##### Document Text Contents

Page 1

College Algebra

Scottsdale Community College

First Edition, OER | 2012

Page 2

College Algebra

An Investigation of Functions

Custom Print for Scottsdale Community College

1st Edition

David Lippman

Melonie Rasmussen

These books are also available to read free online at

http://www.opentextbookstore.com/precalc/ and http://www.stitz-

zeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus.html

If you want a printed copy, buying from the lulu.com is cheaper than printing yourself.

http://www.opentextbookstore.com/precalc/

http://www.stitz-zeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus.html

http://www.stitz-zeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus.html

Page 275

Section 4.5 Graphs of Logarithmic Functions

265

Example 3

Sketch a graph of )2ln()( xxf

This is a horizontal shift to the left by 2 units. Notice that none of our logarithm rules

allow us rewrite this in another form, so the effect of this transformation is unique.

Shifting the graph,

Notice that due to the horizontal shift, the vertical asymptote shifted as well, to x = -2

Combining these transformations,

Example 4

Sketch a graph of )2log(5)( xxf

Factoring the inside as ))2(log(5)( xxf reveals that this graph is that of the

common logarithm, horizontally reflected, vertically stretched by a factor of 5, and

shifted to the right by 2 units.

The vertical asymptote will have been

shifted to x = 2, and the graph will be

defined for x < 2. A rough sketch can be

created by using the vertical asymptote

along with a couple points on the graph,

such as

5)10log(5)2)8(log(5)8(

0)1log(5)21log(5)1(

f

f

Try it Now

2. Sketch a graph of the function 1)2log(3)( xxf

275

Page 276

Chapter 4

266

Example 5

Find an equation for the logarithmic function graphed below

This graph has a vertical asymptote at x = -2 and has been vertically reflected. We do

not know yet the vertical shift (equivalent to horizontal stretch) or the vertical stretch

(equivalent to a change of base). We know so far that the equation will have form

kxaxf )2log()(

It appears the graph passes through the points (-1,1) and (2,-1). Substituting in (-1,1),

k

ka

ka

1

)1log(1

)21log(1

Next substituting in (2,-1),

)4log(

2

)4log(2

1)22log(1

a

a

a

This gives us the equation 1)2log(

)4log(

2

)( xxf

Flashback

3. Using the graph above write the Domain & Range and describe the long run

behavior.

276

Page 549

Index 1079

inconsistent, 553

independent, 554

leading variable, 556

linear

n variables, 554

two variables, 550

linear in form, 646

non-linear, 637

overdetermined, 554

parametric solution, 552

triangular form, 556

underdetermined, 554

unknowns matrix, 590

tangent

graph of, 804

of an angle, 744, 752

properties of, 806

terminal side of an angle, 698

Thurstone, Louis Leon, 315

total squared error, 225

transformation

non-rigid, 129

rigid, 129

transformations of function graphs, 120, 135

transverse axis of a hyperbola, 531

Triangle Inequality, 183

triangular form, 556

underdetermined system, 554

uninhibited growth, 472

union of two sets, 4

Unit Circle

definition of, 501

important points, 724

unit vector, 1021

Upper and Lower Bounds Theorem, 274

upper triangular matrix, 593

variable

dependent, 55

independent, 55

variable cost, 159

variation

constant of proportionality, 350

direct, 350

inverse, 350

joint, 350

variations in sign, 273

vector

x-component, 1010

y-component, 1010

addition

associative property, 1013

commutative property, 1013

definition of, 1012

properties of, 1013

additive identity, 1013

additive inverse, 1013, 1016

angle between two, 1033, 1034

component form, 1010

Decomposition Theorem

Generalized, 1038

Principal, 1022

definition of, 1010

direction

definition of, 1018

properties of, 1018

dot product

commutative property of, 1032

definition of, 1032

distributive property of, 1032

geometric interpretation, 1033

properties of, 1032

relation to magnitude, 1032

relation to orthogonality, 1035

work, 1040

head, 1010

initial point, 1010

magnitude

definition of, 1018

properties of, 1018

relation to dot product, 1032

normalization, 1022

orthogonal projection, 1036

549

Page 550

1080 Index

orthogonal vectors, 1035

parallel, 1028

principal unit vectors, ı̂, ̂, 1022

resultant, 1011

scalar multiplication

associative property of, 1016

definition of, 1015

distributive properties, 1016

identity for, 1016

properties of, 1016

zero product property, 1016

scalar product

definition of, 1032

properties of, 1032

scalar projection, 1037

standard position, 1017

tail, 1010

terminal point, 1010

triangle inequality, 1042

unit vector, 1021

velocity

average angular, 707

instantaneous, 707

instantaneous angular, 707

vertex

of a hyperbola, 531

of a parabola, 188, 505

of an angle, 693

of an ellipse, 516

vertical asymptote

formal definition of, 304

intuitive definition of, 304

location of, 306

vertical line, 23

Vertical Line Test (VLT), 43

whole number

definition of, 2

set of, 2

work, 1039

wrapping function, 704

zero

multiplicity of, 244

of a function, 95

upper and lower bounds, 274

550

College Algebra

Scottsdale Community College

First Edition, OER | 2012

Page 2

College Algebra

An Investigation of Functions

Custom Print for Scottsdale Community College

1st Edition

David Lippman

Melonie Rasmussen

These books are also available to read free online at

http://www.opentextbookstore.com/precalc/ and http://www.stitz-

zeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus.html

If you want a printed copy, buying from the lulu.com is cheaper than printing yourself.

http://www.opentextbookstore.com/precalc/

http://www.stitz-zeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus.html

http://www.stitz-zeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus.html

Page 275

Section 4.5 Graphs of Logarithmic Functions

265

Example 3

Sketch a graph of )2ln()( xxf

This is a horizontal shift to the left by 2 units. Notice that none of our logarithm rules

allow us rewrite this in another form, so the effect of this transformation is unique.

Shifting the graph,

Notice that due to the horizontal shift, the vertical asymptote shifted as well, to x = -2

Combining these transformations,

Example 4

Sketch a graph of )2log(5)( xxf

Factoring the inside as ))2(log(5)( xxf reveals that this graph is that of the

common logarithm, horizontally reflected, vertically stretched by a factor of 5, and

shifted to the right by 2 units.

The vertical asymptote will have been

shifted to x = 2, and the graph will be

defined for x < 2. A rough sketch can be

created by using the vertical asymptote

along with a couple points on the graph,

such as

5)10log(5)2)8(log(5)8(

0)1log(5)21log(5)1(

f

f

Try it Now

2. Sketch a graph of the function 1)2log(3)( xxf

275

Page 276

Chapter 4

266

Example 5

Find an equation for the logarithmic function graphed below

This graph has a vertical asymptote at x = -2 and has been vertically reflected. We do

not know yet the vertical shift (equivalent to horizontal stretch) or the vertical stretch

(equivalent to a change of base). We know so far that the equation will have form

kxaxf )2log()(

It appears the graph passes through the points (-1,1) and (2,-1). Substituting in (-1,1),

k

ka

ka

1

)1log(1

)21log(1

Next substituting in (2,-1),

)4log(

2

)4log(2

1)22log(1

a

a

a

This gives us the equation 1)2log(

)4log(

2

)( xxf

Flashback

3. Using the graph above write the Domain & Range and describe the long run

behavior.

276

Page 549

Index 1079

inconsistent, 553

independent, 554

leading variable, 556

linear

n variables, 554

two variables, 550

linear in form, 646

non-linear, 637

overdetermined, 554

parametric solution, 552

triangular form, 556

underdetermined, 554

unknowns matrix, 590

tangent

graph of, 804

of an angle, 744, 752

properties of, 806

terminal side of an angle, 698

Thurstone, Louis Leon, 315

total squared error, 225

transformation

non-rigid, 129

rigid, 129

transformations of function graphs, 120, 135

transverse axis of a hyperbola, 531

Triangle Inequality, 183

triangular form, 556

underdetermined system, 554

uninhibited growth, 472

union of two sets, 4

Unit Circle

definition of, 501

important points, 724

unit vector, 1021

Upper and Lower Bounds Theorem, 274

upper triangular matrix, 593

variable

dependent, 55

independent, 55

variable cost, 159

variation

constant of proportionality, 350

direct, 350

inverse, 350

joint, 350

variations in sign, 273

vector

x-component, 1010

y-component, 1010

addition

associative property, 1013

commutative property, 1013

definition of, 1012

properties of, 1013

additive identity, 1013

additive inverse, 1013, 1016

angle between two, 1033, 1034

component form, 1010

Decomposition Theorem

Generalized, 1038

Principal, 1022

definition of, 1010

direction

definition of, 1018

properties of, 1018

dot product

commutative property of, 1032

definition of, 1032

distributive property of, 1032

geometric interpretation, 1033

properties of, 1032

relation to magnitude, 1032

relation to orthogonality, 1035

work, 1040

head, 1010

initial point, 1010

magnitude

definition of, 1018

properties of, 1018

relation to dot product, 1032

normalization, 1022

orthogonal projection, 1036

549

Page 550

1080 Index

orthogonal vectors, 1035

parallel, 1028

principal unit vectors, ı̂, ̂, 1022

resultant, 1011

scalar multiplication

associative property of, 1016

definition of, 1015

distributive properties, 1016

identity for, 1016

properties of, 1016

zero product property, 1016

scalar product

definition of, 1032

properties of, 1032

scalar projection, 1037

standard position, 1017

tail, 1010

terminal point, 1010

triangle inequality, 1042

unit vector, 1021

velocity

average angular, 707

instantaneous, 707

instantaneous angular, 707

vertex

of a hyperbola, 531

of a parabola, 188, 505

of an angle, 693

of an ellipse, 516

vertical asymptote

formal definition of, 304

intuitive definition of, 304

location of, 306

vertical line, 23

Vertical Line Test (VLT), 43

whole number

definition of, 2

set of, 2

work, 1039

wrapping function, 704

zero

multiplicity of, 244

of a function, 95

upper and lower bounds, 274

550