This page focuses on horizontal transformations and their effects on function graphs, particularly with scaling factors.

Vocabulary: Horizontal stretch occurs when |c| > 1 in f(cx), while compression happens when |c| < 1.

Definition: For f(cx), when c > 1, the graph compresses horizontally by a factor of 1/c.

Example: For f$x$ = x², f(½x) results in a horizontal stretch, while f(2x) creates a horizontal compression.

The content explores vertical transformations and introduces the concept of increasing and decreasing functions.

Definition: A function f is increasing on an interval I if f(x₁) < f(x₂) when x₁ < x₂ in I.

Example: For y = cf$x$, when c > 1, the graph stretches vertically by a factor of c.

Highlight: Vertical transformations affect the y-coordinates of a function's graph.

This section details the analysis of function behavior, particularly focusing on intervals where functions increase or decrease.

Example: A function can have multiple intervals where it's increasing or decreasing:

• Increasing on $∞,-1$
• Decreasing on $-1,4$
• Increasing on [4,∞)

Vocabulary: Relative extrema are points where the function changes from increasing to decreasing or vice versa.

The page covers methods for identifying domain and range of functions from their graphs.

Definition: Domain is all possible x-values, while range is all possible y-values of a function.

Example: For the given function:

• Domain: (-∞,∞)
• Range: [-10,∞)

Highlight: Relative maximum and minimum points help determine the range of a function.

This section introduces polynomial functions and methods for finding their zeros.

Definition: A polynomial function has the form P$x$ = anx^n + an-1x^n-1 + ... + a₁x + a₀

Example: Solving x² + 4 = 0 yields complex roots ±2i

Vocabulary: Zeros or roots are x-values where P$x$ = 0

The page explains the Factor Theorem and its applications in polynomial analysis.

Definition: If $x-a$ is a factor of polynomial P$x$, then P$a$ = 0

Example: For P$x$ = x² - 3x + 2, testing if $x-2$ is a factor by evaluating P(2)

Highlight: The Factor Theorem provides a method to verify potential factors of polynomials.

This final page covers the Remainder Theorem and its applications in polynomial division.

Definition: When a polynomial P$x$ is divided by $x-a$, the remainder equals P$a$

Example: Solving 2x³ + 7x² - 4x² - 27x - 18 = 0 using factoring techniques

Highlight: The Remainder Theorem simplifies the process of polynomial division and factor verification.

This page introduces fundamental concepts of function transformations using f$x$ = x². The content explores various ways functions can be transformed through shifts and reflections.

Definition: Function transformations are ways to manipulate the graph of a function while maintaining its basic shape.

Example: For y = f$x+2$, the graph shifts 2 units left from the original function.

Highlight: Key transformations covered include:

• Vertical shifts: y = f$x$ ± k
• Horizontal shifts: y = f(x ± h)
• Reflections over x and y axes: y = -f$x$ and y = f$-x$