Algebra II Individual May Assessment Study Guide

Trigonometric Function Graphs and Transformations

• Core Objectives: Mastery of the visual and mathematical properties of sine and cosine functions. This includes the ability to understand and perform transformations on basic trigonometric graphs.

• Primary Functions:

  • $y = \sin(x)$

  • $y = \cos(x)$

• Required Skills:

  • Recognizing the parent graphs of sine and cosine.

  • Applying transformations including vertical shifts, horizontal shifts (phase shifts), vertical stretches (amplitude changes), and horizontal stretches or compressions (period changes).

• Example Practice Problems:

  • The following problems are identified as key practice for these skills: 7-24, 7-36, 7-65, 7-66, 7-116, 7-117, 7-129, 7-130, 7-144, 7-145, 7-147, 7-158, 7-159, 7-160, and 7-167.

Multiple Representations of Polynomials

• Core Objectives: Ability to move seamlessly between different mathematical forms of a polynomial function.

• Interchangeable Representations:

  • Graphical Representation: Identifying the curve on a Cartesian plane.

  • Algebraic Representation: Understanding the equation in both standard and factored forms.

  • Tabular Representation: Interpreting $(x, y)$ data sets that define the polynomial.

  • Verbal/Contextual Description: Describing the polynomial based on its characteristics.

• Example Practice Problems:

  • Review problems 8-56, 8-105, and 8-107.

  • Consult problem CL 8-180.

  • Complete the Chapter 8 Check In Packet, specifically problems 1 through 4.

Key Features of Polynomial Functions

• Core Objectives: Identifying and analyzing the specific structural components and behaviors of polynomial functions.

• Essential Features to Determine:

  • Roots: Identifying both Real Roots (x-intercepts) and Complex Roots (imaginary components).

  • Types of Roots: Distinguishing between single, double (tangent to x-axis), and triple roots based on multiplicity.

  • Degree: Determining the highest power of the variable, which dictates the maximum number of roots.

  • Orientation: Determining the direction of the graph based on the leading coefficient.

  • End Behavior: Describing the direction the function approaches as $x \rightarrow \infty$ and $x \rightarrow -\infty$.

• Example Practice Problems:

  • Reference problems 8-37, 8-39, 8-104, and 8-143.

  • Refer to the Chapter 8 Check In Packet, problems 1 through 4.

Arithmetic Operations with Complex Numbers

• Core Objectives: Performing standard mathematical operations on numbers containing an imaginary unit $i$, where $i = \sqrt{-1}$.

• Required Proficiency:

  • Addition: Combining the real parts and the imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$.

  • Subtraction: Distributing the negative sign and combining like parts: $(a + bi) - (c + di) = (a - c) + (b - d)i$.

  • Multiplication: Using the FOIL method or distributive property and simplifying by substituting $i^2 = -1$.

• Example Practice Problems:

  • Utilize problems 8-70, 8-72, 8-76, 8-87, 8-111, and 8-176.

  • Consult the Chapter 8 Check In Packet, specifically problems 5, 6, and 8.

Investigation of Functions vs. Non-Functions

• Core Objectives: Categorizing mathematical relations and understanding the criteria that define a function.

• Key Concepts:

  • Determining if an input ($x$) corresponds to exactly one output ($y$).

  • Applying the Vertical Line Test on graphs.

  • Investigating logarithmic relationships and their properties.

• Example Practice Problems and Materials:

  • Review problems 6-27, 6-98, 6-116, and 6-158.

  • Study the "Investigating Logs Worksheet".

  • Review the general "Investigating Worksheet".

Unit Circle Fundamentals (April Re-assessment Content)

• Objective 1: Locating Angles:

  • Ability to accurately place angles on the unit circle in standard position.

  • Practice Problems: 7-25, 7-56, 7-62, 7-93, 7-104ab.

• Objective 2: Sine and Cosine Values:

  • Identifying Exact Values (e.g., using special right triangles like $45-45-90$ or $30-60-90$).

  • Identifying Approximate Values (decimal representations).

  • Understanding coordinates as $(\cos(\theta), \sin(\theta))$; however, note that tangent values are excluded from this specific assessment objective.

  • Practice Problems: 7-53, 7-54, 7-55, 7-64, 7-78, 7-90, 7-91, 7-104c, 7-162acd, 7-169, 7-170abde, and 7-171.

Important Assessment Note

• Instructional Guidance: The individual problems listed (such as 7-24 or 8-37) are provided as examples and pre-assessment study tools. They represent the concepts to be tested but are not intended to be the exact problems that will appear on the May Assessment.