Lernzettel: Mastering Algebra and Trigonometry Fundamentals

📋 Course Outline

1. Algebra and quadratics
2. Functions and graphing
3. Trigonometry
4. Probability and combinatorics

📖 1. Algebra and quadratics

🔑 Key Concepts & Definitions

• Surds : Surds are radicals that cannot be simplified into rational powers, such as square roots that remain non-rational after simplification.
• Discriminant : The discriminant is a value computed from a quadratic that predicts how many real solutions the quadratic has.
• Completing the square : Completing the square is a method that rewrites a quadratic in a squared form to reveal key features like the vertex.

📝 Essential Points

• Simplifying surds involves extracting perfect-square factors so the radical part becomes as simple as possible.
• For a quadratic $ax^2+bx+c$, the discriminant is b^2-4ac and its sign determines whether there are 2, 1, or 0 real roots.
• Completing the square converts $ax^2+bx+c$ into a form related to $(x-h)^2$ so the axis of symmetry and vertex can be read off.

💡 Memory Hook

Discriminant sign: negative means no real x-intercepts, zero means one repeated root, positive means two distinct roots.

📖 2. Functions and graphing

🔑 Key Concepts & Definitions

• Functions and relations : A relation pairs inputs and outputs, while a function assigns exactly one output to each input.
• Domain and range : Domain is the set of allowed inputs, and range is the set of resulting outputs produced by the relation or function.
• Piecewise functions : Piecewise functions define different rules on different parts of the domain using cases.

📝 Essential Points

• Set notation can be used to describe domain and range sets as explicit collections of allowed inputs or possible outputs.
• Graphing reciprocal functions uses the idea that the function is undefined where the denominator is 0, creating asymptotic behavior near those x-values.
• Graphing parabolas, cubics, hyperbolas, and circles uses the defining equation to determine key points and shape features on the coordinate plane.

💡 Memory Hook

Piecewise means “different formula on different x-intervals.”

📖 3. Trigonometry

🔑 Key Concepts & Definitions

• Radians : Radians measure angles by comparing arc length to the radius, with $2\pi$ radians completing one full turn.
• Sine, cosine, tangent : Sine, cosine, and tangent are trigonometric ratios that describe angles using side lengths in a right triangle or corresponding unit-circle values.
• Exact values of sin, cos, tan : Exact values are specific trigonometric outputs for standard angles written exactly rather than approximately.

📝 Essential Points

• Trig function graphs are periodic and reflect how sine and cosine repeat their values, while tangent repeats with vertical asymptotes where it is undefined.
• Solving trig functions typically uses known trig identities or inverse-trig reasoning to find all angles within a chosen interval.
• Applications of trig use trigonometric ratios to solve real measurements like lengths and angles from right-triangle or modeled geometry.

💡 Memory Hook

Sine ↔ y, cosine ↔ x, tangent ↔ y/x on the unit circle idea.

📖 4. Probability and combinatorics

🔑 Key Concepts & Definitions

• Pascal's triangle : Pascal’s triangle is an arrangement of numbers where rows generate the binomial coefficients used in expansions and binomial probabilities.
• Binomial distribution : The binomial distribution models the number of successes in a fixed number of independent trials with the same success probability.
• Conditional probability : Conditional probability is the probability of an event occurring given that another event has already happened.

📝 Essential Points

• The addition rule gives probabilities for the union of events, with care needed when events overlap.
• Independent events satisfy that the occurrence of one does not change the probability of the other.
• Combinations count how many groups can be formed without regard to order, often used in counting for probability problems.

💡 Memory Hook

Independence: P(A|B)=P(A), so knowing B doesn’t change A.

⚠️ Common Pitfalls & Confusions

1. Students often confuse domain with range when describing function outputs and inputs.
2. Students sometimes treat relations as functions even when an input maps to multiple outputs.
3. A common error with surds is failing to extract perfect-square factors before simplifying the radical.
4. Students may mix up degree and radian measures, leading to wrong trig graph angles.
5. When solving trig equations, students often forget that trig solutions repeat periodically and must include all required angles.
6. With reciprocal functions, students may incorrectly graph points at x-values where the function is undefined (denominator 0).

✅ Exam Checklist

1. Simplify surds by extracting perfect-square factors and leaving the radical in simplest form.
2. Expand and factorise quadratics to put expressions into solvable or standard forms.
3. Use the discriminant to determine how many real roots a quadratic has.
4. Apply completing the square to rewrite a quadratic into squared form for identifying features.
5. Identify domain and range from a given relation or function and express them using set notation.
6. Determine whether a mapping is a function or a relation by checking each input has exactly one output.
7. Graph key types of functions, including parabolas, cubics, reciprocal functions, hyperbolas, and circles.
8. Define and use piecewise functions to describe different rules over different domain parts.
9. Convert and use radians when working with trig functions.
10. Graph sin, cos, and tan functions and interpret their periodicity and where they are undefined.
11. Evaluate exact values of sin, cos, and tan for standard angles rather than using decimals.
12. Solve trig equations and include all solutions needed for the required interval.
13. Use probability tools including the addition rule, conditional probability, and independent events.
14. Compute binomial probabilities using Pascal’s triangle ideas and apply combinations for counting without order.