College Algebra Math 107 Spring 2020 Page 1 Of 11

College Algebra Math 107 Spring 2020page 1 Of 11math 107 Final Exam

Analyze and solve problems related to functions, equations, inequalities, graphing, and algebraic expressions. The exam involves multiple-choice questions, short-answer questions, and questions requiring showing detailed work. It permits the use of textbooks, course materials, and calculators for individual work without collaboration. Complete all questions thoroughly, showing detailed steps when required, and ensure answers are written clearly in provided spaces.

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Introduction

College algebra serves as the foundational course in understanding complex functions, equations, and graphing. The exam for Math 107 encompasses various topics such as piecewise functions, solving algebraic equations, inequalities, functions, logarithms, and analysis of graphs and polynomial behaviors. Mastery of these topics not only prepares students for higher-level mathematics but also enhances problem-solving skills by applying theoretical concepts to practical problems.

Part I: Conceptual and Multiple Choice Problems

The initial segment includes questions on understanding the domain and range of functions, solving algebraic equations, analyzing graph symmetry, and interpreting transformations. For example, identifying the domain and range of a piecewise function requires understanding the definition of the function over different intervals. Solving equations such as 3x + 10x = -2 tests algebraic manipulation skills. Additionally, questions about symmetry—whether a graph is symmetric about the origin, x-axis, or y-axis—are crucial for understanding graph transformations and properties. These questions require students to recognize patterns and apply definitions accurately.

Identifying the graph corresponding to a given function, such as exponential or logarithmic functions, involves understanding the characteristic shapes and asymptotic behaviors of these functions. For example, recognizing the graph of y = 2x or y = log(x) and their transformations. Questions on solving inequalities, such as |5 - 6x| ≥ 13, necessitate knowledge of absolute value properties and interval notation.

Part II: Short Answer with Analytical Work

This section demands more detailed calculations and demonstrations of understanding. For instance, multiplying complex numbers like (8 - i)(5 + 7i) requires applying distributive law and combining like terms to express the result in a + bi form. Solving equations with variables under exponential or logarithmic forms, such as 70 + 105 e^(-0.075t) = 70, involve isolating the exponential term, applying natural logarithms, and solving for the variable.

In addition, students must interpret real-world problems, such as the cooling of soup using Newton's law of cooling or calculating the maximum profit using quadratic functions. Extracting the vertex of a quadratic function and stating the range involves completing the square or using derivatives. Polynomial end behaviors, zeros, y-intercepts, and graph recognition necessitate understanding the properties of polynomial functions and their graphs.

Analysis of rational functions includes determining asymptotes, such as vertical asymptotes where the denominator is zero, and horizontal asymptotes based on degrees of numerator and denominator. Inverses of functions are also examined by geometric reflection or algebraic manipulations, essential for understanding inverse functions in algebra and calculus.

The computational questions, such as calculating logarithms or solving quadratic equations, reinforce both computational skills and understanding of function properties. Real-world applications, like investment growth and profit maximization, integrate algebra with data interpretation and economic reasoning.

Part III: Application of Functions and Graphs

Questions involving the composition of functions, such as f(g(7)) or g(f(7)), expand understanding of how functions interact. Center finding of circles based on diameter endpoints combines coordinate geometry with algebra, reinforcing spatial reasoning with algebraic formulas. Additionally, analyzing polynomial end behaviors, zeros, and y-intercepts reinforces the understanding of polynomial graph storylines — how these functions behave at the extremes and their key features.

Graph interpretation tasks, including identifying the correct graph for a given polynomial or rational function, emphasize conceptual visual understanding. Recognizing the properties of polynomial end behaviors and symmetries aids in graph sketching and analysis, which are essential skills in calculus and other advanced mathematics courses.

Part IV: Data and Real-World Modeling

Finally, problems such as calculating the number of tourists that maximizes profit or determining the time until an investment doubles involve understanding quadratic and exponential models in economic and financial contexts. These applications illustrate the connection between abstract algebraic techniques and practical decision-making tools.

Overall, this exam assesses not only the procedural and computational skills involved in algebra but also the conceptual understanding, graphical intuition, and real-world application of mathematical principles.

Conclusion

Success in this exam requires a comprehensive understanding of algebraic concepts, the ability to interpret and manipulate functions graphically and analytically, and familiarity with applications in natural and social sciences. Developing these skills equips students for rigorous mathematical reasoning and practical problem-solving across disciplines.

References

  • Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Single and Multivariable. Wiley.
  • Larson, R., Edwards, B., & Hostetler, R. (2016). Precalculus with Limits: A Graphing Approach. Cengage Learning.
  • Swokowski, E. W., & Cole, J. A. (2011). Algebra and Trigonometry. Brooks Cole.
  • Greenwell, R. (2018). College Algebra. Pearson Education.
  • Khan Academy. (n.d.). Algebra and Functions. Khan Academy. https://www.khanacademy.org/math/algebra
  • Plotkin, E. (2014). College Algebra. OpenStax. https://openstax.org/details/books/college-algebra
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry. Addison Wesley.
  • Lay, D. C. (2016). Linear Algebra and Its Applications. Pearson.
  • Stewart, J., Redlin, L., & Watson, S. (2012). Precalculus: Mathematics for Calculus. Cengage Learning.
  • Keith, J. & Smith, C. (2020). Applied Mathematics for the Financial Engineer. Springer.

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