Math 119 Final Exam Concept List: College Algebra Study Guide

Review of Algebra

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 percentage of the original value.

• Absolute Change: $\text{Absolute Change} = \text{New Value} - \text{Original Value}$

• Relative Change: $\text{Relative Change} = \frac{\text{Absolute Change}}{\text{Original Value}} \times 100\%$

• Example: If a population increases from 200 to 250, the absolute change is 50, and the relative change is $\frac{50}{200} \times 100\% = 25\%$.

Index Numbers

Index numbers are used to compare values over time or between different items, often as a ratio or percentage relative to a base value.

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

• Example: If the base year price is $50 and the current price is $60, the index is$\frac{60}{50} \times 100 = 120$.

Percentages and Ratios

Percentages and ratios are common ways to express relationships between numbers.

• Percent: A ratio expressed as a fraction of 100.

• Ratio: A comparison of two quantities, often written as $a:b$ or $\frac{a}{b}$.

• Example: 25% is equivalent to the ratio $\frac{25}{100}$ or $1:4$.

Equations & Inequalities

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true.

• Linear Equations: $ax + b = 0$

• Quadratic Equations: $ax^2 + bx + c = 0$

• Example: Solve $2x + 3 = 7$; $x = 2$.

Solving Inequalities

Inequalities compare two expressions and use symbols such as $<, >, \leq, \geq$.

• Example: Solve $2x - 5 > 3$; $x > 4$.

Functions

Definition and Evaluation of Functions

A function is a relation that assigns exactly one output for each input.

• Notation: $f(x)$

• Evaluating: Substitute the input value into the function.

• Example: If $f(x) = 2x + 1$, then $f(3) = 7$.

Domain and Range

The domain is the set of all possible input values, and the range is the set of all possible output values.

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

Linear Functions

Linear functions have the form $f(x) = mx + b$.

• Slope ($m$): Rate of change of the function.

• Y-intercept ($b$): Value when $x = 0$.

• Example: $f(x) = 3x + 2$ has slope 3 and y-intercept 2.

Graphs of Equations

Graphing Linear and Exponential Functions

Graphing helps visualize the behavior of functions.

• Linear Function: Straight line; slope and intercept determine position.

• Exponential Function: $f(x) = ab^x$; shows rapid growth or decay.

• Example: $f(x) = 2^x$ is an exponential growth function.

Polynomial and Rational Functions

Polynomial Functions

Polynomials are sums of terms with non-negative integer exponents.

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0$

• Example: $f(x) = x^2 - 4x + 3$

Rational Functions

Rational functions are ratios of polynomials.

• General Form: $f(x) = \frac{P(x)}{Q(x)}$

• Domain: All real numbers except where $Q(x) = 0$.

Exponential & Logarithmic Functions

Exponential Growth and Decay

Exponential functions model situations where quantities grow or decay at a constant percentage rate.

• Exponential Growth: $y = a(1 + r)^t$

• Exponential Decay: $y = a(1 - r)^t$

• Example: Population doubling every period: $y = 100 \times 2^t$

Evaluating and Creating Exponential Functions

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

• Example: For $f(x) = 3^x$, $f(0) = 1$, $f(1) = 3$, $f(2) = 9$.

Systems of Equations & Matrices

Multi-Step Problems and Applications

Systems of equations can be used to solve real-world problems involving multiple variables.

• Example: Solving for two unknowns using two equations.

Sequences, Series, & Induction

Future Value and Present Value

These concepts are used in finance to determine the value of investments over time.

• Future Value Formula: $FV = PV(1 + r)^n$

• Present Value Formula: $PV = \frac{FV}{(1 + r)^n}$

• Example: $1000 invested at 5% for 3 years:$FV = 1000(1.05)^3$

Combinatorics & Probability

Counting Principles

Counting principles help determine the number of possible outcomes in a scenario.

• Multiplication Principle: If one event can occur in $m$ ways and another in $n$ ways, both can occur in $m \times n$ ways.

• Example: 3 shirts and 2 pants: $3 \times 2 = 6$ combinations.

Probability

Probability measures the likelihood of an event occurring.

• Probability Formula: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Examples: Calculating probability for independent, dependent, overlapping, and non-overlapping events.

Listing Events and Calculating Outcomes

• List Possible Events: Enumerate all outcomes for a scenario.

• Calculate Outcomes: Use counting principles to find the total number of outcomes.

Summary Table: Key Concepts and Applications

Topic	Key Concept	Example/Application
Absolute/Relative Change	Difference and percent change	Population growth
Index Numbers	Comparing values to a base	Price index
Linear Functions	Slope and intercept	Cost modeling
Exponential Functions	Growth/decay	Compound interest
Probability	Likelihood of events	Coin toss, dice roll
Counting	Number of outcomes	Possible outfits

Additional info: Some content was inferred and expanded for completeness, including standard formulas and examples for each topic listed in the concept list.