Math 119 Final Exam Concept List: College Algebra Study Guide

Equations & Inequalities

Absolute Change and Relative Change

Understanding how quantities change is fundamental in algebra. Absolute change measures the difference between two values, while relative change expresses this difference as a proportion of the original value.

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

• Relative Change: The absolute change divided by the initial value, often expressed as a percentage. $\text{Relative Change} = \frac{\text{Absolute Change}}{\text{Initial Value}}$

• Example: If a price increases from $50 to $60, the absolute change is $10, and the relative change is$\frac{10}{50} = 0.2$ or 20%.

Indexing and Data Analysis

Indexes are used to compare values over time or between categories. Calculating an index involves expressing a value relative to a base value.

• Index Calculation: $\text{Index} = \frac{\text{Value}}{\text{Base Value}} \times 100$

• Application: Used in economics to compare prices, population, or other quantities over time.

• Example: If the base year population is 1,000 and the current year is 1,200, the index is $\frac{1200}{1000} \times 100 = 120$.

Graphs of Equations

Graphing and Interpreting Functions

Graphing is a visual way to understand the behavior of equations and functions. Key features include slope, intercepts, and shape.

• Slope: Measures the rate of change of a linear function. $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Y-intercept: The point where the graph crosses the y-axis.

• Example: The line passing through (2, 3) and (4, 7) has slope $m = \frac{7-3}{4-2} = 2$.

Functions

Evaluating and Describing Functions

Functions relate inputs to outputs. Evaluating a function means finding the output for a given input.

• Function Notation: $f(x)$ represents the output when input is $x$.

• Domain and Range: The domain is the set of possible inputs; the range is the set of possible outputs.

• Example: For $f(x) = 2x + 1$, $f(3) = 2 \times 3 + 1 = 7$.

Polynomial Functions

Linear Functions

Linear functions have the form $f(x) = mx + b$. They model constant rates of change.

• Creating Linear Functions: Use two points to determine the slope and intercept.

• Example: Given points (1, 2) and (3, 6), slope $m = \frac{6-2}{3-1} = 2$, so $f(x) = 2x$.

Exponential & Logarithmic Functions

Exponential Growth and Decay

Exponential functions model rapid increases or decreases. The general form is $f(x) = a b^x$.

• Growth: If $b > 1$, the function grows.

• Decay: If $0 < b < 1$, the function decays.

• Example: $f(x) = 3 \times 2^x$ grows as $x$ increases.

Evaluating Exponential Functions

• Table of Values: Substitute values of $x$ to find $f(x)$.

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

Combinatorics & Probability

Counting and Probability

Combinatorics involves counting possible outcomes. Probability measures the likelihood of events.

• Counting Outcomes: Use the multiplication principle for independent events.

• Probability Formula: $P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}$

• Example: If there are 3 shirts and 2 pants, total outfits = $3 \times 2 = 6$.

Types of Events

• Independent Events: The outcome of one does not affect the other.

• Dependent Events: The outcome of one affects the other.

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

• Example: Probability of drawing a red card from a deck: $P(\text{red}) = \frac{26}{52} = 0.5$.

Sequences, Series, & Induction

Future Value and Savings Plans

Algebra is used to calculate future values in savings and investments using formulas for sequences and series.

• Future Value Formula (Compound Interest): $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Application: Used to determine how much money will accumulate over time.

• Example: $P = 1000$, $r = 0.05$, $n = 1$, $t = 3$ years: $A = 1000(1 + 0.05)^3 = 1157.63$

Systems of Equations & Matrices

Multi-Unit and Scenario Analysis

Systems of equations can be used to solve problems involving multiple variables or units.

• Setting Up Equations: Assign variables to unknowns and write equations based on the scenario.

• Solving: Use substitution or elimination methods.

• Example: If $x + y = 10$ and $2x - y = 3$, solve for $x$ and $y$.

Summary Table: Key Concepts and Applications

Topic	Key Concept	Formula/Example
Absolute Change	Difference between values	$\text{Final} - \text{Initial}$
Relative Change	Proportional difference	$\frac{\text{Change}}{\text{Initial}}$
Linear Function	Slope-intercept form	$y = mx + b$
Exponential Function	Growth/Decay	$f(x) = ab^x$
Probability	Likelihood of event	$P(E) = \frac{\text{Favorable}}{\text{Total}}$
Future Value	Compound interest	$A = P(1 + \frac{r}{n})^{nt}$

Additional info: Some context and examples have been inferred and expanded for completeness and clarity, based on standard College Algebra curriculum.