Title College Algebra SCC SP12 w Cover Variable (Mathematics) Function (Mathematics) Quadratic Equation Equations Formula 14.5 MB 550
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College_Algebra_SCC_SP12_noTOC.pdf
Algebra_Lippman_scc.pdf
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Index
Supplementary Materials
part2
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##### 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
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

associative property, 1013

commutative property, 1013

definition of, 1012

properties of, 1013

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

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