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1.3
Models Section and Applications
Problem Solving
with Linear Equations
Strategy for Solving Word Problems
Step 1: Read the problem carefully. Attempt to state the
problem in your own words and state what the problem is
looking for. Let any variable represent one of the
quantities in the problem.
Step2: If necessary, write expressions for any other
unknown quantities in the problem in terms of x.
Step 3: Write an equation in x that models the verbal
conditions of the problem.
Step 4: Solve the equation and answer the problems
question.
Step 5: Check the solution in the original wording of the
problem, not in the equation obtained from the words.
s with
em
probl
be
ord
ing w ceeds can
fy
Solv
identi n
ex
l to
ord
the w It is helpfu as x, the
.
ity
tricky ller quant
e
a
ent th eight
the sm t to repres ms h
i
Ti
s
add to uantity. If
inche
y
q
larger s Toms by , x, and
r
horte
xce ed
e
ss
Tom i is x+y.
then
ht
s heig
Tim
Example
Spice Drops candy calorie count exceeds Smarties
candy calorie count by 70 calories per serving. If the
sum of one serving of each candy equals 170 calories
find the calorie count of each kind of candy.
Step 1: Represent one of the quantities
Step 2: Represent the other quantity.
Step 3:Write an equation in x that models the conditions.
Step 4: Solve the equation and answer the question.
Step 5: check the proposed solution.
Example
The percentage of women in the labor
force and the percentage of men in the
labor force is illustrated in the graph at
left. The decrease yearly of men in the
labor force is % and the increase in
women in the labor force is %. If there
are presently 70 million men and 60
million women in the labor force, when
will the number of both sexes be
equal?
Graphing Calculator
Solving the previous problem using intersection. y1
be Let the left side of the equation and y2 be the right side of
the equation.
y 2 = 60 + .005 ( 60 ) x
y1=70-.0025 ( 70) x
Example
A woman who was going to retire had $100,000 that she
invested in her local bank. She put some of the money in a
money market account at 3 % and some in a certificate of
deposit at 4%. If the first years interest is $3850, how much
money did she put in each account?
%
Money
market
CD
amount Interest
Example
A local telephone company charges $11 for local
phone service and an additional $ .10 for each long
distance phone call. A second local telephone
company charges $14 for local service and an
additional $ .05 for each long distance phone call.
For how many minutes of long-distance calls will the
costs for the two companies be the same?
Example
In 2002, the median annual income for people with
an advanced college degree was $73,000. This is a
170% increase over the median income in 1982 of
people with an advanced degree. What were people
with an advanced college degree making in 1982?
Solving a Formula
for One of Its Variables
The formula for the perimeter of a rectangle is given below.
Solve for the length of the rectangle.
Example
Solve for r :
C =2r
The formula below describes the amount A, that a principal
of P dollars is worth after t years when invested at a simple
annual interest rate, r. Solve this formula for P.
Example
Solve for h:
A = hb
1
2
1
1+
2
hb 2
If you invest a total of $20,000 in two accounts.
One account pays 5% and another account
pays 3%. If you make $900 in interest the first
year, how much did you invest at each
percent?
(a) $15,500- 5%, $4,500 -3%
(b)$12,000-5%, $8,000-3%
(c)$15,000-5%, $5,000-3%
(d)$10,000-5%, $10,000-3%
Solve for h.
(a)
(b)
(c)
(d)
V
l+w
lw
V
Vlw
V
lw
V = lwh

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UTEM Chile - FKP - bmfp

Section 1.4Complex NumbersThe Imaginary Unit iThe Imaginary Unit iThe imaginary unit i is defined asi = -1, where i 2 = 1.Complex Numbers and Imaginary NumbersThe set of all numbers in the forma+biwith real numbers a and b, and i, the imaginary u

UTEM Chile - FKP - bmfp

Section 1.5Quadratic EquationsDefinition of a Quadratic EquationA quadratic equation in x is an equation that can be writtenin the general formax 2 + bx + x = 0,where a, b, and c are real numbers, with a 0. A quadraticequation in x is also called a

UTEM Chile - FKP - bmfp

Section 1.6Other Types of EquationsPolynomial EquationsA polynomial equation is the result of setting twopolynomials equal to each other. The equation is ingeneral form if one side is 0 and the polynomial onthe other side is in descending powers of

UTEM Chile - FKP - bmfp

Section 1.7Linear InequalitiesandAbsolute Value InequalitiesInterval NotationExampleExpress the interval in set builder notationand graph:( 3, 2][ 0, 4]( , 2 )Intersections andUnions of IntervalsExampleFind the set:( 2,3]U [ 0,4 )Example

UTEM Chile - FKP - bmfp

Section 2.1Functions and GraphsRelationsDomain: cfw_ sitting, walking, aerobics, tennis, running, swimmingRange: cfw_ 80,325,505,720,790Do not list 505 twice.ExampleFind the domain and the range.cfw_ ( 98.6, Felicia ) , ( 98.3,Gabriella ) , ( 99.1

UTEM Chile - FKP - bmfp

Section 2.2More on Functionsand Their GraphsIncreasing andDecreasing FunctionsThe open intervalsdescribing wherefunctions increase,decrease, or are constant,use x-coordinates andnot the y-coordinates.Find where the graph is increasing?Where is

UTEM Chile - FKP - bmfp

Section 2.3Linear Functions and SlopesThe Slope of a LineFind the slope of the line that passesthrough (-2,5) and (3,-1)change in y 5 1 66m===or change in x 2 3 55ExampleFind the slope of the line passing through thepair of points. (5,-2) a

UTEM Chile - FKP - bmfp

Section 2.4More on SlopeParallel and Perpendicular Linesy=2x+7Parallel LinesFind a line parallel to -x+6y=8 and passing through (-2,3).-x+6y=8Solve for y6y=x+8141y= x +Use the slope6361y-3= ( x 2)Substitute into the Point-Slope form61

UTEM Chile - FKP - bmfp

Section 2.5Transformation of FunctionsGraphs of Common FunctionsReciprocal Functiony43f ( x) =21x1x4321123451234Domain: ( -,0 ) ( 0, )Range: ( -,0 ) ( 0, )Decreasing on ( -,0 ) and ( 0, )Odd functionVertical ShiftsVertical S

UTEM Chile - FKP - bmfp

Section 2.6Combinations of Functions:Composite FunctionsThe Domain of a Function3x+5x2 4x 5x2 4x 5 = 0g(x)=( x 5) ( x + 1) = 0( x 5) = 0 ( x + 1) = 0x=5x=-1Domain: ( -,-1) ( 1,5 ) ( 5, )h( x ) = 2 x 52x 5 02x 55x25Domain : , )2Domain

UTEM Chile - FKP - bmfp

Section 2.7Inverse FunctionsInverse FunctionsThe function f is a set of ordered pairs, (x,y), then thechanges produced by f can be undone by reversingcomponents of all the ordered pairs. The resultingrelation (y,x), may or may not be a function. Inv

UTEM Chile - FKP - bmfp

Section 2.8Distance and Midpoint Formulas;CirclesThe Distance Formulay9Find the Distance between (-4,2) and (3,-7)87( x 2 x1 )2+ ( y2 y1 )( 3 4 ) + ( 7 2 )26254322176543211149 + 81130 = 11.423456782345678Exam

UTEM Chile - FKP - bmfp

Section 3.1Quadratic FunctionsGraphs of Quadratic FunctionsGraphs of Quadratic Functions2Parabolas f ( x) = ax + bx + cVertexMinimumMaximumAxis of symmetryy9y9887766554433221987654321112341x23456789109

UTEM Chile - FKP - bmfp

Section 3.2Polynomial Functions andTheir GraphsSmooth, Continuous GraphsPolynomial functions of degree 2 or higher have graphs thatare smooth and continuous. By smooth, we mean that thegraphs contain only rounded curves with no sharp corners.By con

UTEM Chile - FKP - bmfp

Section 3.3Dividing Polynomials;Remainder and Factor TheoremsLong Division of PolynomialsandThe Division AlgorithmLong Division of PolynomialsLong Division of Polynomials3x + 43x 2 9 x + 6 x + 529x 6x12 x + 5213+3x 212 x 813Long Divisio

UTEM Chile - FKP - bmfp

Section 3.4Zeros of Polynomial FunctionsThe Rational Zero TheoremExampleList all possible rational zeros of f(x)=x 3-3x2-4x+12Find one of the zeros of the function using synthetic division, thenfactor the remaining polynomial. What are all of the ze

UTEM Chile - FKP - bmfp

Section 3.5Rational FunctionsandTheir GraphsRational FunctionsRational Functions are quotients of polynomialfunctions. This means that rational functions canp( x)be expressed as f(x)=where p and q areq( x)polynomial functions and q(x) 0. The do

UTEM Chile - FKP - bmfp

Section 3.6Polynomials andRational InequalitiesSolving Polynomial InequalitiesExampleSolve and graph the solution set on a real numberline: x 2 x 1210-50510ExampleSolve and graph the solution set on a real number line:x + x 17 x 153210-5

UTEM Chile - FKP - bmfp

Section 3.7Modeling Using VariationDirect VariationExampleThe volume of a sphere varies directly as thecube of the radius. If the volume of a sphereis 523.6 cubic inches when the radius is 5inches, what is the radius when the volume is33.5 cubic i

UTEM Chile - FKP - bmfp

Section 4.1Exponential FunctionsExampleThe exponential equation f ( x ) = 13.49 ( .967 ) 1 predicts the number of O-ringsxthat are expected to fail at the temperature x o F on the space shuttles. TheO-rings were used to seal the connections between

UTEM Chile - FKP - bmfp

Section 4.2Logarithmic FunctionsThe Definition of LogarithmicFunctionsTo change from logarithmic form to the morefamiliar exponential form, use the pattern;y=log b x means b y = xExampleWrite each equation in the equivalent exponential form.a. 4=

UTEM Chile - FKP - bmfp

Section 4.3Properties of LogarithmsThe Product RuleExampleUse the product rule to expand each logarithmic expression.log 3 (9 5)log (1000x)The Quotient RuleExampleUse the quotient rule to expand each logarithmic expression. 25 log 5 x xlog

UTEM Chile - FKP - bmfp

Section 4.4Exponential and LogarithmicEquationsExponential EquationsExampleSolve for x: 643 x = 322+ xExampleSolve for x: 3x = 21ExampleSolve for x: 3x + 2 = 73 x 1Logarithmic EquationsLogarithmic expressions are defined only for logarithms of

UTEM Chile - FKP - bmfp

Section 4.5Exponential Growth and Decay;Modeling DataExponential Growth and DecayExampleThe equation A=A 0 e kt models growth of a deerpopulation in a small local preserve. If the initial populationis 14 deer and the population grows to 20 in 4 yea

UTEM Chile - FKP - bmfp

Section5.1Angles and Their MeasureAnglesA ray is a part of a line that has only one endpointand extends forever in the opposite direction. An angleis formed by two rays that have a common endpoint. Oneray is called the initial side and the other the

UTEM Chile - FKP - bmfp

Section 5.1Systems of Linear Equations inTwo VariablesSystems of Linear Equationsand Their SolutionsTwo linear equations are called a system of linearequations. A solution to a system of linear equations intwo variables is an ordered pair that sati

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Section 5.2Right Triangle TrigonometryRight Triangle Definitions ofTrigonometric FunctionsTrigonometry values for a given angle arealways the same no matter how large thetriangle isExampleFind the value of each of the six trigonometric functionso

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Section 5.3Trigonometric Functionsof Any AngleTrigonometric FunctionsofAny AngleExampleEvaluate the cosine function and the cotangent functionat the following four quadrantal angles:a. =00 = 0b. = 90 =20c. =1800 = 3d. =270 =20The Signs o

UTEM Chile - FKP - bmfp

Section 5.3Partial FractionsThe Idea Behind PartialFraction DecompositionEach of the two fractions on the right is called a partial fraction. Thesum of these fractions is called the partial fraction decomposition of therational expression on the lef

UTEM Chile - FKP - bmfp

Section 5.4Trigonometric Functionsof Real Numbers;Periodic FunctionsTrigonometric Functions ofReal NumbersExampleUse the figure at right to find the trigonometricfunctions at t.1 3P ,2 2 Domain and Range of Sine andCosine FunctionsEven and O

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Section 5.4Systems of Nonlinear Equatonsin Two VariablesSystems of NonlinearEquations and Their SolutionsA system of two nonlinear equations in two variables, alsocalled a nonlinear system, contains at least one equationthat cannot be expressed in

UTEM Chile - FKP - bmfp

Section 5.5Graphs of Sine and CosineFunctionsThe Graph of y=sin xContinuation of the previous problem showing 3 cyclesGraphing Variations of y=sin xGraphing y=2 Sin xGraphing y= Sin xGraphing y=-2 Sin xGraphing y=3 Sin 2xThe Effect of Horizontal

UTEM Chile - FKP - bmfp

Section 5.5Systems of InequalitiesLinear Inequalities in TwoVariables and Their SolutionsA solution of an inequality in two variables, x and y, is anordered pair of real numbers with the following property:When the x-coordinate is substituted for x

UTEM Chile - FKP - bmfp

Section 5.6Graphs of Other TrigonometricFunctionsThe Graph of y=tan xGraphing Variations of y=tan xTwo Examples of Variation of the y= tan x GraphsExampleGraph the following equation: y= - tan 2xExampleGraph the equation y=tan x - 2The Graph of

UTEM Chile - FKP - bmfp

Section 5.6Linear ProgrammingObjective Functions in LinearProgrammingWe will look at the important application ofsystems of linear inequalities. Such systems arisein linear programming, a method for solvingproblems in which a particular quantity th

UTEM Chile - FKP - bmfp

Section 5.7Inverse Trigonometric FunctionsThe Inverse Sine FunctionA Function and Its InverseExampleFind the exact value of sin 1 0.ExampleFind the exact value of sin 1 1.Example3Find the exact value of sin.21The Inverse Cosine FunctionExam

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Section 5.8Applications of TrigonometricFunctionsSolving Right TrianglesExampleFind the height of the tower given that it is 150 feetfrom the line of sight of the individual who is looking upat the top of the tower at a 60 degree angle.600150 fee

UTEM Chile - FKP - bmfp

Section 6.1Verifying Trigonometric IdentitiesThe Fundamental IdentitiesUsing Fundamental Identities toVerify Other IdentitiesExampleSimplify the given expression: cos x(sec x - cos x)ExampleVerify the identity: sec x - sin x gtan x = cos xExample

UTEM Chile - FKP - bmfp

Section 6.1Matrix Solutions toLinear SystemsSolving Linear SystemsUsing MatricesThis rectangular array of 24 numbers, arranged in rows andcolumns and placed in red brackets, is an example of a matrix.The numbers inside the brackets are called eleme

UTEM Chile - FKP - bmfp

Section 6.2Sum and Difference FormulasThe Cosine of the Difference ofTwo AnglesVerify that cos = sin 2ExampleFind the exact value of cos 1000 cos 550 + sin 1000 sin 550ExampleVerify the identity: cos ( - ) = cos Sum and Difference Formulasfor C

UTEM Chile - FKP - bmfp

Section 6.2Inconsistent and DependentSystems and Their ApplicationsGaussian Elimination toSystems Without UniqueSolutionsPossible Positions for Three PlanesInconsistent SystemsDependent SystemsPossible Positions for Three PlanesLinear systems ca

UTEM Chile - FKP - bmfp

Section 6.3Double-Angle, Power-Reducing,and Half-Angle FormulasDouble-Angle FormulasExample3If sin = and lies in Quadrant II, find the exact5value of each of the following.a. sin 2b. cos 2c. tan 2ExampleVerify the identity: sin 4x = 4 sin x c

UTEM Chile - FKP - bmfp

Section 6.3Matrix Operations and TheirApplicationsNotations for MatricesMatrix NotationWe can represent a matrix in two different ways.1. A capital letter, such as A, B, or C, can denote a matrix.2. A lowercase letter enclosed in brackets, such as

UTEM Chile - FKP - bmfp

Section 6.4Product-to-Sum andSum-to-Product FormulasThe Product-to-Sum FormulasExampleExpress each of the following products as a sum or difference:a. sin 2x cos 3xb. sin 3x sin 2xExampleExpress each of the following as a sum or difference:a. 2

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Section 6.4Multiplicative Inverses of Maticesand Matrix EquationsThe Multiplicative IdentityMatrix1 0The Multiplicative Identity matrix is I= for 2 2 matrices.0 1That means that AI=A and IA=AThe Multiplicative Inverseof a MatrixIf a square ma

UTEM Chile - FKP - bmfp

Section 6.5Trigonometric EquationsTrigonometric Equations andTheir SolutionsEquations Involving A SingleTrigonometric FunctionExampleSolve for x where 0 x<2 :3 sin x = 3 + sin xExampleSolve for x where 0 x < 2 :x2 sin=12ExampleSolve for x

UTEM Chile - FKP - bmfp

Section 6.5Determinants and Cramers RuleThe Determinant of a 2 x 2MatrixExampleEvaluate the determinant of each of the following matices: 2 3a. 5 1 3 2 b. 4 1Solving Systems of LinearEquations in Two VariablesUsing DeterminantsExampleUse C

UTEM Chile - FKP - bmfp

Section 7.1The Law of SinesThe Law of Sines and ItsDerivationDeriving the Law of SinesAn oblique triangle is a triangle that does not contain a right angle.Solving Oblique TrianglesSolving an SAA Triangle Using the Law of SinesSolve the triangle s

UTEM Chile - FKP - bmfp

Section 7.1The EllipseObtaining Conic Sections byIntersecting a Plane and a ConeDefinition of an EllipseStandard Form of the Equationof an Ellipse4The relationship of a, b and c in an Ellipse3Used to Find the Foci, c.(0,b)2a1b432xc11

UTEM Chile - FKP - bmfp

Section 7.2The Law of CosinesThe Law of Cosines and ItsDerivationDeriving the Law of CosinesSolving Oblique TrianglesExampleFind the unknown side, and twounknown angles.b = 8 inches1200c = 10 inchesExampleb = 12 inchesc = 4 inchesFind all t

UTEM Chile - FKP - bmfp

Section 7.2The HyperbolaDefinition of a HyperbolaStandard Form of the Equationof a Hyperbolay6The Relationship of a, b, c in a Hyperbola5432bc1x6(-c,0)5432111a23(c,0)4567234Using the Pythagorean theorem you can see that

UTEM Chile - FKP - bmfp

Section 7.3Polar CoordinatesPlotting Points in the PolarCoordinate SystemExamplePlot the ponnts with the following polar coordinates:a. (4,30 )y 3 b. 3, 4c. ( 2,1200 )x12345Multiple Representations ofPoints in the Polar CoordinateSyst

UTEM Chile - FKP - bmfp

Section 7.3The ParabolaDefinition of a ParabolaExampleFind the vertex and the axis of symmetry of the parabola givenby y = 2( x 3) 2 4. Does it open up or down?Standard Form of the Equationof a ParabolaExampleFind the focus and the directrix of t

UTEM Chile - FKP - bmfp

Section 7.4Graphs of Polar EquationsUsing Polar Grids to GraphPolar EquationsA polar equation is an equation whose variables are r and . Thegraph of a polar equation is the set of all points whose polarcoordinates satisfy the equation. We use polar

UTEM Chile - FKP - bmfp

Section 7.5Complex Numbers in Polar Form:DeMoives TheoremThe Complex PlaneExampleDetermine the absolute value of each of the following:a. z=2+4ib. z=1-6ic. z=7iPolar Form of aComplex NumberExamplePlot z=3-2i in the complex plane. Then write z

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Section 7.6VectorsDirected Line SegmentsandGeometric VectorsExampleShow that u=v.y4(2,3)32u1x43211(2, 1)1v23(4, 4)42345Vectors in the RectangularCoordinate SystemExampleSketch the vector v=-2i+j andfind its magnitude.y

UTEM Chile - FKP - bmfp

Section 7.7The Dot ProductThe Dot Product of Two VectorsExampleIf v=4i-2j and u=-2i+3ja. find vgub. find vgvThe Angle betweenTwo VectorsExampleFind the angle between v=2i-2j and w=-3i+5j.Round to the nearest tenth of a degree.5y4321543

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Section 8.1Sequences andSummation NotationSequencesThe graph of a sequence is a set of discrete points. The graph11of the sequence a n = is similar to f(x)= except it only containsnxthe points whose x-coordinates are positive integers.Technolog

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Section 8.2Arithmetic SequencesArithmetic SequencesThe graph of each arithmetic sequence forms a set of discrete pointslying on a straight line. An arithmetic sequence is a linear functionwhose domain is the set of positive integers.If the first ter

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Section 8.3Geometric Sequences and SeriesGeometric SequencesExampleIn a geometric sequence the first term is 5 and thesecond term is 10, what is the common ratio? To finda2the common ratio r = .a1ExampleIf in a geometric sequence the first term

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Section 8.4Mathematical InductionThe Principle of MathematicalInductionVisualizing Summation FormulasExampleFor the given statement Sn , write the three statementsS1, S k , S K +1.n(2n 1)(2n + 1)Sn : 1 + 3 + 5 + 7 + . + (2n 1) =322222Provi