Glossary
Break-even Point
The point at which total costs and total revenue are equal, meaning there is no net loss or gain.
Example:
A lemonade stand reaches its break-even point when the money earned from selling lemonade exactly covers the cost of lemons, sugar, and cups.
Dependent Variable
The output variable in a relationship, whose value depends on the changes in the independent variable. It is typically represented by 'y' in linear equations.
Example:
In the equation for calculating total cost, the total cost is the dependent variable because it changes based on the number of items purchased.
Independent Variable
The input variable in a relationship, whose value is chosen or controlled and determines the value of the dependent variable. It is typically represented by 'x' in linear equations.
Example:
When tracking plant growth, the amount of water given is the independent variable because you control how much water the plant receives.
Linear Equation (General Form)
An algebraic equation that represents a straight line on a graph, commonly expressed as y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
Example:
The equation y = 2x + 5 is a linear equation that could model the cost of a service with a 2 per unit.
Linear Relationships
A relationship between two variables that, when plotted on a graph, forms a straight line, indicating a constant rate of change.
Example:
The relationship between the number of hours you study and your SAT score might show a linear relationship if each additional hour consistently boosts your score by a certain amount.
Point-Slope Form
A specific form of a linear equation, y - y₁ = m(x - x₁), used to construct the equation of a line when given its slope and one point it passes through.
Example:
If you know a line has a slope of 3 and passes through (2, 1), you can use the point-slope form to quickly write its equation as y - 1 = 3(x - 2).
Rate of Change
How one quantity changes in relation to another quantity, often expressed as a ratio. In linear relationships, this is constant and equivalent to the slope.
Example:
If a car's speed is 60 miles per hour, its rate of change in distance over time is 60 mph.
Slope
The measure of the steepness of a line, representing the rate at which the dependent variable changes for every one-unit increase in the independent variable.
Example:
If a runner's distance increases by 5 miles every hour, the slope of their distance-time graph is 5 miles per hour.
Y-intercept
The point where a line crosses the y-axis, representing the value of the dependent variable when the independent variable is zero.
Example:
In a savings account, the y-intercept would be the initial amount of money you deposited before any interest or withdrawals.