Unit 1a Study Guide

Fundamental Concepts of Algebra

Order of Operations

The order of operations is essential for evaluating mathematical expressions correctly. The standard order is Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right), often remembered by the acronym PEMDAS.

• Parentheses: Calculate expressions inside parentheses first.

• Exponents: Evaluate powers and roots next.

• Multiplication/Division: Perform these operations from left to right.

• Addition/Subtraction: Complete these last, also from left to right.

• Example: $2(4+3) = 2 \times 7 = 14$

Evaluating Expressions

Apply the order of operations to simplify algebraic expressions and solve equations.

• Combine like terms and use distributive property as needed.

• Example: $3x + 2x = 5x$

Equations and Inequalities

Solving Linear Equations

Linear equations can be solved by isolating the variable using inverse operations.

• Steps:

  1. Combine like terms on each side.

  2. Use addition/subtraction to isolate the variable term.

  3. Use multiplication/division to solve for the variable.

• Example: $2x + 3 = 11 \implies 2x = 8 \implies x = 4$

Translating Word Problems

Translate verbal statements into algebraic equations using key words for operations.

Operation	Key Words	Example	Expression
Addition	increased by, more than	a number increased by 2	$x+2$
Subtraction	decreased by, less than	a number decreased by 2	$x-2$
Multiplication	product, times	the product of 2 and a number	$2x$
Division	quotient, ratio	the quotient of a number and 2	$\frac{x}{2}$
Equals	is, equals	the sum of a number and 2 is 10	$x+2=10$

Graphs

Graphing Linear Equations

Linear equations can be graphed using slope-intercept, point-slope, or standard form.

• Slope-Intercept Form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Standard Form: $Ax + By = C$

• Example: For $y = 2x + 3$, slope $m = 2$, y-intercept $b = 3$.

Vertical and Horizontal Lines

• Horizontal Line: $y = c$ (slope $m = 0$)

• Vertical Line: $x = c$ (undefined slope)

Functions & Graphs

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Set Notation: $\{x | a \leq x \leq b\}$

• Interval Notation: $[a, b]$ for included endpoints, $(a, b)$ for excluded endpoints.

• Example: For $f(x) = x^2$, domain $(-\infty, \infty)$, range $[0, \infty)$.

Types of Numbers

Number Set	Symbol	Definition
Real	R	All numbers on the number line
Rational	Q	Numbers expressible as $\frac{a}{b}$, $b \neq 0$
Irrational	I	Cannot be written as $\frac{a}{b}$
Integers	Z	Whole numbers and negatives
Whole	W	Non-negative integers
Natural	N	Counting numbers

Polynomial Functions

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Factor out the largest common factor.

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

• Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$

• Example: $x^2 - 9 = (x - 3)(x + 3)$

Quadratic Functions

Standard, Vertex, and Intercept Forms

Quadratic functions can be written in several forms, each useful for different purposes.

• Standard Form: $y = ax^2 + bx + c$

• Vertex Form: $y = a(x - h)^2 + k$

• Intercept Form: $y = a(x - p)(x - q)$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Vertex: $(h, k)$ in vertex form, or $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$ in standard form

Graphing Quadratic Functions

To graph a quadratic function, find the axis of symmetry, vertex, and plot additional points using symmetry.

• Opening Direction: If $a > 0$, opens up; if $a < 0$, opens down.

• Domain: $(-\infty, \infty)$

• Range: $[k, \infty)$ if opens up, $(-\infty, k]$ if opens down

Maximum and Minimum Values

The maximum or minimum value of a quadratic function occurs at the vertex.

• Maximum: If $a < 0$, vertex is the maximum.

• Minimum: If $a > 0$, vertex is the minimum.

Applications: Projectile Motion

Quadratic Models in Physics

Projectile motion and falling objects can be modeled using quadratic functions.

• General Form: $h(t) = -16t^2 + vt + h_0$ (height as a function of time)

• Vertex: Gives the maximum height and time to reach it.

• Example: For $h(t) = -16t^2 + 48t + 160$, vertex at $t = 1.5$ sec, $h = 196$ ft.

Exponent Rules

Properties of Exponents

Exponent rules are used to simplify expressions involving powers.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

Set Notation and Interval Notation

Expressing Sets of Numbers

Set notation and interval notation are used to describe domains and ranges of functions.

• Set Notation: $\{x | a < x < b\}$

• Interval Notation: $(a, b)$, $[a, b]$, $(-\infty, b]$, etc.

• Parentheses: Exclude endpoint; Brackets: Include endpoint.

Additional info:

• These notes cover foundational algebra, functions, graphing, and quadratic equations, which are essential for Precalculus and align with chapters 0-4, 19, and 20 of a standard Precalculus curriculum.