Chapter 1 Functions and Their Graphs - Cengagecollege.cengage.com/mathematics/larson/precalculus_limits/1e2/...aaaaaaaaa Example 1: Complete the table. ... aaaaaa aaaaaaaaa aaaaaa

Name______________________________________________

Chapter 1 Functions and Their Graphs Section 1.1 Rectangular Coordinates Objective: In this lesson you learned how to plot points in the

coordinate plane and use the Distance and Midpoint Formulas.

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Important Vocabulary Define each term or concept. Rectangular coordinate system a aaaaa aaaaaaaaaa aaaaaa aaaa aa aaaaaaaaaaa aaaaaaaaa aaaaaaa aaaaa aa aaaa aaaaaaaa Ordered pair aaa aaaa aaaaaaa a aaa aa aaaaaaa aaa aaa aaaaa aaaaaaaaa a aaaaa aa aaa aaaaaaaaa aaaaaa

What you should learn How to plot points in the Cartesian plane

. The Cartesian Plane (Pages 2−3)

he Cartesian plane, named after the French mathematician

ené Descartes, is formed by . . . aaaaa aaa aaaa aaaaaa aaaaa

aaaaaaaaaaa aa aaaaa aaaaaaa

n the Cartesian plane, the horizontal real number line is usually

alled the aaaaaa , and the vertical real number line is

sually called the aaaaaa . The origin is the aaa a

a aaaaaaaaaaaa of these two axes, and the two axes divide

he plane into four parts called aaaaaaaaa .

n the Cartesian plane shown below, label the x-axis, the y-axis, he origin, Quadrant I, Quadrant II, Quadrant III, and Quadrant V.

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arson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide opyright © Houghton Mifflin Company. All rights reserved. 1

2 Chapter 1 • Functions and Their Graphs

To sketch a scatter plot of paired data given in a table, . . .

aaaaaaaaa aaaa aaaa aa aaaaaa aa aa aaaaaaa aaaa aaa aaaa aaa

aaaaaaaaa aaaaaaa

Example 1: Explain how to plot the ordered pair (3, − 2), and then plot it on the Cartesian plane provided.

aaaaaaa a aaaaaaaa aaaa aaaaaaa a aa aaa aaaaaa aaa a aaaaaaaaaa aaaa aaaaaaa a a aa aaa aaaaaaa aaa aaaaaaaaaaaa aa aaaaa aaa aaaaa aa aaa aaaaa aaa a aaa

y

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II. The Pythagorean Theorem and the Distance Formula

What you should learn How to use the Distance Formula to find the distance between two points

(Pages 4−5) The Pythagorean Theorem states that for a right triangle with

hypotenuse of length c and sides of lengths a and b, the

mathematical relationship between a, b, and c is . . .

a a a a aa a a.

The Distance Formula states that . . . aaa aaaaaaaa a aaaaaaa

aaa aaaaaa aa a a a aaa aa a a a aa aaa aaaaa aaa a a a

a a a aa a a a a aa a a a aa aa

a aa

Example 2: Explain how to use the Distance Formula to find

the distance between the points (4, 2) and (5, − 1). Then find the distance and round to the nearest hundredth.

aaaaaaaaaaaa aaaa aaaaa aaa a a a aaa a a aa aaa aaa a a a aaa a a a aa aaaa aaaaaaaaaa aaaaa aaaaaa aaaa aaa aaaaaaaa aaaaaaa aaa aaaaaaaaa

a a

a a

aaaa

Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide Copyright © Houghton Mifflin Company. All rights reserved.

Section 1.1 • Rectangular Coordinates 3 Name______________________________________________

III. The Midpoint Formula (Page 5) What you should learn How to use the Midpoint Formula to find the midpoint of a line segment

To find the midpoint of a line segment that joins two points in a

coordinate plane, simply . . . aaaa aaa aaaaaaa aaaaaa aa aaa

aaaaaaaaaa aaaaaaaaaaa aa aaa aaa aaaaaaaaa aaaaa aaa aaaaaaaa

aaaaaaaa

The Midpoint Formula gives the midpoint of the segment

joining the points (x1, y1) and (x2, y2) as . . .

a a a a a a a a a a a aaaaaaaaa a a aaaa a aaaa a a a a a Example 3: Explain how to find the midpoint of the line segment with

endpoints at (− 8, 2) and (6, − 10). Then find the coordinates of the midpoint.

aaaaaaaaaaaa aaaa aaaaa aaaa aaa aaaaaaa aa aaa aaa aaaaaaaaaaaaa aaa aaaa aaa aaaaaaa aa aaa aaa aaaaaaaaaaaaaa aaaaa aaaaaaaa aaaa aaa aaaaaaaaaaa aa aaa aaaaaaaaa aa aa a aaa

What you should learn How to use a coordinate plane and geometric formulas to model and solve real-life problems

IV. Applications of the Coordinate Plane (Pages 6−8) To shift a figure plotted in the rectangular coordinate system by

a units to the left and b units upward, . . . aaaaaaaa a

aaaa aaa aaaaaaaaaaaa aa aaaa aaaaa aa aaa aaaaaa aaa aaa a aa

aaa aaaaaaaaaaaa aa aaaa aaaaa aa aaa aaaaaaa

Give an example of a real-life situation in which representing data graphically would be useful. aaaaaaa aaaa aaaaa Describe a real-life situation in which the Distance Formula could be used to solve a problem. aaaaaaa aaaa aaaaa



Complete the following list of common formulas for basic geometric figures. Perimeter/Circumference Rectangle with width w and length l: P = aa a aa a

Triangle with sides a, b, and c: P = a a a a a a

Circle with radius r: C = aa a

Area Rectangle with width w and length l: A = aa a

Triangle with base b and height h: A = aa a

Circle with radius r: A = aa a

Volume Rectangular solid with width w, length l, and height h: V = aa a

Circular cylinder with radius r and height h V = aa a

Sphere with radius r: V = aa a

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Homework Assignment Page(s) Exercises


Section 1.2 • Graphs of Equations 5 Name______________________________________________

Section 1.2 Graphs of Equations Objective: In this lesson you learned how to sketch the graph of an

equation. I T

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Important Vocabulary Define each term or concept. Equation in two variables a aaa aa aaaaaaaaaa a aaaaaaaaaaaa aaaaaaa aaa aaaaaaaaaaaSolution of equation in two variables aa aaaaaaa aaaa aaa aa aa a aaaaaaaa aa aa aaaaaaaa aa a aaa a aa aaa aaaaaaaa aa aaaa aaaa a aa aaaaaaaaaaa aaa a aaa a aa aaaaaaaaaaa aaa aaGraph of an equation aaa aaa aa aaa aaaaaa aaaa aaa aaaaaaaaa aa aaa aaaaaaaaa Intercepts aaa aaaaaa aa aaaaa a aaaaa aaaaaaaaaa aa aaaaaaa aaa aa aa aaaaaaa Symmetry aa a aaaaa aa aaaaaa aaaaa a aaaaaaaa aaaa aaa aaa aaaaaaa aa aaa aaaaa aa aaa aaaa aa aaa aaaaaaaa aaaa aaaaaaaaa aaaa aaa aaaaaaa aa aaa aaaaa aa aaa aaaaa aaaa aa aaa aaaaaaaa aaaaa aaaa aaa aaaaa aa aaaa aa aaaa aaaaaaaaaCircle aaa aaa aa aaaaaa aaaa aaa aaaaaaaaaaa aaaa a aaaaa aaaaaa aaa aaa aaaaaa aaa aaaaaaa

What you should learn How to sketch graphs of equations

. The Graph of an Equation (Pages 14−17)

o sketch the graph of an equation in two variables using the

oint-plotting method, . . . aa aaaaaaaaa aaaaaaa aaa aaaaaaaa

a aaaa aaa aa aaa aaaaaaaaa aa aaaaaaaa aa aaa aaaa aa aaa

aaaaaaaa aaaa aaaa a aaaaa aa aaaaaa aaaaaaa aaaaaaa aaaaaaaa

aaaaaa aaaa aaaaa aaaaaa aa a aaaaaaaaaaa aaaaaaaaaa aaaaaaa

aaaaaaa aaaaaaa aaa aaaaaa aaaa a aaaaaa aaaaa aa aaaaa

shortcoming of the point-plotting method is . . . aaaa aaaa

aa aaa aaaaaaaa aaaaaaa aaa aaa aaaaaaaaaaaa aaa aaaaa aa aa

aaaaaaaa

xample 1: Complete the table. Then use the resulting solution points to sketch the graph of the equation y = 3 − 0.5x.

xy

arson/Hostetler Popyright © Hough

− 4 − 2 0 2 4 a a a a a

recalculus/Precalculus with Limits Notetaking Guide ton Mifflin Company. All rights reserved.


y

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To graph an equation involving x and y on a graphing utility, . . .

aaaaaaa aaa aaaaaaaa aa aaaa a aa aaaaaaaa aa aaa aaaa aaaaa

aaaaa aaa aaaaaaaa aaaa aaa aaaaaaaa aaaaaaaa aaaaaaaaa a

aaaaaaa aaaaaa aaaa aaaaa aaa aaaaaaaaa aaaaaaaa aa aaa aaaaaa

aaa aaaaa aaa aaaaaaaaa

What you should learn How to find x- and y-intercepts of graphs of equations

II. Intercepts of a Graph (Page 17) An x-intercept is written as the ordered pair aaa aa ,

and a y-intercept is written as the ordered pair aaa aa .

To identify the x-intercepts of a graph, . . . aaaaaaa aaa aaaaa

aa aaaa aaa aaaaaa aaaaa a aa aaaaa

To identify the y-intercepts of a graph, . . . aaaaaaa aaa aaaaa

aa aaaa aaa aaaaaa aaaaa a aa aaaaa

What you should learn How to use symmetry to sketch graphs of equations

III. Symmetry (Pages 18−20) The three types of symmetry that a graph can exhibit are . . .

aaaaaa aaaaaaaaa aaaaaa aaaaaaaaa aa aaaaaa aaaaaaaaa

Knowing the symmetry of a graph before attempting to sketch it

is helpful because . . . aaaa aaa aaaa aaaa aaaa aa aaaa

aaaaaaaa aaaaaa aa aaaaaa aaa aaaaaa

A graph is symmetric with respect to the x-axis if, whenever

(x, y) is on the graph, aaa a aa is also on the graph. A

graph is symmetric with respect to the y-axis if, whenever (x, y)


Section 1.2 • Graphs of Equations 7 Name______________________________________________

is on the graph, aa aa aa is also on the graph. A graph is

symmetric with respect to the origin if, whenever (x, y) is on the

graph, aa aa a aa is also on the graph.

The graph of an equation is symmetric with respect to the x-axis

if . . . aaaaaaaaa a aaaa a a aaaaaa aa aaaaaaaaaa aaaaaaaaa

The graph of an equation is symmetric with respect to the y-axis

if . . . aaaaaaaaa a aaaa a a aaaaaa aa aaaaaaaaaa aaaaaaaaa

The graph of an equation is symmetric with respect to the origin

if . . . aaaaaaaaa a aaaa a a aaa a aaaa a a aaaaaa aa aaaaaaaaaa

aaaaaaaaa

Example 2: Use symmetry to sketch the graph of the equation

. 22 2 += xy

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IV. Circles (Page 20) What you should learn

How to find equations of and sketch graphs of circles

The standard form of the equation of a circle with center

(h, k) and radius r is aa a aa a aa a aa a aa a a .

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The standard form of the equation of a circle with radius r and its center at the origin is a a a a aa a a . Example 3: For the equation , find the

center and radius of the circle and then sketch the graph of the equation.

4)1()2( 22 =−++ yx

aaaaaaa aa aa aa aaaaaaa a



V. Applications of Graphs of Equations (Page 21) What you should learn How to use graphs of equations in solving real-life problems

List and describe three common approaches to solving a problem. 1) aaaaaaaaa aaaaaaaaa aaaaaaaaa aaa aaa a aaaaa

2) aaaaaaaaa aaaaaaaaa aaaa aaa aaa a aaaaa

3) aaaaaaaaa aaaaaaaaa aaa aaa aaaaa aa aaaaaaa

Describe a real-life situation in which a graphical solution approach would be helpful. aaaaaa aaaa aaaaa Additional notes

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Section 1.3 • Linear Equations in Two Variables 9 Name______________________________________________

Section 1.3 Linear Equations in Two Variables Objective: In this lesson you learned how to find and use the slopes of lines to write and graph linear equations in two variables.

Important Vocabulary Define each term or concept. Slope aaa aaaaaa aa aaaaa a aaaaaaaaaaa aaaa aaaaa aaa aaaaaa aaaaaaaaaa aaa aaaa aaaa aa aaaaaaaaaa aaaaaa aaaa aaaa aa aaaaaaParallel aaa aaaaaaaa aaaaaaaaaaa aaaaa aaa aaaaaaaa aa aaa aaaa aa aaaaa aaaaaa aaa aaaaaa aaaa aaa a a a aa aPerpendicular aaa aaaaaaaaaaa aaaaa aaa aaaaaaaaaaaaa aa aaa aaaa aa aaaaa aaaaaa aaa aaaaaaaa aaaaaaaaaaa aa aaaa aaaaaa aaaa aaa a a a aaa aa a

What you should learn How to use slope to graph linear equations in two variables

I. Using Slope (Pages 25−26) The equation y = mx + b is called a linear equation in two

variables because . . . aaa aaaaa aa aaaa aaaaaaaa aa aaa aaa

aaaaaaaaa a aaa a aa a aaaaaaaa aaaaa

A line whose slope is positive aaaaa from left to right.

A line whose slope is negative aaaaa from left to right.

The slope-intercept form of the equation of a line is

a a aa a a , where m is the aaaaa and the

y-intercept is ( a , a ).

A vertical line has an equation of the form a a a . The

equation of a vertical line cannot be written in the form

y = mx + b because . . . aaa aaaaa aa a aaaaaaaa aaaa aa

aaaaaaaaaa

Example 1: Explain how to graph the linear equation y = − 2/3x − 4. Then sketch its graph.

aaaaaaa a a a aa aaa aaaaaaaaaaa aa aaa a aaa aaaaaaa aaa aaaaa aa a aaaa aaa aaaa aaaaa a aaaaa aaa aaaaa a aaaaa aaa aaaa aaaaa aa aaa aaaaaa

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Example 2: Sketch and label the graph of (a) y = − 1 and (b) x = 3.

y y

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(a) (b)

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II. Finding the Slope of a Line (Pages 27−28)

The formula for the slope of a line passing through the points

(x1, y1) and (x2, y2) is m = aa a a aaaa a a aa a a a .

To find the slope of the line through the points (− 2, 5) and

(4, − 3), . . . aaaaaaaa a a aaaa a aaa aaaaaa aaaa aaaaaa aa aaa

aaaaaaaaaa aa a a aaa aa

If a line falls from left to right, it has aaaaaaaa slope. If a

line is horizontal, it has aaaa slope. If a line is

vertical, it has aaaaaaaaa slope. If a line rises from left to

right, it has aaaaaaaa slope.

III. Writing Linear Equations in Two Variables (Page 29) The point-slope form of the equation of a line is

a a a a aaa a a a a a .

The point-slope form is most useful for . . . aaaaaaa aaa

aaaaaaaa aa a aaaaa

The two-point form of the equation of a line is

a a a a aa a a aaaa a a aaaa a a a a a a a a a .

Larson/Hostetler Precalculus/Precalc Copyright © Houghton

What you should learn How to find slopes of lines

What you should learn How to write linear equations in two variables

ulus with Limits Notetaking Guide Mifflin Company. All rights reserved.

Section 1.3 • Linear Equations in Two Variables 11 Name______________________________________________

Example 3: Find an equation of the line that passes through the points (1, 5) and (− 3, 7) using (a) the point-slope form and (b) the slope-intercept form.

aaa a a a a a aaaaa a aa aaa a a a aaaa a aaa IV. Parallel and Perpendicular Lines (Page 30) 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 . . . aaaa aaa aaaaaa aaa aaaaaa

The relationship between the slopes of two lines that are

perpendicular is . . . aaaa aaa aaaaaa aaa aaaaaaaa aaaaaaaaaaa

aa aaaa aaaaaa

A line that is parallel to a line whose slope is 2 has slope a .

A line that is perpendicular to a line whose slope is 2 has slope

a aaa .

What you should learn How to use linear equations in two variables to model and solve real-life problems

V. Applications of Slope (Pages 31−33) In real-life problems, the slope of a line can be interpreted as

either a(n) aaaaa or a(n) aaaa . If the x-

axis and y-axis have the same unit of measure, then the slope has

no units and is a aaaaa . If the x-axis and y-axis have

different units of measure, then the slope is a aaaa aa aaaa

aa aaaaaa .

Describe a real-life situation in which slope is a ratio. aaaaaaa aaaa aaaaa Describe a real-life situation in which slope is a rate of change. aaaaaaa aaaa aaaaa Linear or straight-line depreciation is . . . aaaa aaaa aaa aaaa

aaaaaa aa aaaaaaaaaaaa aa aaaaaaa aaaa aaaa aaaa aaa aaaaaa

aaaa aa aaaaaaaaa



The prediction method in which an estimated point does not lie

between the given points is called aaaaaa aaaaaaaaaaa .

When the estimated point lies between two given points, the

procedure is called aaaaaa aaaaaaaaaaaaa .

Every line has an equation that can be written in general form,

which is given as aa a aa a a a a , where A

and B are not both zero.

Additional notes

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Section 1.4 • Functions 13 Name______________________________________________

Section 1.4 Functions Objective: In this lesson you learned how to evaluate functions and find

their domains. I A

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Important Vocabulary Define each term or concept. Function a aaaaaaaa a aaaa a aaa a aa a aaa a aa a aaaaaaaa aaaa aaaaaaa aa aaaa aaaaaaa a aa aaa aaa a aaaaaaa aaa aaaaaaa a aa aaa aaa aaDomain aaa aaa aa aaaaaa aa aaa aaaaaaaa aa Range aaa aaa aa aaa aaaaaaa aaa aaa aaaaa aaa aa aaaaaa aa aaa aaaaaaaa aa Independent variable a aaaaaaaa aa aa aaaaaaaa aaaa aaaaaaaaaa a aaaaaaaa aaaa aaa aaaa aa aaa aaaaa aaa aaaaa aaa aaaaaaaa aa aaaaaaaaDependent variable a aaaaaaaa aa aa aaaaaaaa aaaa aaaaaaaaaa a aaaaaaaa aaaaa aaaaa aaaaaaa aa aaa aaaaa aa aaa aaaaaaaaaaa aaaaaaaaa

What you should learn How to determine whether relations between two variables are functions

. Introduction to Functions (Pages 40−42)

rule of correspondence that relates two quantities is a

aaaaaaaa .

n functions that can be represented by ordered pairs, the first

oordinate in each ordered pair (the x-value) is the aaaa a

nd the second coordinate (y-value) is the aaaaaa .

ome characteristics of a function from set A to set B are . . .

1) aaaa aaaaaaa aa a aaaa aa aaaaaaa aaaa aa aaaaaaa aa aa

2) aaaa aaaaaaaa aa a aaa aaa aa aaaaaaa aaaa aaa aaaaaaa

aa aa

3) aaa aa aaaa aaaaaaaa aa a aaa aa aaaaaaa aaaa aaa aaaa

4) 4)aaaaaaa aaaa aaa aaaaaaaaa aaaaaaaa aa aa

ome common ways to represent functions are . . .

1) aaaaaaaa aa a aaaaaaaa

2) aaaaaaaaaaa aa a aaaaa aa aaaa aa aaaaaaa aaaaa

3) aaaaaaaaaaa aa aaaaaa aa a aaaaa aa aaa aaaaaaaaaa

4) aaaaaaaaaaaaa aa aa aaaaaaaa aa aaa aaaaaaaaa

arson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide opyright © Houghton Mifflin Company. All rights reserved.


To determine whether or not a relation is a function, . . . aaaaaa

aaaaaaa aaaa aaaaa aaaaa aa aaaaaaa aaaa aaaaaaa aaa aaaaaa

aaaaaa

If any input value of a relation is matched with two or more

output values, . . . aaa aaaaaaaa aa aaa a aaaaaaaaa

Example 1: Decide whether the table represents y as a function

of x. x − 3 − 1 0 2 4 y 5 − 12 5 3 14

aaaa aaaa aaaaa aaaaaaaaaa a aa a aaaaaaaa aa aa II. Function Notation (Pages 42−43)

What you should learn How to use function notation and evaluate functions

The symbol aaaa is function notation for the value

of f at x or f of x, used to describe y as a function of x. In this

case, a is the name of the function and aaa is

the value of the function at x.

Example 2: If , describe how to

find . 13754)( 23 +−−= wwwwf

)2(−f aaaaaaa aaaa aaaaaaaaaa aa a aa aaa aaaaaaaa aa

a a aaa aaaaaaaa aaa aaaaaaaaa aaaaaaaaa aaaaaaaaaaa

A piecewise-defined function is . . . a aaaaaaaa aaaa aa

aaaaaaa aa aaa aa aaaa aaaaaaaaaa aaaa aaaa a aaaaaaaaa aaaaaaa

III. The Domain of a Function (Page 44) What you should learn

How to find the domains of functions

The implied domain of a function defined by an algebraic

expression is . . . aaa aaa aa aaa aaaa aaaaaaa aaa aaaaa aaa

aaaaaaaaaa aa aaaaaaaa

For example, the implied domain of the function 85)( −= xxf

is . . . aaa aaa aa aaa aaaa aaaaaaa aaaaaaa aaaa aa aaaaa aa

aaaa aa aaaaa aaa


Section 1.4 • Functions 15 Name______________________________________________

What you should learn How to use functions to model and solve real-life problems

IV. Applications of Functions (Pages 45−46) Describe a real-life situation that can be represented by a function. aaaaaaa aaaa aaaaa

What you should learn How to evaluate difference quotients

V. Difference Quotients (Pages 46−47) A difference quotient is defined as . . .

aaaa a aa a aaaaaaaa a a aa

Where are difference quotients seen? aaa aa aaa aaaaa aaaaaaaaaaa aa aaaaaaaa aaaaaaa a aaaaaaaaaa aaaaaaaaaa Additional notes

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Additional notes

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Section 1.5 • Analyzing Graphs of Functions 17 Name______________________________________________

Section 1.5 Analyzing Graphs of Functions Objective: In this lesson you learned how to analyze graphs of

functions. I T

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Important Vocabulary Define each term or concept. Graph of a function aaa aaaaaaaaaa aa aaaaaaa aaaaa aaa aaaaa aaaa aaaa a aa aa aaa aaaaaa aa aaEven function a aaaaaaaa a a aaaa aa aaaa aaa aaa aaaa a aa aaa aaaaaa aa aa aaa aa a aaaaaOdd function a aaaaaaaa a a aaaa aa aaa aaa aaa aaaa a aa aaa aaaaaa aa aa aaa aa a a aaaaa

What you should learn How to use the Vertical Line Test for functions

. The Graph of a Function (Pages 54−55)

o find the domain of a function from its graph, . . . aaaaaaa

aa aaaaa aa aaa aaa aaaaa aaaaaaaa aaa aaaaa aa aaaaaaaa

o find the range of a function from its graph, . . . aaaaaaa aaa

aaaa aa aaa aaa aaaaa aaaaaaaa aaa aaaaa aa aaaaaaaa

he Vertical Line Test for functions states that. . . a aaa aa

aaaaa aa a aaaaaaaaaa aaaaa aa aaa aaaaa aa a aa a aaaaaaaa aa

aa aaa aaaa aa aa aaaaaaaa aaaa aaaaaaaaaa aaa aaaaa aa aaaa

aaa aaa aaaaaa

xample 1: Decide whether each graph represents y as a function of x.

a) (b) y

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aa aaaa aaaaaaaaaa a aaaaaaaa aaa aaa aaaa aaa aaaaaaaaa a aaaaaaaa

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II. Zeros of a Function (Page 56) What you should learn How to find the zeros of functions

If the graph of a function of x has an x-intercept at (a, 0), then a

is a aaaa of the function.

The zeros of a function f of x are . . . aaa aaaaaaaa aaa

aaaaa aaaa a aa

To find the zeros of a function, . . . aaa aaa aaaaaaaa aaaaa

aa aaaa aaa aaaaa aaa aaa aaaaaaaaaaa aaaaaaaaa

Example 2: Find the zeros of the function

. 5194)( 2 −+= xxxf aaa aaaaa aaa a a aaa aaaa

What you should learn How to determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions

III. Increasing and Decreasing Functions (Pages 57−58) A function f is increasing on an interval if, for any x1 and x2 in

the interval, . . . a a a aaaaaaa aaa a a aaa aaa a a a

A function f is decreasing on an interval if, for any x1 and x2 in

the interval, . . . a a a aaaaaaa aaa a a aaa aaa a a a

A function f is constant on an interval if, for any x1 and x2 in the

interval,. . . aaa a a aaa aaa a

A function value f(a) is called a relative minimum of f if . . .

aaaaa aaaaaa aa aaaaaaaa aa a a a aaaa aaaaaaaa a aaaa aaaa

a a a a a aaaaaaa aaaa a aaaaaa a

a a

A function value f(a) is called a relative maximum of f if . . .

aaaaa aaaaaa aa aaaaaaaa aa a a a aaaa aaaaaaaa a aaaa aaaa

a a a a a aaaaaaa aaaa a aaaaaa a

a a

To approximate the relative minimum or maximum of a function

using a graphing utility, . . . aaa aaa aaaa aaa aaaaa aaaaaaaa

aa aaa aaaaaaa aaaaaaa aa aaaaaaaa aaa aaaaaa aa aaaaaaa aaaaa

aa aaa aaaaaa aa aaa aaa aaaaa aaaaaaa aa aaaaaaaaaaa

aaaaaaaaaaa aaa aaaaaaaa aaaaaaa aa aaaaaaaa


Section 1.5 • Analyzing Graphs of Functions 19 Name______________________________________________

IV. Average Rate of Change (Page 59) What you should learn How to determine the average rate of change of a function

For a nonlinear graph whose slope changes at each point, the

average rate of change between any two points is . . . aaa

aaaaa aa aaa aaaa aaaaaaa aaa aaa aaaaaaa

The line through the two points is called the

aaaaaa aaaa , and the slope of this line is denoted as

aaaa .

Let (a, f(a)) and (b, f(b)) be two points on the graph of a nonlinear function f. The average rate of change of f from a to b is given by: aaaaa a aaaaaaaa a aa a aaaaaaa aa aaaaaaaaaa aa aaa V. Even and Odd Functions (Page 60) What you should learn

How to identify even and odd functions

A function whose graph is symmetric with respect to the y-axis

is a(n) aaaa function. A function whose graph is

symmetric with respect to the origin is a(n) aaa a

function.

Can the graph of a nonzero function be symmetric with respect

to the x-axis? aa

Example 3: Decide whether the function

is even, odd, or neither. 134)( 2 +−= xxxf

aaaaaaa



Additional notes

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Section 1.6 • A Library of Parent Functions 21 Name______________________________________________

Section 1.6 A Library of Parent Functions Objective: In this lesson you learned how to identify and graph various

functions.

What you should learn How to identify and graph linear and squaring functions

I. Linear and Squaring Functions (Pages 66−67) The graph of the linear function f(x) = ax + b is a line with slope

a a a and y-intercept at aaa aa .

List several important features of the graph of the linear function f(x) = ax + b. A constant function is a special type of linear function having

the form aaaa a a . The domain of this function is

aaa aaaa aaaaaaa and the range consists of

a aaaaaa aaaa aaaaaa a .

The identity function is a special type of linear function having

the form aaaa a a . The domain of this function is

aaa aaaa aaaaaaa and the range consists of

aaa aaaa aaaaaaa . The identity function has a

slope of a a a and a y-intercept of aaa aa .

The graph of the identity function is a line for which . . .

aaaa aaaaaaaaaaaa aaaaaa aaa aaaaaaaaaaaaa aaaaaaaaaaaaa

List several important features of the U-shaped graph of the squaring function f(x) = x2.



II. Cubic, Square Root, and Reciprocal Functions What you should learn

How to identify and graph cubic, square root, and reciprocal functions

(Page 68) List several important features of the graph of the cubic function f(x) = x3. List several important features of the graph of the square root function xxf =)( . List several important features of the graph of the reciprocal

function x

xf 1)( = .

What you should learn How to identify and graph step and other piecewise-defined functions

III. Step and Piecewise-Defined Functions (Pages 69−70) Describe the graph of a step function. aaa aaaaa aa a aaaa aaaaaaaa aaaaaaaaa a aaa aa aaaaaaaaaaaa


Section 1.6 • A Library of Parent Functions 23 Name______________________________________________

The greatest integer function, xxf =)( , is defined as . . .

aaa aaaaaaaa aaaaaaa aaaa aaaa aa aaaaa aa aa

Example 1: Let xxf =)( , the greatest integer function. Find

. )74.3(f a List several important features of the graph of the greatest integer function. A piecewise-defined function is defined by . . . aaa aa aaaa

aaaaaaaaa aaaa a aaaaaaaaa aaaaaaa

To graph of a piecewise-defined function, . . . aaaaa aaaa

aaaaaaaa aa aaa aaaaaaaaaaaaaaaaa aaaaaaaa aaaaaaaaaa aaaa aaa

aaaaaaaaaaa aaaaaaa aa aaa aaaaaaa

IV. Parent Functions (Page 70) What you should learn

How to recognize graphs of common functions

Sketch an example of each of the following most commonly used functions in algebra. Constant Function Identity Function

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Absolute Value Function Square Root Function

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Quadratic Function Cubic Function y

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Reciprocal Function Greatest Integer Function

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Section 1.7 • Transformations of Functions 25 Name______________________________________________

Section 1.7 Transformations of Functions Objective: In this lesson you learned how to identify and graph rigid

and nonrigid transformations.

Important Vocabulary Define each term or concept. Vertical shift a aaaaaaaaaaaaaa aa aaa aaaaa aa a a aaaaa aaaaaaaaaaa aa aaaa a aaaa a aa aa aaaaa aaa aaaaa aa aaaaaaa aaaaaa aa aaaaaaaa a aaaaa aaaaaaaaaaaa aa aa a aaaaaaaa aaaa aaaaaaaaHorizontal shift a aaaaaaaaaaaaaa aa aaa aaaaa aa a a aaaaa aaaaaaaaaaa aa aaaa a aaa a aaa aa aaaaa aaa aaaaa aa aaaaaaa aa aaa aaaa aa aa aaa aaaaa a aaaaa aaaaaaaaaaaa aa aa a aaaaaaaa aaaa aaaaaaaaNonrigid transformations a aaaaaaaaaaaaaa aa a aaaaa aaaa aaaaa a aaaaaaaaaaaa aaaaaa aa aaa aaaaa aa aaa aaaaaa

I. Shifting Graphs (Pages 74−75) What you should learn How to use vertical and horizontal shifts to sketch graphs of functions

Let c be a positive real number. Complete the following representations of shifts in the graph of )(xfy = : 1) Vertical shift c units upward: aaaa a aaaa a a a

2) Vertical shift c units downward: aaaa a aaa a a a

3) Horizontal shift c units to the right: aaaa a aa a aa a

4) Horizontal shift c units to the left: aaaa a aaa a aa a

Example 1: Let xxf =)( . 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 . )(xf

aaaa a a a a a a a a A family of functions is . . . a aaa aa aaaaaa aa aaaaaaaaaa

aaaaaaaaa aa aaaaaaaa aaa aaaaaaaaaa aaaaaaa aaaa aaaa aaa

aaaa aaaaa aaa aaa aa aaaaaaaaa aaaaaaaaa aa aaa aaaaaa

What you should learn How to use reflections to sketch graphs of functions

II. Reflecting Graphs (Pages 76−77) A reflection in the x-axis is a type of transformation of the graph

of y = f(x) represented by h(x) = a aaaa . A reflection in

the y-axis is a type of transformation of the graph of y = f(x)

represented by h(x) = aaa aa .



Example 2: Let xxf =)( . Describe the graph of xxg −=)( in terms of . f

aaa aaaaa aa a aa a aaaaaaaaaa aa aaa aaaaa aa a aa aaa aaaaaaa

III. Nonrigid Transformations (Page 78) What you should learn

How to use nonrigid transformations to sketch graphs of functions

A rigid transformation is . . . a aaaaaaaaaaaaaa aa aaaaa aaa

aaaaa aaaaa aa aaa aaaaa aa aaaaaaaaaa

Rigid transformations change only the aaaaaaaa of the

graph in the coordinate plane.

Name three types of rigid transformations:

1) aaaaaaaaaa aaaaaa

2) aaaaaaaa aaaaaa

3) aaaaaaaaaaa

The four types of nonrigid transformations are the . . .

aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaa aaaaaaaaaa aaaaaaaa aaa

aaaaaaaaaa aaaaaaa

For y = f(x) and the real number c,

• A vertical stretch is represented by aaaa a aaaa ,

where a a a .

• A vertical shrink is represented by aaaa a aaaa ,

where a a a a a .

• A horizontal shrink is represented by aaaa a aaaa ,

where a a a .

• A horizontal stretch is represented by aaaa aaaaa ,

where a a a a a .



Section 1.8 • Combinations of Functions: Composite Functions 27 Name______________________________________________

Section 1.8 Combinations of Functions: Composite Functions

Objective: In this lesson you learned how to find arithmetic

combinations and compositions of functions.

What you should learn How to add, subtract, multiply, and divide functions

I. Arithmetic Combinations of Functions (Pages 84−85) Just as two real numbers can be combined with arithmetic

operations, two functions can be combined by the operations of

aaaaaaaaa aaaaaaaaaaaa aaaaaaaaaa aaa aaaaaaaa a

to create new functions. A combined function like this is called

an arithmetic combination of functions.

The domain of an arithmetic combination of functions f and g

consists of . . . aaa aaaa aaaaaaa aaaa aaa aaaaaa aa aaa

aaaaaaa aa a aaa aa aa aaa aaaa aa aaa aaaaaaaa aaaaaaaaaa

aaaaa aa aaa aaaaaaa aaaaaaaaaaa aaaa aaaa a aa

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: =+ ))(( xgf aaaa a aaaa a

2) Difference: =− ))(( xgf aaaa a aaaa a

3) Product: =))(( xfg aaaa a aaaa a

4) Quotient: =⎟⎟⎠

⎞⎜⎜⎝

⎛)(x

gf aaaa a aa aaaa a a a

Example 1: Let and 57)( −= xxf xxg 23)( −= . Find

. )4)(( gf − aa II. Composition of Functions (Pages 86−87) What you should learn

How to find the composition of one function with another function

The composition of the function f with the function g is defined

as =))(( xgf aaaaaaa .



For the composition of the function f with g, the domain of

is . . . )( gf aaa aaa aa aaa a aa aaa aaaaaa aa a aaaa aaaa

aaaa aa aa aaa aaaaaa aa aa

For two functions f and g, to find , . . . ))(( xgf aaaaaaa aaaa

aaaaaaaaaa aa a aa a aaaa aaa aaaaaaaaa aaaaaaaaaa aaaaa

aaaaaaa aaaaa aaaa aaaaaaaaa

Example 2: Let and let . Find

(a) and (b) . 43)( += xxf 12)( 2 −= xxg

))(( xgf ))(( xfg aaa aa a a aaa aaa a aaa a aaa a

To “decompose” a composite function, . . . aaaa aaa aa

aaaaaaa aaaaaaaa aaa aa aaaaaaa aaaaaaaaa

Example 3: Write the function given by 152)( +−= xxh

as a composition of two functions.

aaaaaaa aaa aaaaa aaa aaaaaaaaaaa aa aa aaa aaaaa a aa a a aaa aaa aaaa a a a a a aa aaaa aaaa a aaaaaaaa

III. Applications of Combinations of Functions (Page 88) What you should learn

How to use combinations of functions to model and solve real-life problems

The function represents the sales tax owed on a purchase with a price tag of x dollars and the function

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.

xxf 06.0)( =

xxg 75.0)( =

aa aaaaa a aa aaaaa a aaaaaa



Section 1.9 • Inverse Functions 29 Name______________________________________________

Section 1.9 Inverse Functions Objective: In this lesson you learned how to find inverse functions

graphically and algebraically. I F

t

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T

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a

a

a

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I I

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Important Vocabulary Define each term or concept. Inverse function aaa a aaa a aa aaa aaaaaaaaaa aa aaaaaaa a a aaa aaaaa a aa aaa aaaaaa aa a aaa aaaaaaa a a aaa aaaaa a aa aaa aaaaaa aa aa aaaa a aa aaa aaaaaaa aaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaaaa a aa aaaaaaa aa a aaa

Horizontal Line Test a aaaaaaaa a aaa aa aaaaaaa aa aaa aaaa aa aa aaaaaaaaaa aaaa aaaaaaaaaa aaa aaaaa aa a aa aaaa aaaa aaa aaaaaa

What you should learn How to find inverse functions informally and verify that two functions are inverse functions of each other

. Inverse Functions (Pages 93−94)

or a function f that is defined by a set of ordered pairs, to form

he inverse function of f, . . . aaaaaaaaaaa aaa aaaaa aaa

aaaaa aaaaaaaaaaa aa aaaa aa aaaaa aaaaaaa aaaaaa

or a function f and its inverse f −1, the domain of f is equal to

_______________________, and the range of f is equal to

_______________________.

o verify that two functions, f and g, are inverse functions of

ach other, . . . aaaa aaaaaaa aaa aaaaaaaa aa aaaa aa aaaaa

aaaaaaaaaaa aaa aaaaa aa aaa aaaaaaaa aaaaaaaa a aaa aaaaa a

a aaa aaaaaa aa aaa aaaaa aaaaaaaaa aaaa aaa aaaaaaaaa aaa

aaaaaaa aa aaaa aaaaaa

xample 1: Verify that the functions 32)( −= xxf and

23)( +

=xxg are inverse functions of each other.

What you should learn How to use graphs of functions to determine whether functions have inverse functions

I. The Graph of an Inverse Function (Page 95)

f the point (a, b) lies on the graph of f, then the point

a a a ) must lie on the graph of f −1 and vice versa. The

raph of f −1 is a reflection of the graph of f in the line

a a a .



III. One-to-One Functions (Page 96) What you should learn How to use the Horizontal Line Test to determine if functions are one-to-one

To tell whether a function has an inverse function from its

graph, . . . aaaaaa aaa aaa aaaaaaaaaa aaaa aaaaa aaaa aaa

aaaaa aa aaa aaaa aa aaaaaaaaaa aaaa aaaaaaaaaa aaa aaaaa aa

aaa aaaaaaaa aa aaaa aaaa aaa aaaaaa

A function f is one-to-one if . . . aaaa aaaaa aa aaa aaaaaaaaa

aaaaaaaa aaaaaaaaaaa aa aaaaaaa aaa aaaaa aa aaa aaaaaaaaaaa

aaaaaaaaa

y

-5

-3

-1

1

3

5

-5 -3 -1 1 3 5x

A function f has an inverse function if and only if f is

aaaaaaaaaa .

Example 2: Does the graph of the function at the right have

an inverse function? Explain. aaa aa aaaaaaa aaaa aaa aaaaaaaaaa aaaa aaaaa

What you should learn How to find inverse functions algebraically

IV. Finding Inverse Functions Algebraically (Pages 97−98) To find the inverse of a function f algebraically, . . .

1) aaa aaa aaaaaaaaaa aaaa aaaa aa aaaaaa aaaaaaa a aaa aa

aaaaaaa aaaaaaaaa

2) aa aaa aaaaaaaa aaa aaaaa aaaaaaa aaaa aa aa

3) aaaaaaaaaaa aaa aaaaa aa a aaa aa aaa aaaaa aaa aa

4) aaaaaaa a aa a aaa aa aaa aaa aaaaaaaaaaa

5) aaaaaa aaaa a aaa a aaa aaaaaaa aaaaaaaaa aa aaaa aaaaa aa

aaaaaaa aaaa aaa aaaaaa aa a aa aaaaa aa aaa aaaaa aa a a

aaa aaaaa aa a aa aaaaa aa aaa aaaaaa aa a a aaa

aaa aaaa a a a a aaaaaaa

aa

aa

aa

aa aa

Example 3: Find the inverse (if it exists) of 54)( −= xxf . a aaa a aaaaa a aaaaaa



Section 1.10 • Mathematical Modeling and Variation 31 Name______________________________________________

Section 1.10 Mathematical Modeling and Variation Objective: In this lesson you learned how to write mathematical models

for direct, inverse, and joint variation. I D aaaaa I T

a

a

a

a

T

v

LC

Important Vocabulary Define each term or concept. Directly proportional aa a a aa aaa aaaa aaaaaaa aaaaaaaa aa aaaa a aa aaaaaaaa aaaaaaaaaaaa aa aa aa aa aaaa aa aaaa aaaaaaaa aa aa Sum of square differences aaa aaa aa aaa aaaaaaa aa aaa aaaaaaaaaaa aaaaaaa aaaaaa aaaa aaaaaa aaa aaaaa aaaaaaaLeast squares regression line aaa aaaa aaaaaaa aaaaaa aaaaa aaaa aaa aaaaa aaa aa aaaaaa aaaaaaaaaaaa

. Introduction (Page 103) What you should learn How to use mathematical models to approximate sets of data points

escribe what is meant by “fitting a model to data.”

aaaaaa aaa aaaaa aaa aaaaaaaaa aaaaaaa a aaaaa aa aaaa aaaaa aaaaaa a aaaaaaaaaaaa aaaaaa aaaa aa a aaaaaaaaaa aaaaaaaa aa aaaaaaaa aaaaaaaaa aaaa aaaaaaaa a aaaa aaa aaa aaaaa aaaaa a aaaaa aaaaaaa a aaaaa aa a aaaaa aaaaa aaaaaaa aaaaaa aaaaaa a aaa aaaaaaaaa aaaaaaaa aa aaa aaaaaa aaaaa aa aaa aaaaaa

I. Least Squares Regression and Graphing Utilities (Page 104) What you should learn How to use the regression feature of a graphing utility to find the equation of a least squares regression line

o find the least squares regression line for a set of data, . . .

aaaaaaaaaa aa aaaaaaaa aa aaaaaaaa aaa aaaa aaaaaa aaa

aaaaaa aaa aaaa aaaa aaaaaaa aa aaa aaaaa aa aaaaa aaa aaaa

aaaaa aaaa a aaaaaaaaaa aa aaaaaaaa aaa aaa aaa aaaaaaaaaa aa

aaaaaaaaa aaaaaa aaaaaaaaaa aaaaaaaa

he correlation coefficient r of a set of data gives a measure of

aaa aaaa aaa aaaaa aaaa aaa aaaa . The closer the

alue of | r | is to 1, the better . . . aaa aaaa



Example 1: The numbers (in thousands) of U.S. Air Force female personnel p on active duty for the years 2000 through 2004 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 = 0 corresponding to 2000. Year 2000 2001 2002 2003 2004 p 66.8 67.6 71.5 73.5 73.8

(Source: U.S. Department of Defense)

a a aaaaa a aaaaa III. Direct Variation (Page 105) What you should learn

How to write mathematical models for direct variation

When a variable y is directly proportional to a variable x, the

constant of variation is . . . aaa aaaaaaa aaaaaaaa a aa aaa

aaaaaaaa a a aa . Another

name for the constant of variation is the aaaaaaaa aa

aaaaaaaaaaaaaaa .

In a direct variation model, the y-intercept of the model is

aaaa .

Example 2: If y varies directly as x, and y is 6 when x is 4, find

the value of y when x is 20. a a aa IV. Direct Variation as an nth Power (Page 106) What you should learn

How to write mathematical models for direct variation as an nth power

If y = kxn for some nonzero constant k, then describe the relationship between y and x in two different ways. a aaaaaa aaaaaaaa aa aaa aaa aaaaa aa aa aa a aa aaaaaaaa aaaaaaaaaaaa aa aaa aaa aaaaa aa a Example 3: If y is directly proportional to the third power of x,

and y is 750 when x is 10, find the value of y when x is 8.

a a aaa


Section 1.10 • Mathematical Modeling and Variation 33 Name______________________________________________

IV. Inverse Variation (Page 107) What you should learn How to write mathematical models for inverse variation

If y varies inversely as x, then x and y are related by an equation

of the form a a aaa , where k is some nonzero

constant.

If y varies inversely as x, then another way to describe this

relationship is that y is aaaaaaaaa aaaaaaaaaaaa a to x.

If x and y are related by an equation of the form y = k/xn, then y

aaaaaa aaaaaaaaa aa aaa aaa aaaaa aa a a

or y aa aaaaaaaaa aaaaaaaaaa aa aaa aaa aaaaa aa a .

Example 4: If y varies inversely as x, and y is 4 when x is 16,

find the value of y when x is 10. a a aaa V. Joint Variation (Page 108) What you should learn

How to write mathematical models for joint variation

If z varies jointly as x and y, then z = aaa aaa aaaa aaaaaaaa

a .

Another way to say that z varies jointly as x and y is . . .

aa aaa aaaa a aa aaaaaaa aaaaaaaaaaaa aa a aaa aa

Example 5: If z varies jointly as x and y, and if z = 10 when

x = 4 when y = 15, find the value of z when x = 12 and y = 7.

a a aa



Additional notes

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Documents

Chapter 1 Functions and Their Graphs - Cengagecollege.cengage.com/mathematics/larson/precalculus_limits/1e2/...aaaaaaaaa Example 1: Complete the table. ... aaaaaa aaaaaaaaa aaaaaa