College Algebra Study Guide: Numbers, Data, and Graphing

Section 1.1 – Numbers, Data, and Problem Solving

Recognizing Common Sets of Numbers

This topic introduces the fundamental sets of numbers used in algebra and mathematics. Understanding these sets is essential for classifying numbers and solving problems.

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).

• Whole Numbers: Natural numbers plus zero (0, 1, 2, 3, ...).

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).

• Rational Numbers: Numbers that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Irrational Numbers: Numbers that cannot be written as a simple fraction (e.g., $\sqrt{2}$, $\pi$).

• Real Numbers: All rational and irrational numbers.

Example: Classify the number $-\frac{3}{4}$ as rational and real.

Evaluating Expressions Using the Order of Operations

Correctly evaluating arithmetic expressions requires following the order of operations, often remembered by the acronym PEMDAS.

• P: Parentheses

• E: Exponents

• M/D: Multiplication and Division (left to right)

• A/S: Addition and Subtraction (left to right)

Example: Evaluate $2 + 3 \times (4^2 - 1)$.

Solution: $4^2 = 16$, $16 - 1 = 15$, $3 \times 15 = 45$, $2 + 45 = 47$.

Converting Between Scientific and Decimal Notation

Scientific notation is a way to express very large or very small numbers using powers of ten.

• Format: $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.

• Example: $4,500 = 4.5 \times 10^3$

• Decimal to Scientific: Move the decimal point to create a number between 1 and 10, count the moves for the exponent.

Performing Operations Using Scientific Notation

Operations with numbers in scientific notation follow specific rules for multiplication and division.

• Multiplication: Multiply the coefficients and add the exponents: $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$

• Division: Divide the coefficients and subtract the exponents: $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$

Example: $(2 \times 10^4) \times (3 \times 10^2) = 6 \times 10^6$

Calculating Percent Change

Percent change measures the relative increase or decrease between two values.

• Formula: $\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$

• Application: Used in finance, science, and everyday calculations.

Example: If a price increases from $50 to $60, percent change is$\frac{60-50}{50} \times 100 = 20\%$.

Section 1.2 – Visualizing and Graphing Data

Analyzing One-Variable Data

One-variable data analysis involves summarizing and interpreting data sets with a single variable.

• Mean: The average value, $\text{Mean} = \frac{\sum x_i}{n}$

• Median: The middle value when data is ordered.

• Mode: The most frequently occurring value.

Example: For data set {2, 4, 4, 5, 7}, mean is $\frac{2+4+4+5+7}{5} = 4.4$.

Finding the Domain and Range of a Relation

The domain and range describe the set of possible input and output values for a relation or function.

• Domain: All possible input values (usually $x$ values).

• Range: All possible output values (usually $y$ values).

Example: For $f(x) = x^2$, domain is all real numbers, range is $y \geq 0$.

Calculating Distance Between Two Points

The distance formula calculates the length between two points in the coordinate plane.

• Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example: Between $(1,2)$ and $(4,6)$: $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Finding the Midpoint Between Two Points

The midpoint formula finds the point exactly halfway between two given points.

• Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Example: Between $(2,3)$ and $(4,7)$: $M = (3, 5)$

Finding the Center and Radius of a Circle

The equation of a circle in standard form reveals its center and radius.

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$

• Radius: $r$

Example: $(x - 2)^2 + (y + 3)^2 = 16$ has center $(2, -3)$ and radius $4$.

Finding the Standard Equation of a Circle

To write the equation of a circle, use the center and radius, or complete the square if given in general form.

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$

• Complete the Square: Rearrange to standard form to identify center and radius.

Example: $x^2 + y^2 - 4x + 6y = 12$ can be rewritten by completing the square.

Graphing Circles and Other Equations with a Calculator

Graphing calculators can plot circles and other relations by entering equations and adjusting viewing windows.

• Scatterplots: Used for visualizing data points.

• Line Graphs: Used for linear relations.

Example: Enter $(x - 1)^2 + (y - 2)^2 = 9$ to graph a circle centered at $(1,2)$ with radius $3$.

Quick Summary Table

Exam Question ID	Objective	Textbook Examples	Key Videos
1.1.13, 1.1.9	Recognize sets of numbers	Example 1 (pp. 2–3)	Classifying Numbers
1.1.31, 1.1.33	Convert between notations	Examples 3–4 (pp. 4–5)	Introduction to Scientific Notation
1.1.53	Perform operations with scientific notation	Examples 5–6 (pp. 5–6)	Computing in Scientific Notation with a Calculator I
1.1.83	Calculate percent change	Example 10 (pp. 7–8)	Problem Solving Examples I
1.1.89	Solve applications (percent & notation)	Examples 8–9 (pp. 7–11)	Problem Solving Examples I
1.2.7	Analyze one-variable data	Example 1 (p. 13)	Analyzing a List of Temperatures
1.2.15	Find the domain and range of a relation	Example 2 (p. 14)	Finding the Domain and Range of a Relation
1.2.29	Calculate distance	Examples 5–6 (pp. 16–17)	See the Concept: Distance Between Two Points
1.2.53	Find the midpoint	Examples 7–8 (pp. 18–19)	Finding the Midpoint Between Two Points I
1.2.77	Find center and radius of a circle	Example 9 (pp. 19–20)	Finding the Center and Radius of a Circle I & II
1.2.85	Find the standard equation of a circle	Examples 10–13 (pp. 20–23)	Finding the Center and Radius of a Circle I & II
1.2.101	Graph equations with calculator	Examples 14–16 (pp. 23–24)	Graphing a Function by Hand; Graphing a Relation