A Graph as a Starting Value Plus an Amount per Unit (AF 403)

Any time a cost is built from a flat fee plus an amount per unit, the total follows the same pattern: total = fee + rate times units. That's the equation of a line, y = b + mx, so the graph is always a straight line. The fee doesn't change no matter how many hours or miles or items you add, so it sets where the line crosses the vertical axis. The rate is the extra charge for each additional unit, so it controls how steeply the line rises as you move right.

Once you see the graph this way, reading it backward is easy. The y-intercept tells you the starting value before any work is done or any units are bought. The slope tells you the price per hour, per mile, or per item. A plumber who charges 65 dollars to show up plus 40 dollars an hour has total cost C(h) = 65 + 40h: 65 is the intercept, 40 is the slope. If the plumber instead makes the first hour free, you're only billing h - 1 hours once h reaches 1 or more, so the rule becomes C(h) = 65 + 40(h - 1). The line shifts right by one unit but keeps the same steepness, since the rate never changed.

This connects directly to comparing two linear situations. A steeper line grows faster, but it can still start lower, so for a while it stays cheaper even though it will eventually overtake the flatter line. Deciding which costs more means finding where the two lines cross, not just comparing slopes.