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).
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).
Polynomial, Radical, Rational Expressions: Understanding equivalences and simplifications.
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)
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
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)
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
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.
Problem: Base + Height = 21 cm; maximize area.
Area: Pool area + deck area = 135 m²; find deck width.
Functions:
𝑦 = −2𝑥² − 5𝑥 + 20 and 𝑦 = 6𝑥 - 1
Tasks:
Determine points of intersection and solve.
Find values of k where 4𝑥 + 𝑘 does not intersect the parabola −3𝑥² − 𝑥 + 4.
What is a Function?: Define and identify functions based on outputs.
Domain and Range Analysis:
Various functions given specific outputs for specified inputs.
Evaluate for given functions:
𝑓(−1), 𝑔(1/2), 𝑓(3) + 𝑔(4)
Solve 𝑓(𝑥) = 𝑔(𝑥), 𝑓(𝑥) = 9.
Outline graph transformations from base function y = 𝑓(𝑥).
Parent Functions: y = √𝑥, y = 𝑥², y = 1/𝑥.
Determine mapping rules, transformations, and domain/range assessments.
Steps for determining inverses and plotting graphs.
Work on simplifying expressions with exponential notation and rational exponents.
Create single power forms and evaluate various phrases containing exponential expressions.
Half-life problems.
Connect recursive sequences with Pascal’s triangle.
Explore properties and applications in real-world contexts (e.g., financial calculations).
Compute sums for arithmetic and geometric series.
Evaluate and prove identities relevant to angles recorded in standard position.
Calculate values for common angles and as derived from right triangle relationships.
Steps involved in verifying trigonometric identities.