Positive VS. Negative Correlations - Algebra-1

1. Fundamental Concepts

Correlation refers to the relationship between two variables, describing how they tend to change in relation to each other.
Positive Correlation: When one variable increases, the other variable also tends to increase; conversely, when one variable decreases, the other variable tends to decrease.
Negative Correlation: When one variable increases, the other variable tends to decrease, and vice versa.

The direction of the correlation is determined by the trend of the relationship between the two variables.
A positive correlation indicates a "same direction" change, while a negative correlation indicates an "opposite direction" change.
Correlation does not imply causation, meaning that just because two variables are correlated does not mean one causes the other to change.

Shopping Expense Problem: In a store, the price of 1 kilogram of apples is 8 yuan. Let the weight of apples purchased be x kilograms, and the total cost be y yuan. The relationship between the total cost and the purchased weight can be expressed as \(y = 8x\). When the purchased weight x increases, the total cost y also increases, which is a typical positive correlation.
Running Distance Problem: A person runs at a speed of 200 meters per minute. Let the running time be t minutes, and the running distance be s meters. The relationship between them is \(s = 200t\). As the running time t increases, the running distance s keeps increasing, which is a positive correlation.

Remaining Fuel Problem: A car initially has 50 liters of fuel in its tank, and it consumes 0.1 liters of fuel for every 1 kilometer traveled. Let the distance traveled be x kilometers, and the remaining fuel be y liters. The relationship between the remaining fuel and the distance traveled is \(y = 50 - 0.1x\). When the distance traveled x increases, the remaining fuel y decreases, which is a negative correlation.
Water Temperature Change Problem: A cup of hot water has an initial temperature of 90°C, and when placed in a room temperature environment, its temperature drops by 2°C every minute. Let the placement time be t minutes, and the water temperature be T°C. The relationship between them is \(T = 90 - 2t\). As the placement time t increases, the water temperature T gradually decreases, which is a negative correlation.

Identify the trend: Observe the data points or the equation relating the two variables. If the slope of the line (in a linear relationship) is positive, it's a positive correlation; if the slope is negative, it's a negative correlation.
Analyze real-world scenarios: In practical problems, determine how the two variables behave relative to each other. For example, if studying the relationship between study time and test scores, a positive correlation is likely (more study time leads to higher scores); if studying the relationship between the number of hours spent playing video games and test scores, a negative correlation may exist (more gaming hours lead to lower scores).
Use equations: For given equations, check the coefficient of the independent variable. A positive coefficient indicates a positive correlation, and a negative coefficient indicates a negative correlation, as seen in the examples \(y = 5\) (special case with zero slope, no positive or negative correlation) and \(y=-2.816x - 4\) (negative slope, negative correlation).