Precalculus Course Outline and Topic Overview (Math 112)

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Course Overview

This document outlines the schedule and main topics for a college-level Precalculus course (Math 112). The course covers foundational concepts in algebra, functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations and inequalities, and introductory probability. The following study notes provide a structured overview of the key topics and subtopics as indicated in the course outline.

Functions and Graphs

Basics of Functions and Graphs

Functions are fundamental objects in mathematics that describe relationships between sets. Understanding their properties and how to represent them graphically is essential in Precalculus.

Definition of a Function: A function f from set A to set B assigns to each element in A exactly one element in B.
Domain and Range: The domain is the set of all possible input values; the range is the set of all possible output values.
Graph of a Function: The set of all points (x, f(x)) in the coordinate plane.
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
Example: The function has domain and range .

Transformations of Functions

Transformations alter the appearance of a function's graph without changing its basic shape.

Vertical and Horizontal Shifts: shifts up/down; shifts right/left.
Reflections: reflects over the x-axis; reflects over the y-axis.
Stretching and Compressing: stretches vertically if , compresses if .
Example: is shifted right 3 units, stretched by 2, and up 1 unit.

Combination of Functions

Functions can be combined using arithmetic operations or composition.

Sum, Difference, Product, Quotient: , etc.
Composition: .
Example: If , , then .

Inverse Functions

An inverse function reverses the effect of the original function, if it exists.

Definition: if .
One-to-One Functions: Only one-to-one functions have inverses.
Finding Inverses: Solve for in terms of , then interchange and .
Example:

Polynomial and Rational Functions

Quadratic Functions

Quadratic functions are polynomials of degree 2 and have parabolic graphs.

Standard Form:
Vertex:
Axis of Symmetry:
Example: has vertex at .

Polynomial Functions and Their Graphs

Polynomial functions are sums of terms of the form .

Degree: Highest power of .
End Behavior: Determined by leading term.
Zeros: Values of where .
Example:

Dividing Polynomials; Remainder and Factor Theorems

Polynomials can be divided using long or synthetic division.

Remainder Theorem: The remainder of divided by is .
Factor Theorem: is a factor of if and only if .
Example: Divide by .

Zeros of Polynomial Functions

Finding zeros involves factoring or using the Rational Root Theorem.

Multiplicity: The number of times a zero is repeated.
Rational Root Theorem: Possible rational zeros are , where divides the constant term and divides the leading coefficient.
Example: has zeros at .

Rational Functions and Their Graphs

Rational functions are ratios of polynomials.

Vertical Asymptotes: Occur where denominator is zero.
Horizontal/Oblique Asymptotes: Determined by degrees of numerator and denominator.
Example: has a vertical asymptote at .

Polynomial and Rational Inequalities

Solving inequalities involves finding intervals where the function is positive or negative.

Test Intervals: Use zeros to divide the number line and test each interval.
Example: Solve ; solution: or .

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form , where and .

Growth and Decay: If , function grows; if , function decays.
Example:

Logarithmic Functions

Logarithms are the inverses of exponential functions.

Definition: if and only if .
Domain:
Example:

Properties of Logarithms

Logarithms have several important properties that simplify calculations.

Product Rule:
Quotient Rule:
Power Rule:
Change of Base:

Exponential and Logarithmic Equations

Solving these equations often involves applying logarithmic properties or exponent rules.

Example: Solve ; .
Example: Solve ; .

Exponential Growth & Decay; Modeling Data

Exponential models describe many real-world phenomena.

Growth Model: ,
Decay Model: ,
Example: Population growth, radioactive decay.

Systems of Equations and Inequalities

Systems of Linear Equations in Three Variables

These systems can be solved using substitution, elimination, or matrix methods.

General Form:
Solution: The point that satisfies all equations.

Systems of Nonlinear Equations

Nonlinear systems include at least one equation that is not linear.

Example:
Solution Methods: Substitution, elimination, or graphical methods.

Systems of Inequalities

Systems of inequalities define regions in the plane that satisfy all given inequalities.

Graphical Solution: Shade the region that satisfies all inequalities.
Example:

Conic Sections and Analytic Geometry

Circles

A circle is the set of all points in a plane equidistant from a fixed point (the center).

Standard Equation:
Center: ; Radius:
Example: is a circle with center and radius $4$.

Sequences, Induction, and Probability

Binomial Theorem

The Binomial Theorem provides a formula for expanding powers of binomials.

Formula:
Binomial Coefficient:
Example: Expand using the theorem.

Review and Assessment

The course includes regular reviews and four major tests, each covering several sections. A comprehensive final exam concludes the course.

Test	Sections Covered
Test 1	1.2, 1.3, 1.6, 1.7, 2.2
Test 2	2.3 – 2.7
Test 3	1.8, 3.1 – 3.5
Test 4	1.9, 7.2, 7.4, 7.5, 10.5

Additional info: Some sections (e.g., 1.9 Circles, 7.2, 7.4, 7.5, 10.5) correspond to conic sections, systems of equations/inequalities, and the binomial theorem, which are standard Precalculus topics. The course does not appear to cover trigonometry in detail, but focuses on algebraic and function-based content.