Parallel, perpendicular, and linear models
Slopes reveal how lines relate to each other. Linear functions also model real situations with a constant rate of change, from phone bills to depreciation.
Parallel lines
Non-vertical lines are parallel exactly when they have equal slopes and different intercepts. The lines \(y = 3x - 2\) and \(y = 3x + 5\) never meet. To find the line through (2, 1) parallel to \(y = 3x - 2\), use slope 3: \(y - 1 = 3(x - 2)\), so \(y = 3x - 5\).
Perpendicular lines
Two non-vertical lines are perpendicular exactly when their slopes multiply to -1, that is, each slope is the negative reciprocal of the other. A line perpendicular to \(y = \tfrac{2}{3}x + 1\) has slope \(-\tfrac{3}{2}\). Horizontal and vertical lines are also perpendicular to each other.
Writing a perpendicular line
The line through (4, -2) perpendicular to \(y = 2x + 7\) has slope \(-\tfrac{1}{2}\): \(y + 2 = -\tfrac{1}{2}(x - 4)\), which simplifies to \(y = -\tfrac{1}{2}x\). Perpendicular lines are used to find the shortest distance from a point to a line.
Linear functions
A linear function has the form \(f(x) = mx + b\). Its graph is a line, and its output changes by the same amount, m, for every unit increase in input. If \(f(x) = 4x - 3\), then \(f(5) = 17\). Constant rates of change are the signature of linear relationships.
Building linear models
To model a situation linearly, identify the starting value (the intercept) and the constant rate (the slope). A car bought for $24,000 that loses $2,500 in value each year is worth V(t) = 24,000 - 2,500t after t years. Setting V = 0 shows it reaches zero book value after 9.6 years.
Interpreting a model
Always interpret slope and intercept in context with units. In V(t) = 24,000 - 2,500t, the slope -2,500 means the value falls $2,500 per year, and the intercept 24,000 is the purchase price. Models have limited domains: the car's value cannot be negative, so the model only applies from t = 0 to t = 9.6.
Fitting lines to data
Real data rarely fall exactly on a line. A line of best fit, usually found by the least-squares method, summarises a linear trend; statistics courses study it in detail. Predictions within the data's range (interpolation) are more reliable than predictions beyond it (extrapolation).
Worked example
A phone plan costs $15 a month plus $0.05 per minute, and another costs $35 a month with unlimited minutes. The first costs C = 15 + 0.05m. The plans cost the same when 15 + 0.05m = 35, so m = 400 minutes. Below 400 minutes the first plan is cheaper; above it, the unlimited plan wins.
Key takeaways
Parallel non-vertical lines have equal slopes; perpendicular lines have slopes whose product is -1, negative reciprocals of each other. Use point-slope form with the required slope to write parallel or perpendicular lines through a given point. A linear function f(x) = mx + b has a constant rate of change m. Build models from a starting value and a constant rate, interpret slope and intercept with units, and respect the model's sensible domain. Interpolation is safer than extrapolation.
Flashcard
Equal slopes, different intercepts
Check your understanding
What is the slope of a line perpendicular to \(y = \tfrac{2}{3}x + 1\)?