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Word problems of graphing linear functions

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Linear Function Word Problems

Linear function word problems describe a situation that changes at a constant rate and ask you to model it with a line. Learn to translate the words into y = mx + b by finding the slope (rate of change) and y-intercept (starting value), graph the line, and read answers straight from it, with a worked example.

Turning a word problem into a linear function

Linear function word problems describe a real situation that changes at a constant rate, then ask you to model it with a line and answer a question. The skill is translation: turn the words into an equation of the form y = mx + b, where m is the rate of change (slope) and b is the starting value (y-intercept). From there you can graph it or read off slope-intercept form.

A worked example

Suppose a gym charges a $20 fee to join plus $5 for each visit. The total cost is C = 5x + 20, where x is the number of visits. The $20 fee is the y-intercept (the cost before any visits) and the $5 per visit is the slope (how fast the cost rises).

A linear function word problem as a graph A gym charges a 20 dollar joining fee plus 5 dollars per visit. The cost is C equals 5x plus 20. On the graph the line crosses the cost axis at 20 (the joining fee) and rises 5 dollars for each extra visit (the slope). Cost ($) Visits (x)012345601020304050 $20 join fee (y-intercept) C = 5x + 20 slope = $5 per visit
The word problem becomes the line C = 5x + 20: intercept $20, slope $5 per visit.

Reading the graph

Once the line is drawn, the graph answers questions directly. To find the cost of 4 visits, go to x = 4 and read the height: C = 5(4) + 20 = $40. To find how many visits cost $45, set 45 = 5x + 20 and solve for x. Building the line from a table of points is covered in graphing linear functions using a table of values.

Steps for any linear word problem

  • Identify the constant rate — this is the slope, m.
  • Identify the starting or fixed value — this is the y-intercept, b.
  • Write the equation y = mx + b and graph or solve it.

The same approach handles distance, cost, and savings problems — see more in the application of linear relations.

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