exam review math

Strand A – Characteristics of Functions

1. Understanding Functions

  • Definition: A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

  • Identifying Functions: A relation is a function when each x value corresponds to only one y value.

  • Types of Representations: Functions can be represented algebraically (equations), graphically (graphs), and numerically (tables).

2. Quadratic Functions:

  • Zeros: Solutions to the equation when the function equals zero;

  • Maximum/Minimum: The highest or lowest point of the parabola; determined by vertex calculations.

  • Real-world Applications: Quadratic functions model scenarios such as projectile motion (e.g., basketball shot).

3. Simplifying Expressions

  • Polynomial, Radical, Rational Expressions: Understanding equivalences and simplifications.

4. Expanding and Simplifying Expressions

  • Examples:

  • a) 4(𝑥 − 𝑦)² − (2𝑥 + 3𝑦)(𝑥 − 4𝑦)

  • b) -2𝑥²(3𝑥³ − 7𝑥 + 2) − 𝑥³(5𝑥³ + 2𝑥 − 8)

  • c) (4𝑥 − 1)(5𝑥 + 2)(𝑥 − 3)

  • d) (2𝑥²𝑦 − 3𝑥𝑦²)(4𝑥𝑦² + 5𝑥²𝑦)

5. Factoring Expressions

  • Examples:

  • a) 50𝑥² − 72

  • b) 10𝑎² + 𝑎 − 3

  • c) 𝑎⁴ − 16

  • d) (𝑚 − 𝑛)² − (2𝑚 + 3𝑛)²

  • e) 𝑎² − 𝑏² − 18𝑎 + 81

  • f) 2(𝑎 − 𝑏)² + 5(𝑎 − 𝑏) + 3

Page 2: Simplification and Restrictions

1. Simplifying Rational Expressions

  • Examples:

  • a) 4𝑎²𝑏 / 5𝑎𝑏³ ÷ 6𝑎²𝑏³ × 𝑎² / 𝑎𝑏³

  • b) 4𝑥 / (6𝑥² + 13𝑥 + 6) − 3𝑥 / (4𝑥² − 9)

  • c) 21𝑝 − 3𝑝² / 16𝑝 + 4𝑝² ÷ (14 − 9𝑝 + 𝑝²) / (12 + 7𝑝 + 𝑝²)

  • d) (5𝑚 − 𝑛) / (2𝑚 + 𝑛) − (4𝑚² − 4𝑚𝑛 + 𝑛²) / (4𝑚² − 𝑛²) ÷ (6𝑚² − 𝑚𝑛 − 𝑛²) / (3𝑚 + 15𝑛)

  • e) (𝑝 + 1) / (𝑝² + 2𝑝 − 35) + (𝑝² + 𝑝 − 12) / (𝑝² − 2𝑝 − 24) × (𝑝² − 4𝑝 − 12) / (𝑝² + 2𝑝 − 15)

2. Simplifying Square Roots

  • Examples:

  • a) √45 + √80

  • b) 2√20 − 3√125 + 3√80

  • c) 7√3 − 2√6 + 5√6 − 3√3

  • d) (2√3 + 5)²

  • e) (3√6 + 5√2)(3√6 − 5√2)

  • f) 2√3

Page 3: Applications of Quadratics

1. Basketball Shot Model

  • Equation: ℎ(𝑑) = −0.125𝑑² + 4𝑑 + 2.5

  • Findings:

  • Maximum Height: Evaluate the vertex formula to find maximum h.

  • Horizontal Distance: Calculate d when h is at maximum.

  • Initial Height: h(0) gives height at release.

2. Triangle Area Maximization

  • Problem: Base + Height = 21 cm; maximize area.

3. Rectangular Pool and Deck Problem

  • Area: Pool area + deck area = 135 m²; find deck width.

Page 4: Intersections and Values of Functions

1. Intersection of Functions

  • Functions:

    • 𝑦 = −2𝑥² − 5𝑥 + 20 and 𝑦 = 6𝑥 - 1

  • Tasks:

  • Determine points of intersection and solve.

2. Non-Intersecting Function Values

  • Find values of k where 4𝑥 + 𝑘 does not intersect the parabola −3𝑥² − 𝑥 + 4.

3. Basic Function Definitions

  • What is a Function?: Define and identify functions based on outputs.

  • Domain and Range Analysis:

    • Various functions given specific outputs for specified inputs.

Page 5: Function Evaluations

1. Function Evaluations

  • Evaluate for given functions:

    • 𝑓(−1), 𝑔(1/2), 𝑓(3) + 𝑔(4)

    • Solve 𝑓(𝑥) = 𝑔(𝑥), 𝑓(𝑥) = 9.

2. Graph Transformations of Functions

  • Outline graph transformations from base function y = 𝑓(𝑥).

Page 6: Transformations and Inverses

1. Transformations of Base Functions

  • Parent Functions: y = √𝑥, y = 𝑥², y = 1/𝑥.

  • Determine mapping rules, transformations, and domain/range assessments.

2. Function Inverses

  • Steps for determining inverses and plotting graphs.

Page 8: Exponential Functions

1. Evaluating Powers

  • Work on simplifying expressions with exponential notation and rational exponents.

2. Algebraic Manipulations

  • Create single power forms and evaluate various phrases containing exponential expressions.

3. Exponential Growth and Decay Applications

  • Half-life problems.

Page 11: Discrete Functions and Sequences

1. Recursive Sequences

  • Connect recursive sequences with Pascal’s triangle.

2. Arithmetic and Geometric Sequences

  • Explore properties and applications in real-world contexts (e.g., financial calculations).

3. Series Summation

  • Compute sums for arithmetic and geometric series.

Page 14: Trigonometric Functions

1. Trigonometric Ratios and Angles

  • Evaluate and prove identities relevant to angles recorded in standard position.

2. Exact Values of Trig Functions

  • Calculate values for common angles and as derived from right triangle relationships.

3. Proving Identities

  • Steps involved in verifying trigonometric identities.

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