Linear Applications and Graphs
What you must understand
Linear applications rest on the slope-intercept form y = mx + b, where m is the rate of change and b is the starting value. Word problems about cost, distance, or pay translate into this template: identify the fixed amount (b) and the per-unit amount (m). Slope between two points is (y2 - y1)/(x2 - x1); parallel lines share a slope, and perpendicular slopes are negative reciprocals. You should be able to write an equation from a point and a slope, from two points, or from a description, and to convert among slope-intercept, point-slope, and standard forms. Graphing requires recognizing the y-intercept and using rise over run.
Traps to watch
- Mixing the order of subtraction in the slope formula, which flips the sign.
- Confusing horizontal (slope 0) with vertical (undefined slope) lines.
- Writing a perpendicular slope as the plain reciprocal instead of the negative reciprocal.
- Misreading "per" language: a rate per hour is the slope, not the intercept.
- Forgetting that x-intercepts occur where y = 0.
Worked example
A rental company charges a $40 fee plus $15 per day. Write cost C as a function of days d, then find the cost for 6 days. C = 15d + 40. At d = 6, C = 90 + 40 = $130.
Rule of thumb
Find the rate (slope) and the starting value (intercept) first; the equation then writes itself.