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Algebra 2 9-12 / Week 05 / Wednesday
3/6
Week 05 · Radicals and Inverse Functions

Wednesday

Glass House Lab: the drip cup
// The drip hose, forwards and backwards
⏱ about 20 min

Wednesday: Glass House Lab, the Drip Cup

A grown-up sets the tap over the sink to a slow, steady drip, and Comet slides a measuring cup underneath.

"Lab day. We make our own drip table," she says. "Wren, call the seconds."

Wren watches the stopwatch. "Zero. Five. Ten." Comet reads the cup each time and Nova logs it.

"What do you notice?" Wren asks after a minute. "The same number of milliliters every five seconds."

"So it is linear. Volume equals rate times seconds, plus whatever was in the cup to start."

Nova projects the readings as points. "Would you like a hint? Swap the columns and you have the inverse table."

"Seconds from volume," Comet says. "If a plant wants 60 milliliters, the inverse tells me how long to wait."

"Build the rule, then flip it," Wren says. "The lab writes both for us."

What you need

  • A tap or drip hose set to a slow, steady drip by a grown-up, over a sink or a bucket.
  • A measuring cup marked in milliliters, a stopwatch or kitchen timer, and a towel.
  • Your Glass House Log and graph paper.
Safety first
A grown-up sets the tap and turns it off. Use cold water only; a grown-up handles any hot water.
Work at the sink or on the floor, never on a ladder. Wipe up drips so no one slips.
Nothing heavy is lifted alone; a full bucket is a two-person carry.

Run the lab

  1. Put 10 mL of water in the cup to start, so the rule has a starting value. Note it as V at t = 0.
  2. Start the stopwatch and the drip together. Read the cup every 5 seconds for 20 seconds. Record each reading.
  3. Find the change in volume per second. That is your rate, the slope of your rule.
  4. Write your rule V(t) = rate × t + start. Check it against two of your readings.
  5. Swap the columns of your table. This is the inverse table: time from volume.
  6. Write the inverse rule by solving your rule for t. Check it: pick a volume, find t, and look it up in the table.

The crew's lab table

These are the crew's own readings from Nova's log, made up for the Glass House. Your tap drips at its own rate, so your numbers will differ.

Seconds tVolume V (mL)Inverse: V⁻¹(V) = t
0100
5305
105010
157015
209020

The crew's rate is 4 mL per second and the start is 10 mL, so V(t) = 4t + 10.

Solving for t: V - 10 = 4t, so t = (V - 10)/4. As a function, V⁻¹(x) = 0.25x - 2.5.

Check: V⁻¹(50) = 10, and the table shows V(10) = 50. The inverse undoes the rule.

On graph paper, plot V against t, then plot the inverse table. The two graphs are mirror images across the line y = x.

READ THE INVERSE TABLE
  • Read the question.
  • Tap your answer.
The table gives V: V(0) = 10, V(5) = 30, V(10) = 50, V(15) = 70, V(20) = 90. What is V⁻¹(30)?
The table gives V: V(0) = 10, V(5) = 30, V(10) = 50, V(15) = 70, V(20) = 90. What is V⁻¹(70)?
If V(x) = 4x + 10, what is the inverse function?
A seedling wants 60 mL. With the crew's rule V(t) = 4t + 10, how many seconds should Comet wait? Type the number.
WHY THIS EXERCISEThe inverse rule answers "how long?" directly. Building it once saves solving the same equation again and again.
Using V(t) = 4t + 10, what is V(15)? Type the number.
Using the inverse, when does the cup hold 90 mL? Type the number of seconds.
A rule and its inverse are mirror images across which line? Type it as y = something.
StatementTrue or false?
Swapping the two columns of a function table gives the inverse table.?
4 × 15 + 10 = 70?
(60 - 10) ÷ 4 = 12.5?
The inverse of V(t) = 4t + 10 is V⁻¹(x) = 4x - 10.?
A graph and its inverse are mirror images across the x-axis.?
WHY THIS EXERCISEUndoing steps in reverse order is how every inverse rule is built, from the cup to the square root.
Draw your drip readings as points on graph paper, then the inverse points. Draw the line y = x between them.

Great lab work. Tomorrow you decide which functions have inverses at all, and meet the root graphs.

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