Russia 9 класс Algebra (advanced)

Chapters: 5

1. Numbers and calculations

nth roots and rational exponents

Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.

Transformations

Coming soon

Higher-degree equations · Non-linear systems · Inequalities

Simultaneous Equations – Simultaneous equations are two or more equations that must be true at the same time. Their solution is the set of values that fits every equation. On a graph, each solution is a point where the graphs meet. With one straight line and one curve, put the line into the curve (substitution) to get a quadratic; its discriminant tells you if there are 2, 1 or 0 meeting points. With three linear equations, eliminate one letter at a time.
Quadratic Inequalities – A quadratic inequality asks where ax² + bx + c is above zero (> 0) or below zero (< 0). Find the roots, picture the parabola, and read the answer from the graph. When a > 0 the curve is below zero between the roots and above zero outside them. If there are no real roots (D < 0), the curve is always on one side of the x-axis.

Function properties · Quadratic function

Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).

Sequences · Progressions · Mathematical induction

Sequences and Progressions – A sequence is a list of numbers in a fixed order. We can describe it with a recursive rule (how to get the next term from the last one) or an explicit rule (a formula for term n). In an arithmetic progression (AP) we add the same number every time. In a geometric progression (GP) we multiply by the same number every time. Fractals and the Tower of Hanoi are fun patterns that hide these rules.
Arithmetic Progressions – An arithmetic progression (AP) is a list of numbers where each term is made by adding the same fixed number d (the common difference) to the term before. With first term a: nth term aₙ = a + (n − 1)d; sum of the first n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
Proof by Mathematical Induction – Mathematical induction proves that a statement P(n) is true for every natural number n. Step 1 (base case): show P(1) is true. Step 2 (inductive step): assume P(k) is true for some k, and use it to show P(k + 1) is true. Then, like a line of dominoes, P(1) makes P(2) true, P(2) makes P(3) true, and so on for ever.