Chapter 1
Functions and Their Graphs
Course Number
Section 1.1 Lines in the Plane
Instructor
Objective: In this lesson you learned how to find and use the slope of a
line to write and graph linear equations.
Date
Important Vocabulary
Define each term or concept.
Slope The number of units a line rises or falls vertically for each unit of horizontal
change from left to right.
Parallel Two distinct nonvertical lines are parallel if and only if their slopes are
equal.
Perpendicular Two distinct nonvertical lines are perpendicular if and only if their
slopes are negative reciprocals of each other. That is, m1 = – 1/m2.
I. The Slope of a Line (Pages 3−4)
The formula for the slope of a line passing through the points
(x1, y1) and (x2, y2) is m =
(y2 − y1)/(x2 − x1)
What you should learn
How to find the slopes of
lines
.
To find the slope of the line through the points (− 2, 5) and
(4, − 3), . . .
subtract 5 from − 3 and divide this result by the
difference of 4 and − 2.
A line whose slope is positive
rises
from left to right.
A line whose slope is negative
falls
from left to right.
A line with zero slope is
horizontal
A line with undefined slope is
.
vertical
.
II. The Point-Slope Form of the Equation of a Line
(Pages 5−6)
The point-slope form of the equation of a line is
y − y1 = m(x − x1)
What you should learn
How to write linear
equations given points on
lines and their slopes
.
This form of equation is best used to find the equation of a line
when . . .
you know at least one point that the line passes
through and the slope of the line.
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2
Chapter 1
Functions and Their Graphs
The two-point form of the equation of a line is
y − y1 = (y2 − y1)/(x2 − x1)⋅(x − x1)
.
The two-point form of equation is best used to find the equation
of a line when . . .
you know the coordinates of two points on
the line.
Example 1: Find an equation of the line having slope − 2 that
passes through the point (1, 5).
y = − 2x + 7
The approximation method used to estimate a point between two
given points is called
linear interpolation
. The
approximation method used to estimate a point that does not lie
between two given points is called
linear extrapolation
A linear function has the form
is a
that has slope
line
(0, b)
f(x) = mx + b
m
.
. Its graph
and a y-intercept at
.
III. Sketching Graphs of Lines (Pages 7−8)
The slope-intercept form of the equation of a line is
y = mx + b
y-intercept is (
, where m is the
0
,
b
slope
and the
What you should learn
How to use slopeintercept forms of linear
equations to sketch lines
).
Example 2: Determine the slope and y-intercept of the linear
equation 2 x − y = 4 .
The slope is 2 and the y-intercept is (0, − 4).
The equation of a horizontal line is
horizontal line is
0
. The slope of a
. The y-coordinate of every point on the
graph of a horizontal line is
b
The equation of a vertical line is
vertical line is
y=b
undefined
.
x=a
. The slope of a
. The x-coordinate of every point
on the graph of a vertical line is
.
a
The general form of the equation of a line is
Ax + By + C = 0
.
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Section 1.1
3
Lines in the Plane
Every line has an equation that can be written in
form
.
general
When a graphing utility is used to sketch a straight line, the
graph of the line may not visually appear to have the slope
indicated by its equation because . . .
of the viewing window
used for the graph.
Example 3: Use a graphing utility to graph the linear equation
2 x − y = 4 using (a) a standard viewing window,
and (b) a square window.
(a)
(b)
IV. Parallel and Perpendicular Lines (Pages 9−10)
What you should learn
How to use slope to
identify parallel and
perpendicular lines
The relationship between the slopes of two lines that are parallel
is . . .
that the slopes are equal.
The relationship between the slopes of two lines that are
perpendicular is . . .
that the slopes are negative reciprocals of
each other.
A line that is parallel to a line whose slope is 2 has slope
2
.
A line that is perpendicular to a line whose slope is 2 has slope
− 1/2
.
What must be done to make the graphs of two perpendicular
lines appear to intersect at right angles when they are graphed
using a graphing utility?
The perpendicular lines must be graphed using a square viewing
window.
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Chapter 1
Functions and Their Graphs
Example 4: Use a graphing utility to graph the perpendicular
lines y = 2 x − 3 and y = −0.5 x + 5 using (a) a
standard viewing window, and (b) a square
window.
(a)
(b)
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Section 1.2
5
Functions
Course Number
Section 1.2 Functions
Instructor
Objective: In this lesson you learned how to evaluate functions and find
their domains.
Important Vocabulary
Date
Define each term or concept.
Function A function f from a set A to a set B is a relation that assigns to each element
x in the set A exactly one element y in the set B.
Domain The set of inputs of the function f.
Range The set of all outputs for the given set of inputs of the function f.
Independent variable A variable in an equation that represents a function that can
take on any value for which the function is defined.
Dependent variable A variable in an equation that represents a function whose value
depends on the value of the independent variable.
I. Introduction to Functions (Pages 16−18)
A rule of correspondence that matches quantities from one set
with items from a different set is a
relation
.
What you should learn
How to decide whether a
relation between two
variables represents a
function
In functions that can be represented by ordered pairs, the first
coordinate in each ordered pair is the
second coordinate is the
output
input
and the
.
Some characteristics of a function from Set A to Set B are . . .
1) Each element of A must be matched with an element
of B.
2) Some elements of B may not be matched with any
element of A.
3) Two or more elements of A may be matched with the
same element of B.
4) An element of A (the domain) cannot be matched with
two different elements of B.
To determine whether or not a relation is a function, . . .
decide whether each input value is matched with exactly one
output value.
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Chapter 1
Functions and Their Graphs
If any input value of a relation is matched with two or more
output values, . . .
the relation is not a function.
Some common ways to represent functions are . . .
1) Verbally in a sentence
2) Numerically in a table or list of ordered pairs
3) Graphically by points on a graph in the coordinate plane
4) Algebraically by an equation in two variables
Example 1: Decide whether the table represents y as a function
of x.
0
2
4
x
−3
−1
5
5
3
14
y
− 12
Yes, this table represents y as a function of x.
II. Function Notation (Pages 18−20)
The symbol
f(x)
is function notation for the value
of f at x or simply f of x. The symbol f(x) corresponds to the
y-value
for a given x.
Keep in mind that
whereas
f(x)
What you should learn
How to use function
notation and evaluate
functions
f
is the name of the function,
is the output value of the function at the
input value x.
In function notation, the
variable and the
output
input
is the independent
is the dependent variable.
Example 2: If f ( w) = 4 w 3 − 5w 2 − 7 w + 13 , describe how to
find f (−2) .
Replace each occurrence of w in the function by
− 2 and evaluate the resulting numerical
expression.
A piecewise-defined function is . . .
a function that is defined
by two or more equations over a specified domain.
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Section 1.2
7
Functions
III. The Domain of a Function (Page 20−21)
The implied domain of a function defined by an algebraic
expression is . . .
What you should learn
How to find the domains
of functions
the set of all real numbers for which the
expression is defined.
In general, the domain of a function excludes values that . . .
would cause division by zero or result in the even root of a
negative number.
For example, the implied domain of the function f ( x) = 5 x − 8
is . . .
the set of all real numbers greater than or equal to 8/5, or
[8/5, ∞).
IV. Applications of Functions (Page 22)
Example 3: The price P (in dollars) of a child’s handmade
sweater is given by the function P(s) = 3s + 15,
where s represents the size (size 1, size 2, etc.) of
the sweater. Use this function to find the price of a
child’s size 5 handmade sweater.
$30
V. Difference Quotients (Page 23)
A difference quotient is defined as . . .
What you should learn
How to use functions to
model and solve real-life
problems
What you should learn
How to evaluate
difference quotients
[f(x + h) − f(x)]/h, h ≠ 0.
Describe a real-life situation which can be represented by a
function.
Answers will vary.
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Chapter 1
Functions and Their Graphs
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Section 1.3
9
Graphs of Functions
Course Number
Section 1.3 Graphs of Functions
Instructor
Objective: In this lesson you learned how to analyze the graphs of
functions.
Important Vocabulary
Date
Define each term or concept.
Graph of a function The collection of ordered pairs (x, f(x)) such that x is in the
domain of f.
Greatest integer function The greatest integer function is denoted by [|x|] and is
defined as the greatest integer less than or equal to x.
Step function A function whose graph resembles a set of stair steps.
Even function A function whose graph is symmetric with respect to the y-axis.
Odd function A whose graph is symmetric with respect to the origin.
I. The Graph of a Function (Pages 30−31)
Explain the use of open or closed dots in the graphs of functions.
The use of dots, open or closed, at the extreme left or right points
of a graph indicates that the graph does not extend beyond these
What you should learn
How to find the domains
and ranges of functions
and how to use the
Vertical Line Test for
functions
points. If no such dots are shown, assume that the graph extends
beyond these points.
To find the domain of a function from its graph, . . .
examine
the graph to see for which x-values the graph is plotted.
To find the range of a function from its graph, . . .
examine the
graph to see for which y-values the graph is plotted.
The Vertical Line Test for functions states . . .
that a set of
points in a coordinate plane is the graph of y as a function of x if
and only if no vertical line intersects the graph at more than one
point.
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Chapter 1
Functions and Their Graphs
Example 1: Decide whether each graph represents y as a
function of x.
(a)
(b)
(a) Yes, this is a graph of y as a
function of x
(b) No, this is not a graph of y as
a function of x
II. Increasing and Decreasing Functions (Page 32)
A function f is increasing on an interval if, for any x1 and x2 in
the interval, . . .
x1 < x2 implies f(x1) < f(x2).
A function f is decreasing on an interval if, for any x1 and x2 in
the interval, . . .
What you should learn
How to determine
intervals on which
functions are increasing,
decreasing, or constant
x1 < x2 implies f(x1) > f(x2).
A function f is constant on an interval if, for any x1 and x2 in the
interval,. . .
f(x1) = f(x2).
Given a graph of a function, to find an interval on which the
function is increasing . . .
search for an interval over which
the graph rises from left to right.
Given a graph of a function, to find an interval on which the
function is decreasing . . .
search for an interval over which
the graph falls from left to right.
Given a graph of a function, to find an interval on which the
function is constant . . .
search for an interval over which the
graph is level.
III. Relative Minimum and Maximum Values
(Pages 33−34)
A function value f(a) is called a relative minimum of f if . . .
there exists an interval (x1, x2) that contains a such that
What you should learn
How to determine
relative maximum and
relative minimum values
of functions
x1 < x < x2 implies f(a) ≤ f(x).
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Section 1.3
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Graphs of Functions
A function value f(a) is called a relative maximum of f if . . .
there exists an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a) ≥ f(x).
The point at which a function changes from increasing to
decreasing is a relative
maximum
. The point at which
a function changes from decreasing to increasing is a relative
minimum
.
To approximate the relative minimum or maximum of a function
using a graphing utility, . . .
use the zoom and trace features to
approximate the lowest or highest point on the graph, OR use a
built-in program or feature for finding minimum or maximum
values.
Example 2: Suppose a function C represents the annual
number of cases (in millions) of chicken pox
reported for the year x in the United States from
1960 through 2000. Interpret the meaning of the
function’s minimum at (1998, 3).
The lowest number of cases of chicken pox reported
during this period was 3 million cases in 1998.
IV. Graphing Step Functions and Piecewise-Defined
Functions (Page 35)
Describe the graph of the greatest integer function.
The graph of the greatest integer function jumps vertically one
unit at each integer and is constant (a horizontal line segment)
between each pair of consecutive integers.
What you should learn
How to identify and
graph step functions and
other piecewise-defined
functions
Example 3: Let f ( x) = x , the greatest integer function. Find
f (3.74) .
3
To sketch the graph of a piecewise-defined function, . . .
sketch the graph of each equation of the piecewise-defined
function over the appropriate portion of the domain.
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Chapter 1
V. Even and Odd Functions (Pages 36−37)
What you should learn
How to identify even and
odd functions
A graph is symmetric with respect to the y-axis if, whenever
(x, y) is on the graph,
(− x, y)
Functions and Their Graphs
is also on the graph. A graph
is symmetric with respect to the x-axis if, whenever (x, y) is on
(x, − y)
the graph,
is also on the graph. A graph is
symmetric with respect to the origin if, whenever (x, y) is on the
(− x, − y)
graph,
is also on the graph. A graph that is
symmetric with respect to the x-axis is . . .
not the graph of a function (except for the graph of y = 0).
A function f is even if, for each x in the domain of f,
f(−x) =
.
f(x)
A function f is even if, for each x in the domain of f,
f(−x) =
.
f(x)
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Section 1.4
13
Shifting, Reflecting, and Stretching Graphs
Section 1.4 Shifting, Reflecting, and Stretching Graphs
Course Number
Instructor
Objective: In this lesson you learned how to identify and graph shifts,
reflections, and nonrigid transformations of functions.
Important Vocabulary
Date
Define each term or concept.
Vertical shift A transformation of the graph of y = f(x), represented by h(x) = f(x) ± c,
in which the graph is shifted upward or downward c units respectively (c is a positive
real number).
Horizontal shift A transformation of the graph of y = f(x), represented by
h(x) = f(x ± c), in which the graph is shifted to the left or to the right c units
respectively (c is a positive real number).
Rigid transformations Transformations in which the basic shape of the graph is
unchanged.
Nonrigid transformations Transformations that cause a distortion (a change in the
original shape of the graph).
I. Summary of Graphs of Parent Functions (Page 42)
What you should learn
How to recognize graphs
of parent functions
Sketch an example of each of the six most commonly used
functions in algebra.
Constant Function
Identity Function
y
y
x
Absolute Value Function
x
Square Root Function
y
y
x
x
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Chapter 1
Quadratic Function
Cubic Function
y
y
x
x
II. Vertical and Horizontal Shifts (Pages 43−44)
Let c be a positive real number. Complete the following
representations of shifts in the graph of y = f (x) :
1) Vertical shift c units upward:
h(x) = f(x) + c
What you should learn
How to use vertical and
horizontal shifts and
reflections to graph
functions
h(x) = f(x) − c
2) Vertical shift c units downward:
3) Horizontal shift c units to the right:
4) Horizontal shift c units to the left:
Example 1:
Functions and Their Graphs
h(x) = f(x − c)
h(x) = f(x + c)
Let f ( x) = x . Write the equation for the
function resulting from a vertical shift of 3 units
downward and a horizontal shift of 2 units to the
right of the graph of f (x) .
f(x) = | x − 2 | − 3
III. Reflecting Graphs (Pages 45−46)
A reflection in the x-axis is a type of transformation of the graph
of y = f(x) represented by h(x) =
− f(x)
. A reflection in
the y-axis is a type of transformation of the graph of y = f(x)
represented by h(x) =
f(− x)
.
Example 2: Let f ( x) = x . Describe the graph of g ( x) = − x
in terms of f .
The graph of g is a reflection of the graph of f in
the x-axis.
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Section 1.4
15
Shifting, Reflecting, and Stretching Graphs
IV. Nonrigid Transformations (Page 47)
What you should learn
How to use nonrigid
transformations to graph
functions
Name three types of rigid transformations:
1) Horizontal shifts
2) Vertical shifts
3) Reflections
Rigid transformations change only the
position
of the
graph in the coordinate plane.
Name four types of nonrigid transformations:
1) Vertical stretch
2) Vertical shrink
3) Horizontal shrink
4) Horizontal stretch
A nonrigid transformation y = cf (x) of the graph of y = f (x) is
a
if c > 1 or a
vertical stretch
vertical shrink
if
0 < c < 1. A nonrigid transformation y = f (cx) of the graph of
y = f (x) is a
horizontal shrink
if c > 1 or a
if 0 < c < 1.
horizontal stretch
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Chapter 1
Functions and Their Graphs
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Section 1.5
17
Combinations of Functions
Section 1.5 Combinations of Functions
Course Number
Instructor
Objective: In this lesson you learned how to find arithmetic
combinations and compositions of functions.
I. Arithmetic Combinations of Functions (Pages 51−53)
Just as two real numbers can be combined by the operations of
addition, subtraction, multiplication, and division to form other
Date
What you should learn
How to add, subtract,
multiply, and divide
functions
real numbers, two functions f and g can be combined to create
new functions such as the
quotient
sum, difference, product, and
of f and g to create new functions.
The domain of an arithmetic combination of functions f and g
consists of . . .
all real numbers that are common to the
domains of f and g. In the case of the quotient f(x)/g(x), there is
the further restriction that g(x) ≠ 0.
Let f and g be two functions with overlapping domains.
Complete the following arithmetic combinations of f and g for all
x common to both domains:
1) Sum: ( f + g )( x) =
f(x) + g(x)
2) Difference:
3) Product:
4) Quotient:
( f − g )( x) =
( fg )( x) =
⎛f ⎞
⎜⎜ ⎟⎟(x) =
⎝g⎠
f(x) − g(x)
f(x) · g(x)
f(x) ÷ g(x), g(x) ≠ 0
To use a graphing utility to graph the sum of two functions, . . .
enter the first function as y1 and enter the second function as y2.
Define y3 as y1 + y2, then graph y3.
Example 1:
Let f ( x) = 7 x − 5 and g ( x) = 3 − 2 x . Find
( f − g )(4) .
28
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Chapter 1
II. Compositions of Functions (Pages 53−56)
The composition of the function f with the function g is
( f D g )( x) =
f(g(x))
.
Functions and Their Graphs
What you should learn
How to find
compositions of one
function with another
function
For the composition of the function f with g, the domain of
f D g is . . .
the set of all x in the domain of g such that
g(x) is in the domain of f.
Example 2:
Let f ( x) = 3 x + 4 and let g ( x) = 2 x 2 − 1 . Find
(a) ( f D g )( x) and (b) ( g D f )( x) .
(b) 18x2 + 48x + 31
(a) 6x2 + 1
III. Applications of Combinations of Functions (Page 57)
The function f ( x) = 0.06 x represents the sales tax owed on a
purchase with a price tag of x dollars and the function
g ( x) = 0.75 x represents the sale price of an item with a price tag
of x dollars during a 25% off sale. Using one of the combinations
of functions discussed in this section, write the function that
represents the sales tax owed on an item with a price tag of x
dollars during a 25% off sale.
What you should learn
How to use combinations
of functions to model and
solve real-life problems
(f g)(x) = (g f)(x) = 0.045x
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Section 1.6
19
Inverse Functions
Course Number
Section 1.6 Inverse Functions
Instructor
Objective: In this lesson you learned how to find inverse functions
graphically and algebraically.
Important Vocabulary
Date
Define each term or concept.
Inverse function Let f and g be two functions. If f(g(x)) = x for every x in the domain
of g and g(f(x)) = x for every x in the domain of f, then g is the inverse function of the
function f. The function g is denoted by f −1.
One-to-one A function f is one-to-one if, for a and b in its domain, f(a) = f(b) implies
a = b.
Horizontal Line Test A function is one-to-one if every horizontal line intersects the
graph of the function at most once.
I. Inverse Functions (Pages 62−64)
For a function f that is defined by a set of ordered pairs, to form
the inverse function of f, . . .
interchange the first and second
coordinates of each of these ordered pairs.
What you should learn
How to find inverse
functions informally and
verify that two functions
are inverse functions of
each other
For a function f and its inverse f −1, the domain of f is equal to
the range of f −1
the domain of f −1
, and the range of f is equal to
.
To verify that two functions, f and g, are inverses of each other,
...
find f(g(x)) and g(f(x)). If both of these compositions are
equal to the identity function, then the functions are inverses of
each other.
Example 1:
Verify that the functions f ( x) = 2 x − 3 and
x+3
g ( x) =
are inverses of each other.
2
II. The Graph of an Inverse Function (Page 65)
If the point (a, b) lies on the graph of f, then the point
(
b ,
a
) lies on the graph of f −1 and vice versa. The
graph of f −1 is a reflection of the graph of f in the line
y=x
What you should learn
How to use graphs of
functions to decide
whether functions have
inverse functions
.
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Chapter 1
III. The Existence of an Inverse Function (Page 66)
If a function is one-to-one, that means . . .
that no two
Functions and Their Graphs
What you should learn
How to determine if
functions are one-to-one
elements in the domain of the function correspond to the same
element in the range of the function.
A function f has an inverse f −1 if and only if . . .
f is one-to-one.
To tell whether a function is one-to-one from its graph, . . .
simply use the Horizontal Line Test, that is, check to see that
every horizontal line intersects the graph of the function at most once.
Example 2:
Does the graph of the function at the right have an
inverse function? Explain.
No, it doesn’t pass the Horizontal Line Test.
IV. Finding Inverse Functions Algebraically (Pages 67−68)
To find the inverse of a function f algebraically, . . .
What you should learn
How to find inverse
functions algebraically
1) Use the Horizontal Line Test to decide whether f has an
inverse function.
2) In the equation for f(x), replace f(x) by y.
3) Interchange the roles of x and y, and solve for y.
4) Replace y by f −1(x) in the new equation.
5) Verify that f and f −1 are inverse functions of each other by showing
that the domain of f is equal to the range of f −1, the range of f is equal
to the domain of f −1, and f(f −1(x)) = x and f −1(f(x)) = x.
Example 3:
Find the inverse (if it exists) of f ( x) = 4 x − 5 .
f −1(x) = 0.25x + 1.25
Homework Assignment
Page(s)
Exercises
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section 1.7
21
Linear Models and Scatter Plots
Section 1.7 Linear Models and Scatter Plots
Course Number
Instructor
Objective: In this lesson you learned how to use scatter plots and a
graphing utility to find linear models for data.
Important Vocabulary
Date
Define each term or concept.
Fitting a line to data The process of finding a linear model to represent the
relationship described by a scatter plot.
I. Scatter Plots and Correlation (Pages 73−74)
Many real-life situations involve finding relationships between
two variables. If data is collected and written as a set of ordered
pairs, the graph of such a set is called a
scatter plot
What you should learn
How to construct scatter
plots and interpret
correlation
.
For a collection of ordered pairs of the form (x, y), if y tends to
increase as x increases, the collection is said to have a(n)
positive correlation
. If y tends to decrease as x
increases, the collection is said to have a(n)
correlation
negative
.
II. Fitting a Line to Data (Pages 74−77)
To fit a line to data represented in a scatter plot, . . .
sketch
the line that appears to fit the points, find two points on the line,
and then find the equation of the line that passes through the two
What you should learn
How to use scatter plots
and a graphing utility to
find linear models for
data
points, OR use the linear regression feature of a graphing utility
to find the linear model.
To measure how well a linear model fits the data used to find the
model, . . .
compare the actual values with the values given
by the model.
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
22
Chapter 1
Functions and Their Graphs
The correlation coefficient r of a set of data varies between
−1
and
1
. The closer | r | is to 1, the better . . .
the points can be described by a line.
Example 1: The numbers of U.S. Navy personnel p in
thousands on active duty for the years 2002
through 2006 are shown in the table. Use the
regression capabilities of a graphing utility to find
a linear model for the data. Let t represent the year
with t = 2 corresponding to 2002.
Year
2002 2003 2004 2005 2006
383
381
376
364
353
p
(Source: U.S. Department of Defense)
p = − 7.7t + 402.2
y
y
x
y
y
x
y
x
x
y
x
x
Homework Assignment
Page(s)
Exercises
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.