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Expressions and Equations 6-8 / Week 09 / Monday
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Week 09 · Linear Functions

Monday

Compare two functions
// Rate of change, starting value, and graphs that tell a story
⏱ about 20 min

Monday: Compare Two Functions

Two clear cups sit under two dripping jugs at the Puzzle Post. A kitchen timer ticks between them.

"Cup A, I wrote a table," Raven says. "Every minute I marked the water: 3, 5, 7, 9, 11 centimeters."

"Cup B, I wrote a rule," Rocket says. "Height = 3 × minutes + 1. Mine fills faster."

"What do you notice about how each one started?" Raven asks.

Rocket looks again. "Yours started at 3. Mine started at 1. But mine rises 3 every minute and yours rises 2."

Nova hovers between the cups, her light flicking from table to rule. "Two different forms," she says. "Same two questions."

"How fast, and from where," Raven says. Nova hums. "Rate of change, and initial value."

Rocket, Raven and Nova at the Puzzle Post watching a jug drip into a clear cup beside a kitchen timer.

Two questions for every function

A linear function is a rule y = mx + b. Two numbers tell its whole story.

The rate of change, m, is how much y changes for each 1 of x. It is the slope of the line.

The initial value, b, is the y when x = 0, the starting amount. It is where the line crosses the y-axis.

Different forms, same questions

Cup A comes as a table. Each minute adds 2 cm, so the rate of change is 2. At 0 minutes it had 3 cm, the initial value.

Cup B comes as a rule, h = 3m + 1. The rate of change is 3, the number multiplying m. The initial value is 1.

Cup B has the greater rate of change. Cup A has the greater initial value. Both are linear functions.

A table of minutes and cm for Cup A, from 0 and 3 up to 4 and 11
A grid from 0 to 16 with two lines: Cup A, steady from 3 cm, and Cup B, steeper from 1 cm

On the graph, Cup B is the steeper line because its rate of change is bigger. Cup A starts higher on the axis.

To compare two functions, bring each one down to its two numbers, however it is written.

READ EACH FORM
  • Read the question.
  • Tap your answer.
Cup A's table shows (1, 5) and (3, 9), minutes then cm. What is its rate of change?
Cup B's rule is h = 3m + 1. What is its rate of change?
Cup A's table shows (1, 5) and (2, 7). What was its height at 0 minutes?
Cup B's rule is h = 3m + 1. How many cm after 6 minutes?
WHICH FUNCTION WINS?
  • Read the question.
  • Tap your answer.
Cup A's table rises 2 cm each minute. Cup B's rule is h = 3m + 1. Which has the greater rate of change?
Cup A's table starts at 3 cm. Cup B's rule is h = 3m + 1. Which has the greater initial value?
Cup C is described in words: it rises 1.5 cm every minute from 4 cm. Cup A rises 2 cm a minute. Which has the smaller rate of change?
A graph of Cup D passes through (0, 2) and (2, 10). Cup B's rule is h = 3m + 1. Which has the greater rate of change?
StatementTrue or false?
The rate of change of a linear function is its slope.?
The initial value is the y when x = 0.?
A table and a rule cannot be compared.?
In h = 3m + 1, the initial value is 3.?
WHY THIS EXERCISEComparing functions means comparing their two numbers, not their forms.
Cup B's rule is h = 3m + 1. What is its initial value? Type the number.
WHY THIS EXERCISEThe initial value is the amount already there before the clock starts.
Try it
Pour 3 cm of water into a clear cup. Add one spoonful every 10 seconds and mark the level each time.
Write your rate of change in cm per spoonful and your initial value. Work over a tray.
Draw Cup A and Cup B on one grid, minutes across and cm up. Mark each starting point on the vertical axis.

Two numbers for every function. Tomorrow you meet a function that is not a straight line at all.