Glossary
Break-even Point
The point where total costs equal total revenue, meaning there is no profit or loss, often found by setting two linear equations equal to each other.
Example:
A business reaches its break-even point when the number of units sold generates just enough revenue to cover all production costs.
Domain and Range
The domain refers to all possible input (x) values for a function, while the range refers to all possible output (y) values.
Example:
If a function models the height of a ball thrown in the air, the domain might be the time from launch to landing, and the range would be all possible heights the ball reaches.
Linear Equations
Equations that represent a straight line when graphed, showing a constant relationship between two variables.
Example:
The equation y = 2x + 3 is a linear equation because its graph is a straight line, unlike a curve.
Linear Functions in Real-World
The application of linear equations to model situations where there is a constant rate of change, such as cost per item or distance traveled over time.
Example:
Calculating the total cost of a phone plan that charges a fixed monthly fee plus a constant rate per gigabyte used is an example of linear functions in the real-world.
Parallel Lines
Two or more lines that have the exact same slope but different y-intercepts, meaning they will never intersect.
Example:
The lines y = 3x + 1 and y = 3x - 4 are parallel lines because they both have a slope of 3.
Perpendicular Lines
Two lines that intersect to form a 90-degree angle, where their slopes are negative reciprocals of each other.
Example:
A line with a slope of 2 and another with a slope of -1/2 are perpendicular lines.
Point-Slope Form
A linear equation form, y - y₁ = m(x - x₁), useful when you know the slope 'm' and one point (x₁, y₁) on the line.
Example:
If a line has a slope of -1 and passes through (3, 7), you can write its equation using point-slope form as y - 7 = -1(x - 3).
Slope (m)
The measure of a line's steepness and direction, calculated as the ratio of vertical change (rise) to horizontal change (run) between any two points on the line.
Example:
If a car travels 60 miles in 2 hours, its average speed, or slope, is 30 miles per hour.
Slope-Intercept Form
A common way to write linear equations, expressed as y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
Example:
To quickly graph a line with a slope of 2 and a y-intercept of 5, you'd use the slope-intercept form: y = 2x + 5.
Standard Form
A linear equation form written as Ax + By = C, where A, B, and C are constants, often used for finding intercepts or solving systems of equations.
Example:
The equation 3x + 4y = 12 is in standard form, making it easy to find the x-intercept (set y=0) and y-intercept (set x=0).
X-intercept
The point where a line crosses the x-axis, representing the value of 'x' when 'y' is zero.
Example:
On a graph showing a company's profit over time, the x-intercept would indicate the point where the company's profit becomes zero.
Y-intercept (b)
The point where a line crosses the y-axis, representing the value of 'y' when 'x' is zero.
Example:
In a graph showing the cost of a taxi ride, the y-intercept might represent the initial flat fee before any distance is traveled.