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Precalculus 9-12 / Week 03 / Monday
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Week 03 · Rational Functions

Monday

The wall
// Lap times that climb toward a wall
⏱ about 20 min

Monday: The Wall

Morning on the dock. Comet clicks the stopwatch as the rowboat passes the post, a life vest snug over her jacket.

"Nova, the laps model," Wren says, taping a sheet to the Boathouse wall. "After x weeks of practice, laps per hour is 8x over x + 2."

Nova reads the log. "Week 2 gives 4. Week 6 gives 6. Week 14 gives 7."

"It keeps climbing," Comet says. "What can we make of week 100? Week 1,000?"

"7843137/1000000. Then 7.98," Wren says. "What do you notice? It never reaches 8."

Nova projects a dashed line across the sheet at 8. "Would you like a hint? Divide the top and bottom by x."

"8 over 1 plus 2/x," Comet says. "And 2/x shrinks to nothing. So the curve climbs toward 8 and stops short. A wall."

Morning on the dock: Comet times a rowboat while Wren draws a curve climbing toward a dashed line on a wall sheet.

A rational function

A rational function is one polynomial divided by another: y = p(x)/q(x). The laps model is y = (8x)/(x + 2).

It is not allowed wherever the bottom is 0. Here x = -2 is forbidden, though a week count is never negative anyway.

The crew's model is their own made-up fit from Nova's log, not a fact about rowing. The math is what matters.

The crew's laps model (8x)/(x + 2) for x from 0 to 16, climbing toward the dashed line y = 8.

A solved problem to study

  1. Find the laps per hour after 6 weeks in the crew's model y = (8x)/(x + 2).
  2. Top: 8 × 6 = 48. Bottom: 6 + 2 = 8.
  3. Divide: 48 ÷ 8 = 6 laps per hour.
  4. Check against the wall: 6 is below 8, as every value must be.
  5. Why below? Divide top and bottom by x: y = 8/(1 + 2/x). The bottom is more than 1, so y is less than 8.

The laps model in Nova's log

Weeks xLaps per hourGap below 8
008
244
662
1471
307.50.5
627.750.25

The gap shrinks every row but never reaches 0. The line y = 8 is a horizontal asymptote: the graph approaches it and never touches.

Read the end behavior from the gap. As x grows, y approaches 8 from below.

READ THE LAPS MODEL
  • Read the question.
  • Tap your answer.
In the crew's laps model y = (8x)/(x + 2), how many laps per hour after 6 weeks?
In the crew's laps model y = (8x)/(x + 2), how many laps per hour after 14 weeks?
Graph of y = (8x)/(x + 2), dashed vertical asymptotes x = -2, dashed horizontal asymptote y = 8The crew's laps model is y = (8x)/(x + 2). Which line is the wall, its horizontal asymptote?
As x grows very large, does the laps model approach 8 from above or from below?
In the crew's laps model y = (8x)/(x + 2), which x value is not allowed? Type the number.
WHY THIS EXERCISEA rational function has no value where its bottom is 0. Finding that x is always the first step.
StatementTrue or false?
The laps model reaches 8 laps per hour at some week.?
8 × 6 ÷ (6 + 2) = 6?
A rational function is a polynomial divided by a polynomial.?
The line y = 8 is a horizontal asymptote of the laps model.?
The laps model is allowed at x = -2.?
WHY THIS EXERCISESeeing the wall in a table, a graph and a rule is the heart of reading a rational function.
Try it
Compute the laps model at x = 198 and x = 1,998 on paper. Write both gaps below 8.
Say in one sentence why the gap can get as small as you like but never reach 0.
Draw the laps curve from x = 0 to 20 with the dashed wall at y = 8. Mark the points for weeks 2, 6 and 14.

Strong start. Tomorrow four more functions from the wall sheet, with zeros, asymptotes and one hole.