MAC1140 Precalculus Algebra: Syllabus and Core Study Guide

Course Overview

MAC1140 Pre-Calculus Algebra is a foundational course designed to prepare students for calculus. The course covers essential topics in algebra and precalculus, including polynomial, rational, exponential, and logarithmic functions, as well as systems of equations, matrices, conic sections, sequences, and series. Emphasis is placed on understanding functions and their graphs, solving equations and inequalities, and applying algebraic techniques to real-world problems.

Course Structure and Learning Outcomes

• Recognize and graph polynomial, rational, exponential, and logarithmic functions.

• Solve polynomial, rational, and absolute value equations and inequalities.

• Manipulate and graph equations of conic sections.

• Perform matrix operations and solve systems of equations.

• Apply properties of sequences and series, including the binomial theorem.

Module 1: Polynomial and Rational Functions

1.1 Absolute Value Inequalities

Absolute value inequalities involve expressions where the variable is inside an absolute value. To solve them, consider both the positive and negative scenarios.

• Definition: $|x| < a$ means $-a < x < a$; $|x| > a$ means $x < -a$ or $x > a$.

• Example: Solve $|2x - 3| < 5$.

• Solution: $-5 < 2x - 3 < 5 \implies -2 < x < 4$

1.2 Functions and Their Properties

Understanding the behavior of functions is crucial in precalculus. Key properties include intervals of increase/decrease, maxima/minima, symmetry, and types of functions.

• Intervals: A function increases where its graph rises as $x$ increases, and decreases where it falls.

• Relative Maxima/Minima: Points where the function reaches local highest or lowest values.

• Symmetry: Even functions are symmetric about the $y$-axis; odd functions are symmetric about the origin.

• Piecewise Functions: Defined by different expressions over different intervals.

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$, used to analyze rates of change.

1.3 Transformations of Functions

• Vertical/Horizontal Shifts: $f(x) + k$ shifts up/down; $f(x - h)$ shifts right/left.

• Reflections: $-f(x)$ reflects over the $x$-axis; $f(-x)$ reflects over the $y$-axis.

• Stretching/Shrinking: $af(x)$ stretches vertically if $|a| > 1$, shrinks if $0 < |a| < 1$.

• Example: Graph $f(x) = (x-2)^2 + 3$ (shifted right 2, up 3).

1.4 Polynomial Functions

• Definition: $f(x) = a_nx^n + \ldots + a_0$

• End Behavior: Determined by the leading term $a_nx^n$.

• Zeros: Solutions to $f(x) = 0$; found by factoring or using the Rational Zero Theorem.

• Intermediate Value Theorem: If $f(a)$ and $f(b)$ have opposite signs, there is at least one zero between $a$ and $b$.

• Turning Points: A polynomial of degree $n$ has at most $n-1$ turning points.

1.5 Polynomial Division and Theorems

• Long Division and Synthetic Division: Methods for dividing polynomials.

• Remainder Theorem: The remainder of $f(x)$ divided by $x - c$ is $f(c)$.

• Factor Theorem: $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$.

1.6 Rational Functions and Asymptotes

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is not.

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator.

• Slant Asymptotes: Occur if the numerator's degree is one more than the denominator's.

• Graphing: Use transformations and asymptotes to sketch rational functions.

1.7 Polynomial and Rational Inequalities

• Solving: Set the expression to zero, find critical points, and test intervals.

• Application: Used in modeling real-world constraints.

Module 2: Exponential and Logarithmic Functions

2.1 Exponential Functions

• Definition: $f(x) = a^x$, $a > 0$, $a \neq 1$.

• Graph: Always positive, passes through $(0,1)$, increases if $a > 1$.

• Base $e$: $f(x) = e^x$, where $e \approx 2.718$.

• Compound Interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$ or $A = Pe^{rt}$ for continuous compounding.

2.2 Logarithmic Functions

• Definition: $y = \log_a x$ is equivalent to $a^y = x$.

• Properties: $\log_a (xy) = \log_a x + \log_a y$ \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \log_a (x^r) = r \log_a x

• Change of Base: $\log_a x = \frac{\log_b x}{\log_b a}$

• Common Logarithm: $\log_{10} x$; Natural Logarithm: $\ln x = \log_e x$

2.3 Solving Exponential and Logarithmic Equations

• Exponential Equations: Use like bases or logarithms to solve.

• Logarithmic Equations: Use properties of logarithms and the one-to-one property.

• Applications: Population growth, radioactive decay, and financial modeling.

2.4 Exponential and Logarithmic Models

• Exponential Growth/Decay: $y = y_0 e^{kt}$, $k > 0$ for growth, $k < 0$ for decay.

• Logistic Growth: $P(t) = \frac{M}{1 + Ae^{-kt}}$ models limited growth.

Module 3: Matrices, Determinants, and Systems of Equations

3.1 Partial Fraction Decomposition

• Purpose: Express rational expressions as sums of simpler fractions.

• Cases: Distinct linear factors, repeated linear factors, and irreducible quadratic factors.

3.2 Matrices and Row Operations

• Augmented Matrix: Represents a system of equations.

• Row Operations: Swap, scale, and add rows to solve systems (Gaussian/Gauss-Jordan elimination).

• Matrix Notation: $A = [a_{ij}]$

• Matrix Operations: Addition, subtraction, scalar multiplication, and multiplication.

• Solving Systems: Use row reduction or Cramer's Rule.

3.3 Determinants

• Second-Order Determinant: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

• Third-Order Determinant: Use expansion by minors or Sarrus' Rule.

• Cramer's Rule: Solve $n$ equations in $n$ variables using determinants.

Module 4: Conic Sections, Sequences, and Series

4.1 Conic Sections

• Ellipses: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Hyperbolas: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Parabolas: $(y-k)^2 = 4p(x-h)$ or $(x-h)^2 = 4p(y-k)$

• Applications: Modeling planetary orbits, satellite dishes, etc.

4.2 Sequences and Series

• Sequence: An ordered list of numbers defined by a formula or recursion.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Sum of Arithmetic Sequence: $S_n = \frac{n}{2}(a_1 + a_n)$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Sum of Geometric Sequence: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

• Sum of Infinite Geometric Series: $S = \frac{a_1}{1 - r}$, $|r| < 1$

• Factorial Notation: $n! = n \times (n-1) \times \ldots \times 1$

• Summation Notation: $\sum_{k=1}^n a_k$

Assessment and Grading

Assessment	Percent of Final Grade
Proctored Chapter Tests	60%
Discussions	4%
Homework	21%
Proctored Final Exam	15%

Grading Scale:

Grade	Percentage
A	90 - 100%
B	80 - 89.9%
C	70 - 79.9%
D	60 - 69.9%
F	0 - 59.9%

Required Materials and Technology

• Textbook: Algebra and Trigonometry, 6th Edition (Pearson)

• Graphing calculator (as required by instructor)

• Reliable internet, computer, webcam, microphone, and whiteboard for proctored exams

• Access to MyMathLab and D2L for course content and assessments

Support and Resources

• Math Lab and Academic Success Center for tutoring and study support

• Online tutoring via Brainfuse

• Technical support through the BC Help Desk

• Accessibility resources for students with disabilities

Course Policies

• Attendance and participation are required for success.

• Academic honesty is strictly enforced; violations result in severe penalties.

• Late work is only accepted under qualifying circumstances with documentation.

• Communication should be conducted through D2L email unless otherwise specified.

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