alg complete practice

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COLLEGE ALGEBRAPractice Problems

Paul Dawkins


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College Algebra

Table of Contents

Preface ........................................................................................................................................... iii

Outline ........................................................................................................................................... iv

Preliminaries .................................................................................................................................. 6

Introduction ................................................................................................................................................ 6

Integer Exponents ...................................................................................................................................... 6

Rational Exponents .................................................................................................................................... 7

Real Exponents .......................................................................................................................................... 8Radicals ...................................................................................................................................................... 9Polynomials ...............................................................................................................................................10Factoring Polynomials ..............................................................................................................................10Rational Expressions .................................................................................................................................12Complex Numbers ....................................................................................................................................13

Solving Equations and Inequalities ............................................................................................ 13Introduction ...............................................................................................................................................13

Solutions and Solution Sets .......................................................................................................................14

Linear Equations .......................................................................................................................................15

Application of Linear Equations ...............................................................................................................15Equations With More Than One Variable .................................................................................................16

Quadratic Equations Part I .....................................................................................................................16Quadratic Equations Part II ....................................................................................................................17Solving Quadratic Equations : A Summary ..............................................................................................18Application of Quadratic Equations ..........................................................................................................18Equations Reducible to Quadratic Form ...................................................................................................19Equations with Radicals ............................................................................................................................19Linear Inequalities.....................................................................................................................................20Polynomial Inequalities .............................................................................................................................20

Rational Inequalities .................................................................................................................................20

Absolute Value Equations .........................................................................................................................21

Absolute Value Inequalities ......................................................................................................................22

Graphing and Functions ............................................................................................................. 22Introduction ...............................................................................................................................................22

Graphing ...................................................................................................................................................23Lines ..........................................................................................................................................................24Circles .......................................................................................................................................................24The Definition of a Function .....................................................................................................................25Graphing Functions ...................................................................................................................................27

Combining Functions ................................................................................................................................27

Inverse Functions ......................................................................................................................................28

Common Graphs ......................................................................................................................... 28Introduction ...............................................................................................................................................28Lines, Circles and Piecewise Functions ....................................................................................................29Parabolas ...................................................................................................................................................29

Ellipses ......................................................................................................................................................30

Hyperbolas ................................................................................................................................................30

Miscellaneous Functions ...........................................................................................................................31Transformations ........................................................................................................................................31Symmetry ..................................................................................................................................................32

Rational Functions ....................................................................................................................................32

Polynomial Functions .................................................................................................................. 32Introduction ...............................................................................................................................................32Dividing Polynomials ...............................................................................................................................33Zeroes/Roots of Polynomials ....................................................................................................................34

2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx


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College Algebra

Graphing Polynomials ...............................................................................................................................34Finding Zeroes of Polynomials .................................................................................................................34Partial Fractions ........................................................................................................................................35

Exponential and Logarithm Functions ...................................................................................... 36Introduction ...............................................................................................................................................36

Exponential Functions ...............................................................................................................................36

Logarithm Functions .................................................................................................................................37

Solving Exponential Equations .................................................................................................................38

Solving Logarithm Equations ...................................................................................................................39Applications ..............................................................................................................................................39

Systems of Equations ................................................................................................................... 40Introduction ...............................................................................................................................................40Linear Systems with Two Variables .........................................................................................................41

Linear Systems with Three Variables .......................................................................................................42

Augmented Matrices .................................................................................................................................42

More on the Augmented Matrix ................................................................................................................43

Non-Linear Systems ..................................................................................................................................44

2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx


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College Algebra

Preface

Here are a set of practice problems for my Algebra notes. If you are viewing the pdf version ofthis document (as opposed to viewing it on the web) this document contains only the problemsthemselves and no solutions are included in this document. Solutions can be found in a numberof places on the site.

1.

If youd like a pdf document containing the solutions go to the note page for the sectionyoud like solutions for and select the download solutions link from there. Or,

2.

Go to the download page for the site http://tutorial.math.lamar.edu/download.aspxandselect the section youd like solutions for and a link will be provided there.

3.

If youd like to view the solutions on the web or solutions to an individual problem you

can go to the problem set web page, select the problem you want the solution for. At thispoint I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printableversion.

Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.

2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx


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College Algebra

Outline

Here is a list of sections for which problems have been written.

PreliminariesInteger ExponentsRational ExponentsReal ExponentsRadicalsPolynomialsFactoring PolynomialsRational ExpressionsComplex Numbers

Solving Equations and InequalitiesSolutions and Solution SetsLinear EquationsApplications of Linear EquationsEquations With More Than One VariableQuadratic Equations, Part IQuadratic Equations, Part IIQuadratic Equations : A SummaryApplications of Quadratic EquationsEquations Reducible to Quadratic FormEquations with RadicalsLinear Inequalities

Polynomial InequalitiesRational InequalitiesAbsolute Value EquationsAbsolute Value Inequalities

Graphing and FunctionsGraphingLinesCirclesThe Definition of a FunctionGraphing FunctionsCombining functions

Inverse Functions

Common GraphsLines, Circles and Piecewise FunctionsParabolasEllipsesHyperbolasMiscellaneous FunctionsTransformations

2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx


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College Algebra

SymmetryRational Functions

Polynomial FunctionsDividing PolynomialsZeroes/Roots of Polynomials

Graphing PolynomialsFinding Zeroes of PolynomialsPartial Fractions

Exponential and Logarithm FunctionsExponential FunctionsLogarithm FunctionsSolving Exponential EquationsSolving Logarithm EquationsApplications

Systems of Equations

Linear Systems with Two VariablesLinear Systems with Three VariablesAugmented MatricesMore on the Augmented MatrixNonlinear Systems

2007 Paul Dawkins v http://tutorial.math.lamar.edu/terms.aspx


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College Algebra

Preliminaries

Introduction

Here are a set of practice problems for the Preliminaries chapter of my Algebra notes. If you areviewing the pdf version of this document (as opposed to viewing it on the web) this documentcontains only the problems themselves and no solutions are included in this document. Solutionscan be found in a number of places on the site.

4.


5.


6.

If youd like to view the solutions on the web or solutions to an individual problem youcan go to the problem set web page, select the problem you want the solution for. At thispoint I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printableversion.


Here is a list of topics in this chapter that have practice problems written for them.

Integer ExponentsRational ExponentsReal ExponentsRadicalsPolynomialsFactoring PolynomialsRational ExpressionsComplex Numbers

Integer Exponents

For problems 1 4 evaluate the given expression and write the answer as a single number with noexponents.

1. 2 26 4 3 +

2007 Paul Dawkins 6 http://tutorial.math.lamar.edu/terms.aspx


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College Algebra

2.( )

( )

4

22 2

2

3 2

+

3.

0 2

1 2

4 2

3 4

4.1 12 4 +

For problems 5 9 simplify the given expression and write the answer with only positiveexponents.

5. ( ) 2

4 52w v

6.

4 1

6 3

2x y

x y

7.2 10

7 3

m n

m n

8.( )( )

32 4

1 7

2

6

p q

q p

9.

42 1 3

8 6 4

z y x

x z y

Rational Exponents

For problems 1 6 evaluate the given expression and write the answer as a single number with noexponents.

1.

1

236

2. ( )1

3125

3.

3

216

4.

5

327



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College Algebra

5.

1

29

4

6.

2

38

343


7.

21 3

3 4a b

8.

11

54x x

9.

31 7

3 2

1

3

q p

q p

10.

111 632

2 7

3 4

m n

n m

Real Exponents


1. ( ) 2.4

0.1 0.3x y

2. ( ) ( )3 1.8

0.15 4x y

3.

1.53.2 0.7

6.4 1.9

p q

q p



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College Algebra

Radicals

For problems 1 3 write the expression in exponential form.

1. 7 y

2. 3 2x

3. 6 ab

4.2 3w v

For problems 5 7 evaluate the radical.

5. 4 81

6. 3 512

7. 3 1000

For problems 8 12 simplify each of the following. Assume thatx, yand zare all positive.

8. 3 8x

9. 38y

10. 7 20 114 x y z

11.6 7 23 54x y z

12.3 2 3 54 44 8x y x y z

For problems 13 15 multiply each of the following. Assume thatxis positive.

13. ( )4 3x x

14. ( )( )2 1 3 4x x+

15. ( )( )3 32 23 2 4x x x+

For problems 16 19 rationalize the denominator. Assume thatxandyare both positive.



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College Algebra

16.6

x

17.3

9

2x

18.4

2x y+

19.10

3 5 x

Polynomials

For problems 1 10 perform the indicated operation and identify the degree of the result.

1. Add3 24 2 1x x + to 27 12x x+

2. Subtract 6 24 3 2z z z + from 6 210 7 8z z +

3. Subtract 23 7 8x x + + from 4 37 12 1x x x+

4. ( )4 212 3 7 1y y y +

5. ( ) ( )2

3 1 2 9x x+

6. ( )( )2 22 3w w w+ +

7. ( )( )6 64 3 4 3x x x x +

8. ( )2

33 10 4y

9. ( ) ( )2 22 3 8 7x x x x+

10. Subtract ( )2

23 1x + from 3 26 9 13 4x x x

Factoring Polynomials

For problems 1 4 factor out the greatest common factor from each polynomial.



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College Algebra

1. 7 4 36 3 9x x x+

2.3 8 10 4 5 27 2a b a b a b +

3. ( ) ( )3 5

2 22 1 16 1x x x+ +

4. ( ) ( )2 2 6 4 4 12x x x x +

For problems 5 & 6 factor each of the following by grouping.

5. 3 4 67 7x x x x+ + +

6.4 318 33 6 11x x x+

For problems 7 15 factor each of the following.

7.2 2 8x x

8. 2 10 21z z +

9.2 16 60y y+ +

10.25 14 3x x+

11. 26 19 7t t

12.24 19 12z z+ +

13. 2 14 49x x+ +

14. 24 25w

15.281 36 4x x +

For problems 16 18 factor each of the following.

16.6 3

3 4x x+

17.5 4 33 17 28z z z

18. 14 62 512x x



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College Algebra

Rational Expressions

For problems 1 3 reduce each of the following to lowest terms.

1.

2

2

6 7

10 21

x x

x x

+

2.2

2

6 9

9

x x

x

+ +

3.2

2

2 28

20

x x

x x

For problems 4 7 perform the indicated operation and reduce the answer to lowest terms.

4.2 2

2 2

5 24 4 4

6 8 3

x x x x

x x x x

+ + +

+ +

5.2 2

2 2

49 42

2 3 5 7 6

x x x

x x x x

+ +

6.2 2

2 2

2 8 9 20

2 8 24 11 30

x x x x

x x x x

+

+

7.

2

3

1

411 10

x

xx x

+

++ +

For problems 8 12 perform the indicated operations.

8.3

4 2 7

x

x x+

+

9.2 4

2 1 2

3 9 4x x x +

+

10.2

8

12 36 6

x x

x x x

+ + +

11.2

2

1 1

13 42 6 7

x x

x x x x

++

+



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College Algebra

12.( ) ( )

3 2

10

3 8 3 8

x x

x x

++

+ +

Complex Numbers

Perform the indicated operation and write your answer in standard form.

1. ( )( )4 5 12 11i i +

2. ( ) ( )3 6 7i i

3. ( ) ( )1 4 16 9i i+ +

4. ( )8 10 2i i+

5. ( ) ( )3 9 1 10i i +

6. ( )( )2 7 8 3i i+ +

7.7

2 10

i

i

+

8.1 5

3

i

i

+

9.6 7

8

i

i

+

Solving Equations and Inequalities

Introduction

Here are a set of practice problems for the Solving Equations and Inequalities chapter of myAlgebra notes. If you are viewing the pdf version of this document (as opposed to viewing it onthe web) this document contains only the problems themselves and no solutions are included inthis document. Solutions can be found in a number of places on the site.



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College Algebra

7.


8.


9.




Solutions and Solution SetsLinear EquationsApplications of Linear EquationsEquations With More Than One VariableQuadratic Equations, Part IQuadratic Equations, Part IIQuadratic Equations : A SummaryApplications of Quadratic EquationsEquations Reducible to Quadratic FormEquations with RadicalsLinear InequalitiesPolynomial Inequalities

Rational InequalitiesAbsolute Value EquationsAbsolute Value Inequalities

Solutions and Solution Sets

For each of the following determine if the given number is a solution to the given equation orinequality.

1. Is 6x= a solution to ( )2 5 3 1 22x x = + ?

2. Is 7t= a solution to 2 3 10 4 8t t t+ = + ?

3. Is 3t= a solution to 2 3 10 4 8t t t+ = + ?

4. Is 2w = a solution to2 8 12

02

w w

w

+ +=

+?



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College Algebra

5. Is 4z= a solution to 2 26 3z z z + ?

6. Is 0y= a solution to ( ) ( ) ( )2 7 1 4 1 3 4 10y y y+ < + + + ?

7. Is 1x= a solution to ( )21 3 1x x+ > + ?

Linear Equations

Solve each of the following equations and check your answer.

1. ( )4 7 2 3 2x x x = +

2. ( ) ( )2 3 10 6 32 3w w+ =

3.4 2 3 5

3 4 6

z z=

4.2

4 1

25 5

t

t t=

5.3 4 7

21 1

y

y y

+= +

6. 5 6 53 3 2 3

xx x

+ = +

Application of Linear Equations

1. A widget is being sold in a store for $135.40 and has been marked up 7%. How much did thestore pay for the widget?

2. A store is having a 30% off sale and one item is now being sold for $9.95. What was the

original price of the item?

3. Two planes start out 2800 km apart and move towards each other meeting after 3.5 hours. Oneplane flies at 75 km/hour slower than the other plane. What was the speed of each plane?

4. Mike starts out 35 feet in front of Kim and they both start moving towards the right at the sametime. Mike moves at 2 ft/sec while Kim moves at 3.4 ft/sec. How long will it take for Kim tocatch up with Mike?



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College Algebra

5. A pump can empty a pool in 7 hours and a different pump can empty the same pool in 12hours. How long does it take for both pumps working together to empty the pool?

6. John can paint a house in 28 hours. John and Dave can paint the house in 17 hours workingtogether. How long would it take Dave to paint the house by himself?

7. How much of a 20% acid solution should we add to 20 gallons of a 42% acid solution to get a35% acid solution?

8. We need 100 liters of a 25% saline solution and we only have a 14% solution and a 60%solution. How much of each should we mix together to get the 100 liters of the 25% solution?

9. We want to fence in a field whose length is twice the width and we have 80 feet of fencingmaterial. If we use all the fencing material what would the dimensions of the field be?

Equations With More Than One Variable

1. Solve2

3 4E vr

=

for r.

2. Solve ( )6

4 17

hQ h

s= + for s.

3. Solve ( )6

4 17

hQ h

s= + for h.

4. Solve1 2 4 3

4 5

t tA

p p + = for t.

5. Solve10

3 7y

x=

for x.

6. Solve3

12 9

xy

x

+=

for x.

Quadratic Equations Part I

For problems 1 7 solve the quadratic equation by factoring.

1.2 5 14 0u u =

2. 2 15 50x x+ =



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College Algebra

3.2 11 28y y=

4.219 7 6x x=

5. 26 5w w =

6. 2 16 61 2 20z z z + =

7.212 25x x=

For problems 8 & 9 use factoring to solve the equation.

8. 4 3 22 3 0x x x =

9.5 39t t=

For problems 10 12 use factoring to solve the equation.

10.2 10

4 32

ww w

w

+ =

+

11.2

4 5 6 5

1

z z

z z z z

++ =

+ +

12.2 7 5 8

15 5

x xx

x x

++ =

+ +

For problems 13 16 use the Square Root Property to solve the equation.

13.29 16 0u =

14. 2 15 0x + =

15. ( )2

2 36 0z =

16. ( )2

6 1 3 0t+ + =

Quadratic Equations Part II

For problems 1 3 complete the square.

1. 2 8x x+



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College Algebra

2. 2 11u u

3.22 12z z

For problems 4 8 solve the quadratic equation by completing the square.

4.2 10 34 0t t + =

5. 2 8 9 0v v+ =

6. 2 9 16 0x x+ + =

7. 24 8 5 0u u + =

8. 22 5 3 0x x+ + =

For problems 9 13 use the quadratic formula to solve the quadratic equation.

9. 2 6 4 0x x + =

10. 29 6 101w w =

11.28 5 70 5 7u u u+ + =

12. 2169 20 4 0t t + =

13.2 22 72 2 58z z z z+ = +

Solving Quadratic Equations : A Summary

For problems 1 4 use the discriminant to determine the type of roots for the equation. Do notfind any roots.

1.2169 182 49 0x x + =

2. 2 28 61 0x x+ + =

3. 249 126 102 0x x + =

4.29 151 0x + =

Application of Quadratic Equations



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College Algebra

1. The width of a rectangle is 1 m less than twice the length. If the area of the rectangle is 100 m2what are the dimensions of the rectangle?

2. Two cars start out at the same spot. One car starts to drive north at 40 mph and 3 hours laterthe second car starts driving to the east at 60 mph. How long after the first car starts driving doesit take for the two cars to be 500 miles apart?

3. Two people can paint a house in 14 hours. Working individually one of the people takes 2hours more than it takes the other person to paint the house. How long would it take each personworking individually to paint the house?

Equations Reducible to Quadratic Form

Solve each of the following equations.

1.6 39 8 0x x + =

2. 4 27 18 0x x =

3.

2 1

3 34 21 27 0x x+ + =

4.8 46 7 0x x + =

5.2

2 1721 0

x x+ + =

6. 1 11 18 0x x

+ =

Equations with Radicals


1. 2 3x x= +

2.33 2 1

x x = +

3. 7 39 3x x= +

4. 1 2 2x x= +

5. 1 1 2 4x x+ = +



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College Algebra

Linear Inequalities

For problems 1 6 solve each of the following inequalities. Give the solution in both inequalityand interval notations.

1. ( ) ( )4 2 1 5 7 4z z+ >

2. ( ) ( )1 1 1 1

3 4 6 2 102 3 2 4

t t t

+ +

3. 1 4 2 10x < + <

4. 8 3 5 12z <

5. 0 10 15 23w

6.1 1

2 46 2

x<

7. If 0 3x < determine aand bfor the inequality : 4 1a x b + <

Polynomial Inequalities

Solve each of the following inequalities.

1. 2 4 21u u+

2.2 8 12 0x x+ + <

3. 24 15 17t t

4. 2 34 12z z+ >

5.2 2 1 0y y +

6.4 3 2

12 0t t t+ <

Rational Inequalities




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College Algebra

1.4

03

x

x

>

+

2.2 5

07

z

z

3.2 5 6

03

w w

w

+

4.3 8

21

x

x

+<

5.4

3u

u

6.

3 2

6 02

t tt >

Absolute Value Equations

For problems 1 5 solve each of the equation.

1. 4 7 3p =

2. 2 4 1x =

3. 6 1 3u u= +

4. 2 3 4x x =

5.1

4 4 62

z z+ =

For problems 6 & 7 find all the real valued solutions to the equation.

6. 2 2 15x x+ =

7. 2 4 1x + =



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College Algebra

Absolute Value Inequalities


1. 4 9 3t+ <

2. 6 5 10x

3. 12 1 9x +

4. 2 1 1w <

5. 2 7 1z >

6. 10 3 4w

7. 4 3 7z >

Graphing and Functions

Introduction

Here are a set of practice problems for the Graphing and Functions chapter of my Algebra notes.If you are viewing the pdf version of this document (as opposed to viewing it on the web) thisdocument contains only the problems themselves and no solutions are included in this document.Solutions can be found in a number of places on the site.

10.


11.


12.

If youd like to view the solutions on the web or solutions to an individual problem you

can go to the problem set web page, select the problem you want the solution for. At thispoint I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printableversion.




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College Algebra


GraphingLinesCircles

The Definition of a FunctionGraphing FunctionsCombining functionsInverse Functions.

Graphing

For problems 1 3 construct a table of at least 4 ordered pairs of points on the graph of theequation and use the ordered pairs from the table to sketch the graph of the equation.

1. 3 4y x= +

2. 21y x=

3. 2y x= +

For problems 4 9 determine the x-intercepts and y-intercepts for the equation. Do not sketch thegraph.

4. 3 7 10x y =

5.26y x=

6.

2

6 7y x x= +

7.2 10y x= +

8.2 6 58y x x= + +

9. ( )2

3 8y x= +



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College Algebra

Lines

For problems 1 & 2 determine the slope of the line containing the two points and sketch the graphof the line.

1. ( ) ( )2, 4 , 1,10

2. ( ) ( )8,2 , 14, 7

For problems 3 5 write down the equation of the line that passes through the two points. Giveyour answer in point-slope form and slope-intercept form.

3. ( ) ( )2, 4 , 1,10

4. ( ) ( )8,2 , 14, 7

5. ( ) ( )4,8 , 1, 20

For problems 6 & 7 determine the slope of the line and sketch the graph of the line.

6. 4 8y x+ =

7. 5 2 6x y =

For problems 8 & 9 determine if the two given lines are parallel, perpendicular or neither.

8.3

17

y x= + and 3 7 10y x+ =

9. 8 2x y = and the line containing the two points ( )1,3 and ( )2, 4 .

10. Find the equation of the line through ( )7,2 and is parallel to the line 3 14 4x y = .

11. Find the equation of the line through ( )7,2 and is perpendicular to the line 3 14 4x y = .

Circles

1. Write the equation of the circle with radius 3 and center ( )6,0 .



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College Algebra

2. Write the equation of the circle with radius 7 and center ( )1, 9 .

For problems 3 5 determine the radius and center of the circle and sketch the graph of the circle.

3. ( ) ( )2 2

9 4 25x y + + =

4. ( )22 5 4x y+ =

5. ( ) ( )2 2

1 3 6x y+ + + =

For problems 6 8 determine the radius and center of the circle. If the equation is not theequation of a circle clearly explain why not.

6.2 2 14 8 56 0x y x y+ + + =

7.2 29 9 6 36 107 0x y x y+ =

8. 2 2 8 20 0x y x+ + + =

The Definition of a Function

For problems 1 3 determine if the given relation is a function.

1. ( ) ( ) ( ){ }2,4 , 3, 7 , 6,10

2. ( ) ( ) ( ) ( ){ }1,8 , 4, 7 , 1,6 , 0,0

3. ( ) ( ) ( ) ( ){ }2,1 , 9,10 , 4,10 , 8,1

For problems 4 6 determine if the given equation is a function.

4.1

14

3

y x=

5.23 1y x= +

6.4 2 16y x =

7. Given ( ) 23 2f x x= determine each of the following.



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College Algebra

(a) ( )0f (b) ( )2f (c) ( )4f (d) ( )3f t (e) ( )2f x +

8. Given ( ) 4

1g w

w=

+determine each of the following.

(a)

( )6g (b)

( )2g (c)

( )0g (d)

( )1g t (e)

( )4 3g w +

9. Given ( ) 2 6h t t= + determine each of the following.

(a) ( )0h (b) ( )2h (c) ( )2h (d) ( )h x (e) ( )3h t

10. Given ( ) 23 if 2

1 if 2

z zh z

z z


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College Algebra

18. ( )2

1

8 15f x

x x=

+

Graphing Functions

For problems 1 5 construct a table of at least 4 ordered pairs of points on the graph of thefunction and use the ordered pairs from the table to sketch the graph of the function.

1. ( ) 2 2f x x=

2. ( ) 1f x x= +

3. ( ) 9f x =

4. ( ) 210 2 if 2

2 if 2

x xf x

x x


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College Algebra

(a) ( )( )g f x (b) ( ) ( )f g x (c) ( )( )g f x (d) ( )( )f f x

5. Given ( ) 2R t t= and ( ) ( )2

2A t t= + , 0t compute each of the following.

(a) ( )( )R A t (b) ( )( )A R t

Inverse Functions

1. Given ( ) 5 9h x x= find ( )1h x .

2. Given ( ) 1

72

g x x= + find ( )1g x .

3. Given ( ) ( )3

2 1f x x= + find ( )1

f x

.

4. Given ( ) 5 2 11A x x= + find ( )1A x .

5. Given ( ) 4

5

xf x

x=

find ( )1f x .

6. Given ( ) 1 2

7

xh x

x

+=

+find ( )1h x .

Common Graphs

Introduction

Here are a set of practice problems for the Common Graphs chapter of my Algebra notes. If youare viewing the pdf version of this document (as opposed to viewing it on the web) this documentcontains only the problems themselves and no solutions are included in this document. Solutionscan be found in a number of places on the site.

13.


14.


15.

If youd like to view the solutions on the web or solutions to an individual problem youcan go to the problem set web page, select the problem you want the solution for. At this



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College Algebra

point I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printableversion.

Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problems

although this will vary from section to section.


Lines, Circles and Piecewise FunctionsParabolasEllipsesHyperbolasMiscellaneous FunctionsTransformationsSymmetryRational Functions

Lines, Circles and Piecewise Functions

We looked at these topics in the previous chapter. Problems for these topics can be found in thefollowing sections.

Lines : Graphing and Functions Lines

Circles : Graphing and Functions Circles

Piecewise Functions : Graphing and Functions Graphing Functions

Parabolas

For problems 1 7 sketch the graph of the following parabolas. The graph should contain thevertex, they-intercept,x-intercepts (if any) and at least one point on either side of the vertex.

1. ( ) ( )2

4 3f x x= +

2. ( ) ( )2

5 1 20f x x=

3. ( ) 23 7f x x= +

4. ( ) 2 12 11f x x x= + +



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5. ( ) 22 12 26f x x x= +

6. ( ) 24 4 1f x x x= +

7. ( )

2

3 6 3f x x x= + +

For problems 8 10 convert the following equations into the form ( )2

y a x h k= + .

8. ( ) 2 24 157f x x x= +

9. ( ) 26 12 3f x x x= + +

10. ( ) 2 8 18f x x x=

Ellipses

For problems 1 3 sketch the ellipse.

1.( ) ( )

2 23 5

19 3

x y+ + =

2.

( )2

2 1

14

y

x

+ =

3. ( ) ( )

2

2 44 2 1

4

yx

++ + =

For problems 4 & 5 complete the square on the xand yportions of the equation and write theequation into the standard form of the equation of the ellipse.

4.2 28 3 6 7 0x x y y+ + + =

5. 2 29 126 4 32 469 0x x y y+ + + =

Hyperbolas

For problems 1 3 sketch the hyperbola.



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1.( )

22 21

16 9

xy =

2.( ) ( )

2 23 1

1

4 9

x y+ =

3. ( ) ( )

2

2 13 1 1

2

yx

+ =

For problems 4 & 5 complete the square on the xand yportions of the equation and write theequation into the standard form of the equation of the hyperbola.

4.2 24 32 4 24 0x x y y + =

5.2 225 250 16 32 209 0y y x x+ + =

Miscellaneous Functions

The sole purpose of this section was to get you familiar with the basic shape of somemiscellaneous functions for the next section. As such there are no problems for this section. Youwill see quite a few problems utilizing these functions in the Transformationssection.

Transformations

Use transformations to sketch the graph of the following functions.

1. ( ) 4f x x= +

2. ( ) 3 2f x x=

3. ( ) 2f x x= +

4. ( ) ( )2

5f x x=

5. ( ) 3f x x=

6. ( ) 4 3f x x= +

7. ( ) 7 2f x x= +



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Symmetry

Determine the symmetry of each of the following equations.

1.6 24x y y=

2.2 2

14 9

y x= +

3.2 37 2x y x= +

4.2 6 84y x x x= +

5.5

7 4y x x= +

Rational Functions

Sketch the graph of each of the following functions. Clearly identify all intercepts andasymptotes.

1. ( ) 4

2f x

x

=

2. ( ) 6 21

xf xx

=

3. ( ) 28

6f x

x x=

+

4. ( )2

2

4 36

2 8

xf x

x x

=

Polynomial Functions

Introduction

Here are a set of practice problems for the Polynomial Functions chapter of my Algebra notes. Ifyou are viewing the pdf version of this document (as opposed to viewing it on the web) this



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College Algebra

document contains only the problems themselves and no solutions are included in this document.Solutions can be found in a number of places on the site.

16.


17.


18.




Dividing PolynomialsZeroes/Roots of PolynomialsGraphing PolynomialsFinding Zeroes of PolynomialsPartial Fractions

Dividing Polynomials

For problems 1 3 use long division to perform the indicated division.

1. Divide4 23 5 3x x + by 2x +

2. Divide 3 22 3 4x x x+ + by 7x

3. Divide 5 42 6 9x x x+ + by 2 3 1x x +

For problems 4 6 use synthetic division to perform the indicated division.

4. Divide 3 2 1x x x+ + + by 9x +

5. Divide3

7 1x by 2x +

6. Divide 4 25 8 2x x x+ + by 4x



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Zeroes/Roots of Polynomials

For problems 1 3 list all of the zeros of the polynomial and give their multiplicities.

1. ( ) 22 13 7f x x x= +

2. ( ) ( ) ( )2 36 5 4 3 2

3 6 10 21 9 3 1g x x x x x x x x x x= + + + = +

3.

( )

( ) ( ) ( )( )

8 7 6 5 4 3 2

2 2 3

2 29 76 199 722 261 648 432

1 4 1 3

A x x x x x x x x x

x x x x

= + + + +

= + +

For problems 4 6 x r= is a root of the given polynomial. Find the other two roots and writethe polynomial in fully factored form.

4. ( ) 3 2

6 16P x x x x= ; 2r=

5. ( ) 3 27 6 72P x x x x= + ; 4r=

6. ( ) 3 23 16 33 14P x x x x= + + ; 7r=

Graphing Polynomials

Sketch the graph of each of the following polynomials.

1. ( ) 3 22 24f x x x x=

2. ( ) ( ) ( )23 3 2 1 2g x x x x x= + = +

3. ( ) ( ) ( ) ( )24 3 2

12 4 16 2 1 4h x x x x x x x x= + + + = + +

4. ( ) ( ) ( )25 3 2 2

12 16 2 4P x x x x x x x= = +

Finding Zeroes of Polynomials

Find all the zeroes of the following polynomials.

1. ( ) 3 22 13 3 18f x x x x= + +



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2. ( ) 4 3 23 5 3 4P x x x x x= + +

3. ( ) 4 3 22 7 2 28 24A x x x x x= +

4. ( )

5 4 3 2

8 36 46 7 12 4g x x x x x x= + + +

Partial Fractions

Determine the partial fraction decomposition of each of the following expressions.

1.2

17 53

2 15

x

x x

2. 2

34 12

3 10 8

x

x x

3.( )( )( )

2125 4 9

1 3 4

x x

x x x

+

+ +

4.( )

2

10 35

4

x

x

+

+

5.

( )

2

6 5

2 1

x

x

+

6.( ) ( )

2

2

7 17 38

6 1

x x

x x

+

+

7.( )( )

2

2

4 22 7

2 3 2

x x

x x

+

+

8.

( )

2

2

3 7 28

7

x x

x x x

+ +

+ +

9.

( )

3

22

4 16 7

4

x x

x

+ +

+



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Exponential and Logarithm Functions

Introduction

Here are a set of practice problems for the Exponential and Logarithm Functions chapter of myAlgebra notes. If you are viewing the pdf version of this document (as opposed to viewing it onthe web) this document contains only the problems themselves and no solutions are included inthis document. Solutions can be found in a number of places on the site.

19.


20.


21.




Exponential FunctionsLogarithm FunctionsSolving Exponential EquationsSolving Logarithm EquationsApplications

Exponential Functions

1. Given the function ( ) 4xf x = evaluate each of the following.

(a) ( )2f (b) ( )1

2f (c) ( )0f (d) ( )1f (e) ( )3

2f

2. Given the function ( ) ( )15x

f x = evaluate each of the following.

(a) ( )3f (b) ( )1f (c) ( )0f (d) ( )2f (e) ( )3f

3. Sketch each of the following.

(a) ( ) 6xf x = (b) ( ) 6 9xg x = (c) ( ) 16xg x +=



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4. Sketch the graph of ( ) xf x = e .

5. Sketch the graph of ( ) 3 6xf x = +e .

Logarithm Functions

For problems 1 3 write the expression in logarithmic form.

1. 57 16807=

2.

3

416 8=

3.

21

93

=

For problems 4 6 write the expression in exponential form.

4.2log 32 5=

5. 15

1625log 5=

6. 19 81

log 2=

For problems 7 - 12 determine the exact value of each of the following without using a calculator.

7.3log 81

8.5log 125

9.2

1log

8

10.1

4

log 16

11.4lne

12.1

log100

For problems 13 15 write each of the following in terms of simpler logarithms



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13. ( )4 7log 3x y

14. ( )2 2ln x y z+

15.4 2 5

4log

x

y z

For problems 16 18 combine each of the following into a single logarithm with a coefficient ofone.

16.4 4 4

12log 5log log

2x y z+

17. ( ) ( )3ln 5 4ln 2ln 1t t s+

18.1

log 6log 23

a b +

For problems 19 & 20 use the change of base formula and a calculator to find the value of each ofthe following.

19.12log 35

20.2

3

log 53

For problems 21 23 sketch each of the given functions.

21. ( ) ( )lng x x=

22. ( ) ( )ln 5g x x= +

23. ( ) ( )ln 4g x x=

Solving Exponential Equations


1. 2 1 36 6x x=

2.15 25x =



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3.2 3 108 8x x+=

4.4 4

7 7x x =

5.3

2 10x

=

6. 1 3 17 4x x +=

7.4 69 10 x+=

8. 7 2 3 0x+ =e

9.4 7 11 20x + =e

Solving Logarithm Equations


1. ( ) ( )24 4log 2 log 5 12x x x =

2. ( ) ( ) ( )log 6 log 4 log 3x x =

3. ( ) ( ) ( )ln ln 3 ln 20 5x x x+ + =

4. ( )23log 25 2x =

5. ( ) ( )2 2log 1 log 2 3x x+ =

6. ( ) ( )4 4log log 6 2x x + =

7. ( ) ( )log 2 log 21x x=

8. ( ) ( )ln 1 1 ln 3 2x x = + +

9. ( ) ( )2log log 7 1 0x x =

Applications



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1. We have $10,000 to invest for 44 months. How much money will we have if we put themoney into an account that has an annual interest rate of 5.5% and interest is compounded

(a) quarterly (b) monthly (c) continuously

2. We are starting with $5000 and were going to put it into an account that earns an annual

interest rate of 12%. How long should we leave the money in the account in order to double ourmoney if interest is compounded

(a) quarterly (b) monthly (c) continuously

3. A population of bacteria initially has 250 present and in 5 days there will be 1600 bacteriapresent.

(a) Determine the exponential growth equation for this population.(b) How long will it take for the population to grow from its initial population of 250 to

a population of 2000?

4. We initially have 100 grams of a radioactive element and in 1250 years there will be 80 gramsleft.

(a) Determine the exponential decay equation for this element.(b) How long will it take for half of the element to decay?(c) How long will it take until there is only 1 gram of the element left?

Systems of Equations

Introduction

Here are a set of practice problems for the Systems of Equations chapter of my Algebra notes. Ifyou are viewing the pdf version of this document (as opposed to viewing it on the web) thisdocument contains only the problems themselves and no solutions are included in this document.Solutions can be found in a number of places on the site.

22.


23.

Go to the download page for the site http://tutorial.math.lamar.edu/download.aspxand

select the section youd like solutions for and a link will be provided there.

24.




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Linear Systems with Two VariablesLinear Systems with Three VariablesAugmented MatricesMore on the Augmented MatrixNonlinear Systems

Linear Systems with Two Variables

For problems 1 3 use the Method of Substitution to find the solution to the given system or to determineif the system is inconsistent or dependent.

1. 7 11

5 2 18

x y

x y

=

+ =

2. 7 8 12

4 2 3

x y

x y

=

+ =

3. 3 9 6

4 12 8

x y

x y

+ =

=

For problems 4 6 use the Method of Elimination to find the solution to the given system or to determineif the system is inconsistent or dependent.

4. 6 5 8

12 2 0

x y

x y

=

+ =

5. 2 10 2

5 25 3

x y

x y

+ =

=

6. 2 3 20

7 2 53

x y

x y

+ =

+ =



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Linear Systems with Three Variables

Find the solution to each of the following systems of equations.

1. 2 5 2 38

3 2 4 17

6 7 12

x y z

x y z

x y z

+ + =

+ =

+ =

2. 3 9 33

7 4 15

4 6 5 6

x z

x y z

x y z

=

=

+ + =

Augmented Matrices

1. For the following augmented matrix perform the indicated elementary row operations.

4 1 3 5

0 2 5 9

6 1 3 10

(a)18R (b) 2 3R R (c) 2 1 23R R R+


1 6 2 0

2 8 10 4

3 4 1 2

(a)2

1

2R (b) 1 3R R (c) 1 3 16R R R




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10 1 5 1

4 0 7 1

0 7 2 3

(a) 39R (b)1 2R R (c)

3 1 3R R R

Note :Problems using augmented matrices to solve systems of equations are in the nextsection.

More on the Augmented Matrix

For each of the following systems of equations convert the system into an augmented matrix and use theaugmented matrix techniques to determine the solution to the system or to determine if the system isinconsistent or dependent.

1. 7 11

5 2 18

x y

x y

=

+ =

2. 7 8 12

4 2 3

x y

x y

=

+ =

3. 3 9 6

4 12 8

x y

x y

+ =

=

4. 6 5 8

12 2 0

x y

x y

=

+ =

5.5 25 3

2 10 2

x y

x y

=

+ =

6. 2 3 207 2 53

x yx y

+ =+ =

7. 2 5 2 38

3 2 4 17

6 7 12

x y z

x y z

x y z

+ + =

+ =

+ =



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8. 3 9 33

7 4 15

4 6 5 6

x z

x y z

x y z

=

=

+ + =

Non-Linear Systems

Find the solution to each of the following system of equations.

1.2 6 8

4 7

y x x

y x

= +

= +

2.

22

1 3

14

y x

xy

=

+ =

3.

2 2

4

14 25

xy

x y

=

+ =

4. 2

22

1 2

19

y x

yx

=

=

alg complete practice

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