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College Algebra

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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index
1.

Determine whether the following algebraic equation can be written as a linear function. 2x+3y=7 2x+3y=7

2.

Determine whether the following function is increasing or decreasing. f( x )=2x+5 f( x )=2x+5

3.

Determine whether the following function is increasing or decreasing. f( x )=7x+9 f( x )=7x+9

4.

Find a linear equation that passes through (5, 1) and (3, –9), if possible.

5.

Find a linear equation, that has an x intercept at (–4, 0) and a y-intercept at (0, –6), if possible.

6.

Find the slope of the line in Figure 1.

This image is a graph of a decreasing linear function on an x, y coordinate plane. The x and y-axis range from -6 to 6. The line passes through the points (0,2) and (1,0).
Figure 1
7.

Write an equation for line in Figure 2.

This image is a graph showing a decreasing linear function on an x, y coordinate plane. The x and y axis range from -6 to 6. The line passes through the points (0,-1) and (-.5,0).
Figure 2
8.

Does Table 1 represent a linear function? If so, find a linear equation that models the data.

x x –6 0 2 4
g( x ) g( x ) 14 32 38 44
Table 1
9.

Does Table 2 represent a linear function? If so, find a linear equation that models the data.

x 1 3 7 11
g(x) 4 9 19 12
Table 2
10.

At 6 am, an online company has sold 120 items that day. If the company sells an average of 30 items per hour for the remainder of the day, write an expression to represent the number of items that were sold n n after 6 am.

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

11.

y= 3 4 x9 4x3y=8 y= 3 4 x9 4x3y=8

12.

2x+y=3 3x+ 3 2 y=5 2x+y=3 3x+ 3 2 y=5

13.

Find the x- and y-intercepts of the equation 2x+7y=14. 2x+7y=14.

14.

Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is the pair of lines parallel, perpendicular, or neither?

Line 1: Passes through (−2,−6) (−2,−6) and (3,14) (3,14)

Line 2: Passes through (2,6) (2,6) and (4,14) (4,14)

15.

Write an equation for a line perpendicular to f(x)=4x+3 f(x)=4x+3 and passing through the point (8,10). (8,10).

16.

Sketch a line with a y-intercept of ( 0,5 ) ( 0,5 ) and slope 5 2 . 5 2 .

17.

Graph of the linear function f(x)=x+6. f(x)=x+6.

18.

For the two linear functions, find the point of intersection: x=y+2 2x3y=1 . x=y+2 2x3y=1 .

19.

A car rental company offers two plans for renting a car.

Plan A: $25 per day and $0.10 per mile

Plan B: $40 per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

20.

Find the area of a triangle bounded by the y axis, the line f( x )=124x, f( x )=124x, and the line perpendicular to f f that passes through the origin.

21.

A town’s population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018.

22.

The number of people afflicted with the common cold in the winter months dropped steadily by 25 each year since 2002 until 2012. In 2002, 8,040 people were inflicted. Find the linear function that models the number of people afflicted with the common cold C C as a function of the year, t. t. When will less than 6,000 people be afflicted?

For the following exercises, use the graph in Figure 3, showing the profit, y, y, in thousands of dollars, of a company in a given year, x, x, where x x represents years since 1980.

This image is a graph showing the company's profit from 1985 at around $15,000 to 2010 at about $32,500.  The x-axis goes from 0 to 30 in intervals of 5 and the y-axis goes from 0 to 35,000 in intervals of 5,000.
Figure 3
23.

Find the linear function y, y, where y y depends on x, x, the number of years since 1980.

24.

Find and interpret the y-intercept.

25.

In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly.

  1. How much did the population drop between the year 2004 and 2012?
  2. What is the average population decline per year?
  3. Find an equation for the population, P, of the school t years after 2004.
26.

Draw a scatter plot for the data provided in Table 3. Then determine whether the data appears to be linearly related.

0 2 4 6 8 10
–450 –200 10 265 500 755
Table 3
27.

Draw a best-fit line for the plotted data.

Scatterplot with domain from 2 to 10 and range from 20 from 33.  The points plotted are (2,20); (4,23); (6,26); (8,26); and (10,32).

For the following exercises, use Table 4, which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year.

Year 2000 2002 2005 2007 2010
Percent Graduates 8.5 8.0 7.2 6.7 6.4
Table 4
28.

Determine whether the trend appears linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.

29.

In what year will the percentage drop below 4%?

30.

Based on the set of data given in Table 5, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient. Round to three decimal places of accuracy.

x 16 18 20 24 26
y 106 110 115 120 125
Table 5

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:

(4,500,2000);(4,700,2001);(5,200,2003);(5,800,2006) (4,500,2000);(4,700,2001);(5,200,2003);(5,800,2006)

31.

Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy.

32.

Predict when the population will hit 20,000.

33.

What is the correlation coefficient for this model?

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