Ukraine 9 клас Algebra
Chapters: 4
1. Inequalities
Numerical inequalities · Linear inequalities in one variable · Systems of linear inequalities · Quadratic inequalities
- Inequalities: Rules, Intervals and Solving Them – An inequality says one amount is bigger or smaller than another, using <, >, ≤ or ≥. On a number line, the smaller number is on the left. You may add or subtract the same number on both sides, and multiply or divide by the same positive number, and the sign stays. If you multiply or divide by a negative number, the sign flips. The answer is usually a whole set of numbers, written as an interval such as (−∞, 4]. A quadratic inequality is solved from its roots and the shape of its graph. |x| < a means −a < x < a. Some inequalities are true for every number, like x² ≥ 0 and the AM–GM inequality.
- Linear Inequalities in One Variable – An inequality compares two expressions with <, >, ≤ or ≥. A linear inequality in one variable looks like ax + b < c. Its answer is usually a whole range of numbers, not one number. We solve it like an equation: we may add or subtract the same number on both sides, and multiply or divide by the same positive number. If we multiply or divide by a negative number, the sign must flip. We show the answer on a number line: a hollow dot for < or > (end not included) and a filled dot for ≤ or ≥ (end included), with the shaded part showing all solutions. Double inequalities like −1 ≤ x < 3 give a piece of the line. If x must be a natural number or an integer, only the whole numbers in that range count.
- 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.
2. Systems of equations
Systems of two equations in two variables
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
3. Functions
Function and its properties · Quadratic function · Number sequences · Arithmetic and geometric progressions · Finance with sequences
- Properties of Functions: Reading a Graph Like a Story – A function gives exactly one output f(x) for each input x. From its graph we read the domain (allowed x), the range (y values reached), the zeros (where f(x) = 0), the intervals where f is positive or negative, where it increases or decreases, its maximum and minimum, and whether it is even (mirror in the y-axis) or odd (half-turn about the origin).
- 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 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.
- Compound Interest – Interest is money paid for using money. Simple interest is paid only on the starting amount (the principal), so it grows by the same step every year. Compound interest is paid on the principal plus all the interest earned so far, so the steps get bigger: interest earns interest. The amount after n years is A = P(1 + r/100)ⁿ. If interest is added k times a year, use rate r/k and k × n periods: A = P(1 + r/(100k))^(kn). Adding it every instant gives continuous compounding, A = Pe^(rt). The same idea run backwards gives depreciation: A = P(1 − r/100)ⁿ. Compound growth is exponential growth.
4. Mathematical tasks and real-world processes
Systems as models · Combinatorics basics · Random events and probability · Introduction to statistics · Story and competence problems
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
- Statistics: Asking Questions, Averages and Stacked Bar Graphs – Statistics starts with a question that has many possible answers. We collect data, organise it, show it in a graph, and find one number that sums it up. The mean is the fair share, the median is the middle value, and the mode is the most common value. A weighted average gives more importance to some values. Stacked and 100% stacked bar graphs show how a whole is split into parts.
- Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.