Math 119 Final Exam Concept List: College Algebra Study Guide

Math 119 Final Exam Concept List

Overview

This study guide summarizes key concepts for a College Algebra final exam, organized by major topics and subtopics. It covers foundational algebraic skills, functions, equations, applications, and probability, providing definitions, formulas, and examples for each area.

Equations & Inequalities

Absolute Change and Relative Change

• Absolute Change: The difference between the final and initial values of a quantity. Formula: $\text{Absolute Change} = \text{Final Value} - \text{Initial Value}$

• Relative Change: The absolute change expressed as a percentage of the initial value. Formula: $\text{Relative Change} = \frac{\text{Absolute Change}}{\text{Initial Value}} \times 100\%$

• Example: If a price increases from $50 to $60, absolute change is $10, relative change is $20\%$.

Indexing and Rates

• Index: A number that measures change relative to a base value, often set to 100. Formula: $\text{Index} = \frac{\text{Current Value}}{\text{Base Value}} \times 100$

• Rate: A ratio comparing two quantities, such as speed or growth rate.

• Example: Consumer Price Index (CPI) uses a base year to compare price changes over time.

Percentages and Proportions

• Percent: A way to express a number as a fraction of 100. Formula: $\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\%$

• Proportion: An equation stating that two ratios are equal. Formula: $\frac{a}{b} = \frac{c}{d}$

• Example: If 30 out of 120 students passed, percent passing is $25\%$.

Mean, Median, and Range

• Mean: The average value of a data set. Formula: $\text{Mean} = \frac{\sum x_i}{n}$

• Median: The middle value when data are ordered.

• Range: The difference between the largest and smallest values. Formula: $\text{Range} = \text{Max} - \text{Min}$

• Example: For data 2, 4, 7, mean is $4.33$, median is $4$, range is $5$.

Functions

Function Definition and Evaluation

• Function: A relation that assigns each input exactly one output.

• Notation: $f(x)$ denotes the value of function $f$ at input $x$.

• Example: $f(x) = 2x + 3$; if $x = 4$, $f(4) = 11$.

Domain and Range

• Domain: The set of all possible input values ($x$) for a function.

• Range: The set of all possible output values ($f(x)$).

• Example: For $f(x) = \sqrt{x}$, domain is $x \geq 0$, range is $f(x) \geq 0$.

Interpreting Functions in Context

• Functions can model real-world scenarios, such as population growth or cost analysis.

• Example: If $C(x)$ is the cost to produce $x$ items, $C(100)$ gives the cost for 100 items.

Graphs of Equations

Linear Functions and Graphs

• Linear Function: A function of the form $f(x) = mx + b$.

• Slope ($m$): Measures the rate of change; rise over run. Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Y-intercept ($b$): The value of $f(x)$ when $x = 0$.

• Example: For $f(x) = 3x + 2$, slope is $3$, y-intercept is $2$.

Creating Linear Functions from Data

• Given two points $(x_1, y_1)$ and $(x_2, y_2)$, find the equation of the line passing through them.

• Formula: $y - y_1 = m(x - x_1)$

• Example: Points (1, 2) and (3, 6): $m = (6-2)/(3-1) = 2$, so $y = 2x$.

Exponential & Logarithmic Functions

Exponential Growth and Decay

• Exponential Function: $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the growth/decay factor.

• Growth: $b > 1$; Decay: $0 < b < 1$.

• Example: $f(x) = 100 \cdot 1.05^x$ models 5% annual growth.

Evaluating and Tabulating Exponential Functions

• Calculate values for different $x$ to create a table.

• Example: For $f(x) = 2^x$, $f(0) = 1$, $f(1) = 2$, $f(2) = 4$.

Systems of Equations & Matrices

Multi-Step Calculations and Applications

• Apply algebraic methods to solve for unknowns in multi-variable scenarios.

• Example: Use systems of equations to solve for investment returns or cost analysis.

Sequences, Series, & Induction

Future Value and Savings Plans

• Future Value: The value of an investment after a certain period, considering interest. Formula (Simple Interest): $A = P(1 + rt)$ Formula (Compound Interest): $A = P(1 + \frac{r}{n})^{nt}$

• Example: $P = 1000$, $r = 5\%$, $t = 3$ years, $A = 1000(1 + 0.05 \times 3) = 1150$ (simple interest).

Combinatorics & Probability

Counting Outcomes and Probability

• Counting Principle: If one event can occur in $m$ ways and another in $n$ ways, total outcomes are $m \times n$.

• Probability: The likelihood of an event occurring. Formula: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Example: Probability of rolling a 3 on a die: $P = 1/6$.

Types of Events

• Dependent Events: The outcome of one event affects another.

• Independent Events: Outcomes do not affect each other.

• Overlapping/Non-Overlapping Events: Events may or may not share outcomes.

• At Least One Rule: Probability that at least one event occurs: $P(\text{at least one}) = 1 - P(\text{none})$

Probability Scenarios

• List all possible outcomes for a scenario.

• Calculate probability for specific events, including not occurring, dependent/independent, and overlapping events.

• Example: Probability of drawing a red card from a deck: $P = 26/52 = 0.5$.

Additional info:

• Scientific calculators are allowed for calculations, but graphing calculators and smart devices are prohibited.

• Reviewing previous exams and practice problems is recommended for exam preparation.