Mathematics Concepts Overview

Study Notes

A function maps each input value (from its domain) to exactly one output value (in its range).
A vertical line test can visually determine if a graph represents a function: If any vertical line intersects the graph more than once, it's not a function.

The primary characteristic is a constant rate of change (slope); it represents a straight line when graphed.
The general form is often expressed as y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

It solves quadratic equations (equations of the form ax² + bx + c = 0).
It provides the values of x (the roots or zeros) where the parabola intersects the x-axis.

It quantifies the amount of dispersion or spread in a dataset around its mean (average).
A larger standard deviation indicates greater variability; a smaller one implies data points are clustered closer to the mean.

The absolute value function |x| does not satisfy the property of being strictly increasing or strictly decreasing over its entire domain. It's increasing for positive x and decreasing for negative x.

Vast, multi-ethnic empire ruled by Tsars.
Autocratic regime with limited political freedoms.
Primarily agrarian economy with significant industrial potential concentrated in specific regions.
Social inequalities were prevalent; a large peasant class coexisted with a wealthy aristocracy and growing industrial working class.
Series of social and political reforms attempted in the late 19th and early 20th centuries, but resisted by conservative forces.
Political instability characterized by revolutionary movements and failed attempts at modernization.
World War I significantly weakened the Tsarist regime and fueled revolutionary sentiment.

Rapid industrialization: Collectivized agriculture provided resources, and the Five-Year Plans prioritized heavy industry.
Increased literacy rates and expansion of education: Significant improvements in access to education across the population.
Advances in science and technology: Focused investment led to notable advancements, particularly in space exploration and military technology.
Women's rights: Significant legal and social advancements were introduced, although full equality remained a challenge.
Significant territorial expansions after WWII.
Creation of a powerful military capable of contending with major global powers.

Collectivization of agriculture caused widespread famine and food shortages.
Repressive political system: Lack of political freedoms, widespread use of secret police and forced labor camps (gulags).
Cult of personality around Stalin led to arbitrary decision-making and suppression of dissent.
Economic inefficiency: The centrally planned economy faced difficulties in adapting to changing needs and technological advancement.
Significant human rights violations: widespread political purges, executions, and deportations.
Social inequalities persisted despite the rhetoric of equality.
Severe shortages of consumer goods were common.