Precalculus Math 112 Worksheet 3 Review For Final Exam
Precalculusmth 112 Worksheet 3 Review For Final Exam
Precalculus mth 112 Worksheet 3 Review For Final Exam name:__________________________ date:__________ class time:__________ Answer each of the following problems on a Scantron answer sheet. This assignment will be graded. Work all problems on your own paper, neatly and in order. Turn in work paper stapled. Also turn in Scantron answer sheet separately. Do not staple Scantron to work paper! Work paper may also be graded. This worksheet is worth 40 points.
Identify and solve various types of algebraic, logarithmic, exponential, and geometric problems, including functions, equations, inequalities, systems of linear equations, quadratic equations, graph analysis, and applying properties of logarithms and exponents, as well as geometric concepts like the circle's center and radius. You will also be assessing line slopes, asymptotes, and function behavior from graphs, as well as converting between logarithmic and exponential forms, expanding and condensing logarithmic expressions, and solving systems of equations and inequalities, including quadratic inequalities.
Paper For Above instruction
This comprehensive review covers essential topics in precalculus that are vital for the final exam preparation. Students are expected to demonstrate mastery of algebraic manipulations, functions, and their properties, as well as logarithmic and exponential equations, systems of equations, inequalities, and geometric concepts.
Functions and Graphical Analysis
The problem set includes evaluating functions at given points, such as evaluating \(f(2)\) for a quadratic function \(f(x) = x^2 + 3x + 2\), and analyzing the behavior of functions based on their graphs. For example, determining whether a graph depicts a function by checking for the vertical line test, and finding lines passing through given points to establish their slopes and equations.
Understanding the slope formula \(\text{slope} = (y_2 - y_1) / (x_2 - x_1)\) and classifying lines as parallel, perpendicular, or neither based on their slopes are key skills tested. Additionally, students should manipulate equations in standard form and convert between forms, such as from slope-intercept to standard form, to understand relationships between lines better.
Algebra and Equations
Problems include solving quadratic inequalities such as \(45x^2 + x
Dividing polynomials using synthetic division, such as dividing \(x^3 + 2x^2 - 3x - 2\) by a binomial, and finding rational roots of polynomial equations through the Rational Root Theorem, are essential algebraic techniques included. These skills facilitate understanding polynomial behavior and the roots of equations.
Logarithmic and Exponential Functions
The review includes converting between logarithmic and exponential forms (e.g., \(S \times T = R\) becoming \(R = S^T\)), expanding and condensing logarithmic expressions via properties like \(\log xy = \log x + \log y\) and \(\log \frac{x}{y} = \log x - \log y\).
Evaluating logs using calculators with change-of-base formulas, as well as understanding the properties of logarithms and their applications in solving equations like \(\log 18\) or \(\ln 65.35\), will be practiced. These skills are essential for solving exponential growth or decay problems and analyzing functions' behaviors at different points.
Asymptotic Analysis and Function Behavior
Identifying vertical, horizontal, and slant asymptotes of rational functions such as \(\frac{8}{x}\) or analyzing the end behavior of functions like \(f(x) = \frac{4x}{x-1}\) can be tested. Recognizing asymptotes helps in graphing and understanding the domain restrictions of rational functions.
Students will evaluate logs with different bases, such as \(\log_3 3\) and \(\log_{16} 16\), and determine the set of solutions for inequalities and equations involving these functions.
System of Equations and Inequalities
Solving systems of linear equations involves methods like substitution, elimination, or graphing. For example, solving \(\begin{cases} 3x + y = 6 \\ xy = -1 \end{cases}\) using substitution or elimination techniques yields specific solutions, which must be classified accordingly (e.g., ordered pairs).
Quadratic inequalities such as \(45x^2 + x
Analytic Geometry
The worksheet emphasizes calculating the center and radius of circles from their equations, as in \(x^2 + y^2 - 6x + 4y = 25\). This involves completing the square to rewrite the equation in standard form, revealing the circle's geometric properties.
Finding the vertex of parabola, such as \(y = x^2 - 5x + 6\), requires completing the square or using vertex formulas. Additionally, students analyze line slopes and relationships, determine parallel and perpendicular lines, and write their equations in standard form passing through specific points.
Additional Topics
Questions include synthetic division, rational root testing, and identifying asymptotes for rational functions, necessary for polynomial division and understanding function limits at certain points.
Logarithmic identities, such as expanding \(\log(xyz)\) into \(\log x + \log y + \log z\), and condensing expressions like \(\log 6 + \log 5\) into \(\log 30\), are practiced to build intuition for logarithmic manipulation.
Finally, the review encompasses evaluating complex logarithmic expressions with calculators when necessary, emphasizing accuracy and rounding to two decimal places, especially in real-world problems involving growth and decay or data modeling.
References
- Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals. Wiley.
- Blitzer, R. (2014). Algebra and Trigonometry. Pearson.
- Larson, R., & Hostetler, R. (2017). Precalculus with Limits: A Graphing Approach. Cengage Learning.
- McCallum, B. (2018). College Algebra. Cengage.
- Stewart, J. (2015). Precalculus: Mathematics for Calculus. Brooks Cole.
- Swokowski, E. W., & Cole, J. A. (2018). Algebra & Trigonometry. Cengage Learning.
- Thomas, G. B., & Finney, R. L. (2014). Calculus and Analytic Geometry. Addison-Wesley.
- Utschig, R. L., & Utschig, L. (2014). Precalculus Enhanced with Graphing Utilities. Pearson.
- William, C. (2013). Precalculus: Graphical, Numerical, Algebraic. Pearson.
- Zeitz, P. (2015). Precalculus: An Investigation of Functions and Their Graphs. Pearson.