Exam Preparation: Functions 11

Exam Preparation: Functions 11

• Materials Allowed: Pencils (for multiple choice), pens (black or dark blue ink for free response), calculator.
• Materials Prohibited: Electronic devices (cell phones, tablets, etc.), internet access, cameras, food (except bottled water).
• Exam Format: 10 multiple choice, 10 short answer, 7 open-response questions.

Unit 1: Relations and Functions

• Relation: A set of ordered pairs where independent variable values are paired with dependent variable values.
• Function: A relation where each independent variable value corresponds to only one dependent variable value.
• Domain: The set of all possible input values (independent variable).
• Range: The set of all possible output values (dependent variable).
• Vertical Line Test: If a vertical line intersects a graph at more than one point, the graph does not represent a function.
• Function Notation: Using a symbolic representation of a function (e.g., f(x)).
• Parent Functions: Basic functions used to understand transformations. Students should be familiar with the graphs of 5 parent functions.
• Asymptote: A line that a graph approaches but never touches.

Unit 2: Equivalent Algebraic Expressions

• Polynomials: Expressions with variables and constants (addition, subtraction).
• Multiplying Polynomials: Combining terms through multiplication.
• Factoring Polynomials: Breaking down expressions into smaller factors.
• Factoring Quadratics: Techniques for factoring expressions of the form ax² + bx + c (including special cases like perfect square trinomials and difference of squares).
• Rational Expressions: Expressions containing fractions with polynomials in the numerator and denominator. Proficiency in multiplying, dividing, adding, and subtracting these is required.

Unit 3: Quadratic Functions

• Quadratic Functions: Functions represented as an equation with degree 2 (e.g.: f(x) = ax² + bx + c).
• Forms: Students need to be comfortable translating between standard, factored, and vertex forms of quadratic equations.
• Domain and Range: Understanding the possible input (x) and output (y) values for quadratic functions.
• Maximum/Minimum Values: Identifying the highest or lowest point on the graph of a parabola.
• Inverse of a Quadratic Function: Finding the inverse function for a quadratic function.
• Operations with Radicals: Manipulating expressions with roots (square roots, cube roots, etc.).
• Zeros of a Quadratic Function: Finding the x-intercepts of the parabola.
• Solving Quadratic Equations: Techniques used to find the values of x that satisfy a quadratic equation.

Unit 4: Exponential Functions

• Laws of Exponents: Rules for manipulating exponents.
• Rational Exponents: Understanding exponents that are fractions (e.g., x^1/2).
• Exponential Equations: Equations involving exponential functions (solving).
• Properties of Exponential Functions: Characteristics like exponential growth or decay and how they relate to graphs.
• Exponential Growth/Decay Problems: Applications of exponential functions in real-world scenarios.
• Transformations of Exponential Functions: Understanding how constants modify the graph of a general exponential equation like y = a*b^k(x-d)+c.
• Horizontal Asymptotes: Understanding the horizontal lines exponential function approaches.

Unit 5: Trigonometry

• Trigonometric Ratios: Relationships between angles and ratios of sides in a right-angled triangle (sine, cosine, tangent, etc.).
• Trigonometric Equations: Solving equations involving trigonometric functions.
• Sine and Cosine Laws: Rules for finding sides and angles in non-right-angled triangles.
• The Ambiguous Case of the Sine Law: Situations where the Sine Law may produce multiple possible solutions for a triangle.
• Trigonometric Identities: Equations that are true for all possible values of angles.

Unit 6: Properties of Sinusoidal Functions

• Sinusoidal Functions: Graphs characterized by a wave-like pattern.
• Graphs of Sine and Cosine Functions: Understanding standard sine and cosine graphs: Their properties like period, axis of symmetry, and amplitude.
• Transformations of Sinusoidal Graphs: Analyzing how a, k, d, and c affect the graphs of sine and cosine functions (f(x) = a sin(k(x-d) ) + c and transformations).

Unit 7: Sequences and Series

• Arithmetic Sequences: Sequences with a constant difference between consecutive terms. (Calculating common difference, general term, and recursive formulas).
• Geometric Sequences: Sequences with a constant ratio between consecutive terms. (Calculating common ratio, general term, and recursive formulas)
• Arithmetic Series: Sum of terms in an arithmetic sequence.
• Geometric Series: Sum of terms in a geometric sequence.