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.