quadrants, which are numbered I, II, III, and IV counterclockwise.


EXAMPLE Plotting points

Plot the following ordered pairs on the same xy-plane. State the quadrant in which each point is located, if possible.

a. (4, 3) b. (3, 4) c. (1, 0)

Solutiona. (4, 3) Move 4 units to the right of the origin and 3 units up.

b. (3, 4) Move 3 units to the left of the origin and 4 units down.

c. (1, 0) Move 1 unit to the left of the origin.

Quadrant I

Quadrant III

Not in any quadrantTry some of Q: 11-20


EXAMPLE Reading a graph

Frozen pizza makers have improved their pizzas to taste more like homemade. Use the graph to estimate frozen pizza sales in 1994 and 2000.

Solutiona. To estimate sales in 1994,

locate 1994 on the x-axis. Then move upward to the data point and approximate its y-coordinate.

b. To estimate sales in 2000, locate 2000 on the x-axis. Then move upward to the data point and approximate its y-coordinate.

a. about $2.1 billion in sales

b. about $3.0 billion in salesTry some of Q: 39-40

If distinct points are plotted in the xy-plane, then the resulting graph is called a scatterplot.


Scatterplots and Line Graphs


EXAMPLE Making a scatterplot of gasoline prices

The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These price have not been adjusted for inflation.

Year 1975 1980 1985 1990 1995 2000 2005

Cost (per gal in cents)

56.7 119.1 111.5 114.9 120.5 156.3 186.6

The data point (1975, 56.7) can be used to indicate the average cost of a gallon of gasoline in 1975 was 56.7 cents. Plot the data points in the xy-plane.


EXAMPLE Making a scatterplot of gasoline prices

The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These prices have not been adjusted for inflation.

Year 1975 1980 1985 1990 1995 2000 2005

Cost (per gal in cents)

56.7 119.1 111.5 114.9 120.5 156.3 186.6

Try some of Q: 23-32

Line Graphs


Sometimes it is helpful to connect consecutive data points in a scatterplot with line segments.This creates a line graph.


EXAMPLE Making a line graph

Use the data in the table to make a line graph.

x 3 2 1 0 1 2 3

y 3 4 0 3 2 4 3

Plot the points and then connect consecutive points with line segments.



Linear Equations in Two Variables

Basic Concepts

Tables of Solutions

Graphing Linear Equations in Two Variables

3.2


Basic Concepts

Equations can have any number of variables.

A solution to an equation with one variable is one number that makes the statement true.


EXAMPLE Testing solutions to equations

Determine whether the given ordered pair is a solution to the given equation.

a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4)

Solutiona. y = x + 5 b. 2x + 3y = 18

7 = 2 + 5

7 = 7 True

The ordered pair (2, 7) is a solution.

2(3) + 3(4) = 18

6 12 = 18

6 18

The ordered pair (3, 4) is NOT a solution.Try some of Q:9-18

Tables of Solutions


A table can be used to list solutions to an equation.

A table that lists a few solutions is helpful when graphing an equation.


EXAMPLE Completing a table of solutions

Complete the table for the equation y = 3x – 1.

Solution

x 3 1 0 3

y

3x

3 1

3( 3) 1

9 1

10

y x

y

y

y

x 3 1 0 3

y 10

1x

3 1

3( ) 1

3

4

1

1

y x

y

y

y

0x

3 1

3( ) 1

0

1

0

1

y x

y

y

y

3x

3 1

3( ) 1

9

8

3

1

y x

y

y

y

4 1 8

Try some of Q:19-24


EXAMPLE Graphing an equation with two variables

Make a table of values for the equation y = 3x, and then use the table to graph this equation.

SolutionStart by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table.

x y

–1 –3

0 0

1 3

2 6

Plot the points and connect the points with a straight line. Try some of Q: 35-40


EXAMPLE Graphing linear equations

Graph the linear equation.

SolutionBecause this equation can be written in standard form, it is a linear equation. Choose any three values for x.

x y

–4 0

0 1

4 2

Plot the points and connect the points with a straight line.

11

4y x


EXAMPLE Graphing linear equations

Graph the linear equation.

SolutionBecause this equation can be written in standard form, it is a linear equation. Choose any three values for x.

x y

0 5

2 3

5 0


5x y



EXAMPLE Solve for y and then graphing

Graph the linear equation by solving for y first.

SolutionSolve for y.

x y

–2 1

0 2

2 3


3 6 12x y

3 6 12x y 6 3 12y x

12

2y x



More Graphing of Lines

Finding Intercepts

Horizontal Lines

Vertical Lines

3.3


Finding Intercepts

The y-intercept is where the graph intersects the y-axis.

The x-intercept is where the graph intersects the x-axis.


EXAMPLE Using intercepts to graph a line

Use intercepts to graph 3x – 4y = 12.

SolutionThe x-intercept is found by letting y = 0.

The graph passes through the two points (4, 0) and (0, –3).

The y-intercept is found by letting x = 0.

3 4 12

3 4( ) 12

3 12

( , )

0

0

4

4

x y

x

x

x

3 4 12

3( ) 4 12

4 12

(0, 3

3

0

)

x y

y

y

x



EXAMPLE Using a table to find intercepts

Complete the table. Then determine the x-intercept and y-intercept for the graph of the equation x – y = 3.SolutionFind corresponding values of y for the given values of x.

3

3

3

6

6

x y

y

y

y

x 3 1 0 1 3

y

1

3

3

4

4

x y

y

y

y

3

3

3

3

0

x y

y

y

y

3

3

2

2

1

x y

y

y

y

3

3

0

3

0

x y

y

y

y

x 3 1 0 1 3

y 6 4 3 2 0

The x-intercept is (3, 0). The y-intercept is (0, –3).



EXAMPLE Modeling the velocity of a toy rocket

A toy rocket is shot vertically into the air. Its velocity v in feet per second after t seconds is given by v = 320 – 32t. Assume that t ≥ 0 and t ≤ 10.a. Graph the equation by finding the intercepts.b. Interpret each intercept.

Solutiona. Find the intercepts.

320 32

320 32

320

0

2

0

3

1

v t

t

t

t

320 32

320 32(0)

320

v t

v

v

b. The rocket had velocity of 0 feet per second after 10 seconds. The v-intercept indicates that the rocket’s initial velocity was 320 feet per second.



Horizontal Lines


EXAMPLE Graphing a horizontal line

Graph the equation y = 2 and identify its y-intercept.

Solution

The graph of y = 2 is a horizontal line passing through the point (0, 2), as shown below.The y-intercept is 2.

Try some of Q: 47-54a’s


Vertical Lines


EXAMPLE Graphing a vertical line

Graph the equation x = 2, and identify its x-intercept.

Solution

The graph of x = 2 is a vertical line passing through the point (2, 0), as shown below.The x-intercept is 2.

Try some of Q: 47-54 b’s


EXAMPLE Writing equations of horizontal and vertical lines

Write the equation of the line shown in each graph.a. b.

Solutiona. The graph is a horizontal line.

The equation is y = –1.

b. The graph is a vertical line.The equation is x = –1.



EXAMPLE Writing equations of horizontal and vertical lines

Find an equation for a line satisfying the given conditions.a. Vertical, passing through (3, 4).b. Horizontal, passing through (1, 2).c. Perpendicular to x = 2, passing through (1, 2).

Solutiona. The x-intercept is 3. The equation is x = 3.

b. The y-intercept is 2.The equation is y = 2.

c. A line perpendicular to x = 2 is a horizontal line with y-intercept –2. The equation is y = 2.



Slope and Rates of Change

Finding Slopes of Lines

Slope as a Rate of Change

3.4


Slope

The rise, or change in y, is y2 y1, and the run, or change in x, is x2 – x1.


EXAMPLE Calculating the slope of a line

Use the two points to find the slope of the line. Interpret the slope in terms of rise and run.Solution

The rise is 3 units and the run is –4 units.

(–4, 1)

(0, –2)

2 1

2 1

( )

4 0

4

3

2

3

4

1

y ym

x x




Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)

Solution2 1

2 1

0 (

a.

4 3

7

( )

3)

3

y ym

x x




Solution2 1

2 1

3 ( 3)

6

b.

( )

1

3

2 4

2

y ym

x x




Solution2 1

2 1

c.

( )

0

)

4

4

0

2 2

4

(

y ym

x x




Solution2 1

2 1

2 5

d.

( )

undef

4 4

0ed

3in

y ym

x x



SlopePositive slope: rises from left to rightNegative slope: falls from left to right


SlopeZero slope: horizontal lineUndefined slope: vertical line


EXAMPLE Finding slope from a graph

Find the slope of each line. a. b.

Solution

a. The graph rises 2 units for each unit of run m = 2/1 = 2.

b. The line is vertical, so the slope is undefined.



EXAMPLE Sketching a line with a given slope

Sketch a line passing through the point (1, 2) and having slope ¾.

SolutionStart by plotting (1, 2).

The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4.

The line passes through the point (1 + 4, 2 + 3) = (5, 5).



Slope as a Rate of Change

When lines are used to model physical quantities in applications, their slopes provide important information.

Slope measures the rate of change in a quantity.


EXAMPLE Interpreting slope

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.a. Find the y-intercept. What does the y-intercept represent?b. The graph passes through the point (4, 15). Discuss the meaning of this point.c. Find the slope of the line. Interpret the slope as a rate of change.

Solutiona. The y-intercept is 35, so

the boat is initially 35 miles from the dock.




Solutionb. The point (4, 15) means

that after 4 hours the boat is 15 miles from the dock.




Solution

c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

15 05

4 7m



Slope-Intercept Form

Finding Slope-Intercept Form

Parallel and Perpendicular Lines

3.5


Finding Slope-Intercept Form


EXAMPLE Using a graph to write the slope-intercept form

For the graph write the slope-intercept form of the line.

SolutionThe graph intersects the y-axis at 0, so the y-intercept is 0.The graph falls 3 units for each 1 unit increase in x, the slope is –3.The slope intercept-form of the line is y = –3x .



EXAMPLE Sketching a line

Sketch a line with slope ¾ and y-intercept −2. Write its slope-intercept form.

SolutionThe y-intercept is (0, −2). Slope ¾ indicates that the graph rises 3 units for each 4 units run in x. The line passes through the point (4, 1).

32

4y x



EXAMPLE Graphing an equation in slope-intercept form

Write the y = 4 – 3x equation in slope-intercept form and then graph it.

Solution4 3

3 4

y x

y x

Plot the point (0, 4).The line falls 3 units for each 1 unit increase in x.



Parallel and Perpendicular Lines


EXAMPLE Finding parallel lines

Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Sketch a graph of each line.

SolutionThe line has a slope of 3 any parallel line also has slope 3.Slope-intercept form: y = 3x + b. The value of b can be found by substituting the point (2, 1) into the equation. 3

1 3(2)

1 6

5

y x b

b

b

b Try some of Q: 67-74


EXAMPLE Finding perpendicular lines

Find the slope-intercept form of a line passing through the origin that is perpendicular to each line. a. y = 4x b.

Solutiona. The y-intercept is 0. Perpendicular line has a slope of

1.

4

23

5 y x

1

4y x

b. The y-intercept is 0. Perpendicular line has a slope of

5.

25

2y x



Point-Slope Form

Derivation of Point-Slope Form

Finding Point-Slope Form

Applications

3.6


, 0 and 1,xf x a a a

The line with slope m passing through the point (x1, y1) is given by

y – y1 = m(x – x1),

or equivalently, y = m(x – x1) + y1

the point-slope form of a line.

POINT-SLOPE FORM


EXAMPLE

Solution

Finding a point-slope form

Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line?

Let m = 2 and (x1, y1) = (3,1) in the point-slope form.

To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y.

y – y1 = m(x – x1)

y − 1 = 2(x – 3)

3 – 1 ? 2(4 – 3)

2 = 2

The point (4, 3) lies on the line because it satisfies the point-slope form. Try some of Q: 9-24


EXAMPLE

Solution

Finding an equation of a line

Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5).

Before we can apply the point-slope form, we must find the slope.

2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2


EXAMPLE continued

We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following.

y – y1= m(x – x1)

1)3 ( )

2( 2y x

13 ( 2)

2y x

If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes

15 ( 2).

2y x



EXAMPLE

Solution

Finding equations of lines

Find the slope-intercept form of the line perpendicular to passing through the point (4, 6).

The line has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows.

3,y x

6 1( 4)y x

3y x

6 4y x

10y x Try some of Q: 45-54


EXAMPLE Modeling water in a pool

A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons.

a. How fast is the pump removing water?b. Find the slope-intercept form of a line that models

the amount of water in the pool. Interpret the slope.c. Find the y-intercept and the x-intercept. Interpret

each.d. Sketch the graph of the amount of water in the

pool during the first 5 hours.e. The point (2, 6000) lies on the graph. Explain its

meaning.


EXAMPLE continued

a. The pump removes 8000 − 2000 gallons of water in 3 hours, or 2000 gallons per hour.

b. The line passes through the points (1,8000) and (4, 2000), so the slope is

Solution

2000 80002000

4 1

Use the point-slope form to find the slope-intercept form.

y – y1= m(x – x1)

y – 8000 = −2000(x – 1)

y – 8000 = −2000x + 2000

y = −2000x + 10,000

A slope of −2000, means that the pump is removing 2000 gallons per hour.


EXAMPLE continued

c. The y-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the x-intercept let y = 0 in the slope-intercept form.

0 2000 10,000x 2000 10,000x 2000 10,000

2000 2000

x

5x

The x-intercept of 5 indicates that the pool is emptied after 5 hours.


EXAMPLE continued

d. The x-intercept is 5 and the y-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000).

X

Y

1 2 3 4 5 6

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

Wat

er (

gallo

ns)

Time (hours)

e. The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water.



Introduction to Modeling

Basic Concepts

Modeling Linear Data

3.7


Generally, mathematical models are not exact representations of data.


EXAMPLE

Solution

Determining whether a model is exact

A person can vote in the United States at age 18 or over. The table shows the voting-age population P in millions for selected years x. Does the equation P = 3.25x – 6297 model the data exactly? Explain.

To determine whether the equation models the data exactly, let x = 2000, 2002, and 2004 in the given equation.

x 2000 2002 2004

P 203 211 216Source: U.S. Census Bureau

x = 2000: P = 3.25(2000) – 6297 = 203

x = 2002: P = 3.25(2002) – 6297 = 209.5

x = 2004: P = 3.25(2004) – 6297 = 216

The model is not exact because it does not predict the voting-age population of 211 million in 2002.



EXAMPLE Determining gas mileage

The table shows the number of miles y traveled by a motorhome on x gallons of gasoline.

a. Plot the data in the xy-plane. Be sure to label each axis.

b. Sketch a line that models the data.c. Find the equation of the line and interpret the slope

of the line.d. How far could this motorhome travel on 15 gallons

of gasoline?

x 3 6 9

P 24 48 72


EXAMPLE

Solution

Determining gas mileage

x 3 6 9

P 24 48 72

a. Plot the points (3, 24), (6, 48), and (9, 72).

X

Y

1 2 3 4 5 6 7 8 9 10

10

20

30

4050

60

70

80

90

100

0

Dis

tanc

e (m

iles)

Gasoline (gallons)

b. Sketch a line through the points.

X

Y

1 2 3 4 5 6 7 8 9 10

10

20

30

4050

60

70

80

90

100

0

Dis

tanc

e (m

iles)

Gasoline (gallons)

X

Y

1 2 3 4 5 6 7 8 9 10

10

20

30

4050

60

70

80

90

100

0

Dis

tanc

e (m

iles)

Gasoline (gallons)

X

Y

1 2 3 4 5 6 7 8 9 10

10

20

30

4050

60

70

80

90

100

0

Dis

tanc

e (m

iles)

Gasoline (gallons)


EXAMPLE

Solution

continued

c. Find the slope of the line. 72 24

9 3m

48

6 8

Now find the equation of the line passing through (3, 24) with the slope of 8.

y – y1 = m(x – x1)

y – 24 = 8(x – 3)

y – 24 = 8x – 24

y = 8x

The data are modeled by the equation y = 8x. Slope 8 indicates that the mileage of the motorhome is 8 miles per gallon.

d. On 15 gallons of gasoline, the motorhome could go y = 8(15) = 120 miles.Try Q: 53


EXAMPLE Modeling linear data

The table contains ordered pairs that can be modeled approximately by a line.

a. Plot the data. Could a line pass through all five points?

b. Sketch a line that models the data and determine its equation.

x 1 2 3 4 5

y 5 7 9 10 11

Solution

a.

b. One possibility of the line is shown.

X

Y

1 2 3 4 5 6

1

2

3

4

5

6

7

8

9

10

11

12

0

11 5

5 1m

6

4

3

2

y – y1 = m(x – x1)

y – 5 = 3/2(x – 1)3 7

2 2y x



EXAMPLE Modeling with linear equations

Find a linear equation in the form y = mx + b that models the quantity y after x days.

a. A quantity y is initially 750 and increases at a rate of 4 per day.

b. A quantity y is initially 2300 and decreases at a rate of 50 per day.

c. A quantity y is initially 17,875 and remains constant.

SolutionIn the equation y = mx + b, the y-intercept b represents the initial amount and the slope m represents the rate of change.

b. y = −50x + 2300a. y = 4x + 750 c. y = 17,875



Functions and Their Representations

Basic Concepts

Representations of a Function

Definition of a Function

Identifying a Function

Graphing Calculators (Optional)

8.1



The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.

Name

y = f(x)

Output Input

FUNCTION NOTATION


The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.

Representations of a Function

Verbal Representation (Words)

Numerical Representation (Table of Values)

Symbolic Representation (Formula)

Graphical Representation (Graph)

Diagrammatic Representation (Diagram)



EXAMPLE

Solution

Calculating sales tax

Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3).

a. Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars.

b. Numerical Representation x f(x)

$1.00 $0.06

$2.00 $0.12

$3.00 $0.18

$4.00 $0.24


EXAMPLE continued

c. Symbolic Representation f(x) = 0.06x

d. Graphical Representation

X

Y

1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0

e. Diagrammatic Representation

1 ●2 ●3 ●4 ●

● 0.06● 0.12● 0.18● 0.24

f(3) = 0.18



EXAMPLE

Solution

Evaluating symbolic representation (formulas)

Evaluate the function f at the given value of x.

f(x) = 5x – 3 x = −4

f(−4) = 5(−4) – 3

= −20 – 3

= −23



A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair:

(x, y).

Input Output

A relation is a set of ordered pairs, and a function is a special type of relation.


A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value.

Function

The domain of f is the set of all x-values, and the range of f is the set of all y-values.


EXAMPLE

Solution

Finding the domain and range graphically

Use the graph of f to find the function’s domain and range.

X

Y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

0

The arrows at the ends of the graph indicate that the graph extends indefinitely. Thus the domain includes all real numbers. The smallest y-value on the graph is y = −4. Thus the range is y ≥ −4.

Use f(x) to find the domain of f.

a. f(x) = 3x b.


EXAMPLE Finding the domain of a function

1

4f x

x

Solution

Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers.

a.

b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4. Try some of Q: 89-102

Determine whether the table of values represents a function.


EXAMPLE Determining whether a table represents a function

x f(x)

2 −6

3 4

4 2

3 −1

1 0

Solution

The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.



If every vertical line intersects a graph at no more than one point, then the graph represents a function.

Vertical Line Test

Determine whether the graph represents a function.


EXAMPLE Determining whether a graph represents a function

Solution

Any vertical line will cross the graph at most once. Therefore the graph does represent a function.

X

Y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

0



Linear Functions

Basic Concepts

Representations of Linear Functions

Modeling Data with Linear Functions

The Midpoint Formula (Optional)

8.2



A function f defined by f(x) = ax + b, where a and b are constants, is a linear function.

LINEAR FUNCTION


EXAMPLE

Solution

Identifying linear functions

Determine whether f is a linear function. If f is a linear function, find values for a and b so that f(x) = ax + b.

a. f(x) = 6 – 2x b. f(x) = 3x2 – 5

Let a = –2 and b = 6. Then f(x) = −2x + 6, and f is a linear function.

a.

b. Function f is not linear because its formula contains x2. The formula for a linear function cannot contain an x with an exponent other than 1.

Try some of Q: 9-16


EXAMPLE

Solution

Determining linear functions

Use the table of values to determine whether f(x) could represent a linear function. If f could be linear, write the formula for f in the form f(x) = ax + b.

For each unit increase in x, f(x) increases by 7 units so f(x) could be linear with a = 7. Because f(0) = 4, b = 4. thus f(x) = 7x + 4.

x 0 1 2 3

f(x) 4 11 18 25



EXAMPLE

Solution

Graphing a linear function by hand

Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).

Begin by creating a table.

x y

−1 −4

0 −3

1 −2

2 −1

Plot the points and sketch a line through the points.

X

Y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

0


EXAMPLE

Solution

continued

Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).

To evaluate f(4), first find x = 4 on the x-axis. Then find the corresponding y-value. Thus f(4) = 1.

X

Y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

0




The formula f(x) = ax + b may be interpreted as follows.

f(x) = ax + b(New amount) = (Change) + (Fixed amount)

When x represents time, change equals (rate of change) × (time).

f(x) = a × x + b(Future amount) = (Rate of change) × (Time) + (Initial amount)

MODELING DATA WITH A LINEAR FUNCTION


EXAMPLE

Solution

Modeling the cost of a truck rental

Suppose that a moving truck costs $0.25 per mile and a fixed rental fee of $20. Find a formula for a linear function that models the rental fees.

Total cost is found by multiplying $0.25 (rate per mile) by the number of miles driven x and then adding the fixed rental fee (fixed amount) of $20. Thus f(x) = 0.25x + 20.


EXAMPLE Modeling with a constant function

The temperature of a hot tub is recorded at regular intervals.

Discuss the temperature of the water during this time interval.

a.

b. Find a formula for a function f that models these data.

c. Sketch a graph of f together with the data.

Elapsed Time (hours) 0 1 2 3

Temperature 102°F 102°F 102°F 102°F


EXAMPLE continued

The temperature appears to be a constant 102°F. a.

b. Because the temperature is constant, the rate of change is 0. Thus f(x) = 0x + 102 or f(x) = 102.

c. Graphing the data points, gives the following constant function.

Elapsed Time (hours) 0 1 2 3

Temperature 102°F 102°F 102°F 102°F

Solution

0

20

40

60

80

100

120

0 1 2 3

Time (hours)

Te

mp

era

ture

(d

eg

ree

s)


End of week 3

You again have the answers to those problems not assigned

Practice is SOOO important in this course. Work as much as you can with MyMathLab, the

materials in the text, and on my Webpage. Do everything you can scrape time up for, first the

hardest topics then the easiest. You are building a skill like typing, skiing, playing a

game, solving puzzles. NEXT TIME: Exponents and Polynomials

Week 3. Due for this week… Homework 3 (on MyMathLab – via the Materials Link) Monday night at...

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Transcript of Week 3. Due for this week… Homework 3 (on MyMathLab – via the Materials Link) Monday night at...