Key takeaway: A linear function is a function with a constant rate of change, so equal steps in the input always produce equal steps in the output and the graph is a straight line.
Grade 88.F.A.3Functions and Graphs
Definition of Linear Function
A linear function grows or shrinks by the same amount each time the input goes up by one. A plumber who charges a flat call-out fee plus a fixed price for every hour gives you a perfect case: the bill climbs by the same number of dollars each hour.
Because of that steady change, its table has equal output differences for equal input steps, its graph is a straight line, and its rule has the form \(f(x) = mx + b\). The coefficient \(m\) is the constant rate of change and \(b\) is the value when the input is zero. Anything that breaks the pattern, such as a squared input, belongs to the nonlinear family.
Formula and notation
The rule is written with function notation:
\[f(x) = mx + b\]
For the plumber, \(C(h) = 38h + 45\) gives the cost \(C\) after \(h\) hours. The number 38 is the rate and 45 is the starting fee.
Key facts
- Every linear function has equal output steps for equal input steps, which means a constant rate.
- The graph is a straight line, rising if the rate is positive and falling if negative.
- If b is 0, the function describes a proportional relationship through the origin.
- Two points are enough to find the rule.
Linear Function example
A plumber charges a \$45 call-out fee plus \$38 per hour, a linear function of time. Write the rule \(C(h)\), find the cost of a 3-hour job, and find how many hours a bill of \$235 pays for.
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Rate \$38 per hour, starting value \$45: \(C(h) = 38h + 45\).
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Three hours: \(C(3) = 38(3) + 45 = 114 + 45 = 159\).
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Set \(38h + 45 = 235\). Subtract 45: \(38h = 190\). Divide: \(h = 5\).
The 3-hour job costs \$159, and a \$235 bill covers 5 hours.
Common mistake
Watch out
Gia sees the table \(x: 0, 1, 2, 3\) and \(y: 5, 8, 12, 17\) and says the relationship is linear because y keeps going up.
Why it is wrong
Going up is not enough. The steps in y are 3, 4 and 5, which are not equal, so the rate changes and the table is nonlinear. A table is linear only when equal input steps give equal output steps.
Related terms and lessons
- Grade 8 · Linear Functions and ModelingLessonWorksheetQuiz
- Algebra 1 · Linear Functions: Slope-Intercept, Point-Slope and Standard FormLessonWorksheetQuiz
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