L5 · Linear functions clinic
“I can run the full linear-function workup on any example - check linearity, find the slope, write y = mx + b, give both intercepts as ordered pairs, and compare lines - and I can recognize the slope of a line through the origin as the constant of proportionality k from G8U03.”
A rental shop charges for snowboard rentals. The table shows hours h and total cost C in dollars: (0, 20), (1, 28), (2, 36), (3, 44). Step 1 - Linearity: differences are 28 - 20 = 8, 36 - 28 = 8, 44 - 36 = 8. Constant: linear. Step 2 - Slope: the constant difference IS the slope between consecutive points, and any two points agree: (44 - 20)/(3 - 0) = 24/3 = 8. Slope 8 means $8 per hour. Step 3 - Equation: b is the cost at h = 0, which is 20. So C = 8h + 20. Step 4 - Intercepts as ordered pairs: C-intercept (0, 20): the $20 flat fee before a single hour is rented. h-intercept: 0 = 8h + 20 gives h = -2.5 - negative, so this intercept has no meaning in the situation (the model starts at h = 0); it exists on the extended line but not in the story. Step 5 - Use it: at h = 6, C = 8(6) + 20 = $68. Now the contrast. A second shop has no fee: its graph passes through (0, 0) and (2, 24). Slope = 24/2 = 12, equation C = 12h. This line passes through the origin, so the relationship is proportional and the slope 12 IS the constant of proportionality k = 12 dollars per hour - the G8U03 name for the very same number.
The second shop's slope can also be called k because...
A third shop's graph passes through (0, 10) and (4, 42). Write its equation. -> Slope (42 - 10)/(4 - 0) = 8, b = 10: C = 8h + 10.
"For C = 8h + 20, the cost at h = 9 is 8(9) + 20 = ___" -> 92
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.