Exam 1

Section 1.4: Algebraic Expressions and Radicals

Understanding Radicals and Their Properties

Radicals are expressions that involve roots, such as square roots, cube roots, etc. In precalculus, simplifying and manipulating radical expressions is a foundational skill.

• Radical Notation: The square root of a number a is written as $\sqrt{a}$.

• Properties of Radicals:

  • $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

  • $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

  • $\sqrt{a^2} = |a|$

• Simplifying Radicals: Factor the radicand to extract perfect squares.

  • Example: $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

Additional info: The notes contain several radical expressions, indicating practice with simplification and manipulation.

Section 1.5: More on Algebraic Expressions

Combining and Simplifying Expressions

This section likely continues with algebraic manipulation, including combining like terms and working with rational expressions.

• Combining Like Terms: Add or subtract terms with the same variable and exponent.

• Rational Expressions: Expressions of the form $\frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials.

• Example: $\frac{2x}{x^2} + \frac{3}{x^2} = \frac{2x + 3}{x^2}$

Additional info: The original notes are sparse, but these are standard topics in early precalculus chapters.

Section 2.1: Functions and Their Graphs

Introduction to Functions

A function is a relation that assigns each input exactly one output. Functions are often represented by equations, tables, or graphs.

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

• Domain and Range:

  • Domain: Set of all possible input values ($x$).

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

• Graphing Functions: The graph of a function is a visual representation of all $(x, f(x))$ pairs.

• Example: The graph of $f(x) = x^2$ is a parabola opening upwards.

Common Function Graphs

Several graphs are shown, representing different types of functions commonly studied in precalculus.

• Linear Function: $f(x) = mx + b$ (straight line)

• Quadratic Function: $f(x) = ax^2 + bx + c$ (parabola)

• Cubic Function: $f(x) = ax^3 + bx^2 + cx + d$ (S-shaped curve)

• Absolute Value Function: $f(x) = |x|$ (V-shaped graph)

• Square Root Function: $f(x) = \sqrt{x}$ (starts at origin, increases slowly)

• Reciprocal Function: $f(x) = \frac{1}{x}$ (hyperbola, undefined at $x=0$)

Graph Features and Transformations

Understanding how changes to a function's equation affect its graph is essential.

• Vertical Shifts: $f(x) + k$ shifts the graph up by $k$ units.

• Horizontal Shifts: $f(x - h)$ shifts the graph right by $h$ units.

• Reflections: $-f(x)$ reflects the graph over the $x$-axis.

• Stretching/Compressing: $af(x)$ stretches vertically if $|a| > 1$, compresses if $0 < |a| < 1$.

• Example: The graph of $f(x) = (x-2)^2 + 3$ is a parabola shifted right 2 units and up 3 units.

Graph Identification and Analysis

Recognizing Function Types from Graphs

Being able to identify the type of function from its graph is a key skill in precalculus.

• Linear: Straight line, constant rate of change.

• Quadratic: Parabola, symmetric about its vertex.

• Cubic: S-shaped curve, may have inflection point.

• Absolute Value: V-shaped, vertex at origin or shifted.

• Square Root: Starts at a point, increases slowly.

• Reciprocal: Two branches, undefined at $x=0$.

Table: Common Functions and Their Graphs

Function	Equation	Graph Shape
Linear	$f(x) = mx + b$	Straight line
Quadratic	$f(x) = ax^2 + bx + c$	Parabola
Cubic	$f(x) = ax^3 + bx^2 + cx + d$	S-shaped curve
Absolute Value	$f(x) = |x|$	V-shaped
Square Root	$f(x) = \sqrt{x}$	Starts at origin, increases slowly
Reciprocal	$f(x) = \frac{1}{x}$	Hyperbola, undefined at $x=0$

Practice and Review

Sample Questions and Graphs

The file contains several graphs and radical expressions, likely as part of a review or exam practice. Students should be able to:

• Simplify radical expressions using properties of roots.

• Identify function types from their equations and graphs.

• Describe transformations applied to basic functions.

• Sketch graphs of common functions and their transformations.

Additional info: The presence of "Answer Key" and "Review" suggests these are exam review questions, focusing on algebraic manipulation and graph identification.