Last updated .
3.4 — The Water Park
Human review completed .
1. In class: A Solidify Understanding Task
From the draining pools to their equations
Read the pool graph, use the pumping information to label it, and write each draining rule.
Aly and Dayne work at a water park. They each pump water out of a pool, starting at the same time. Let \(x\) be the number of minutes since they started. The amount left in Aly’s pool is \(a(x)\), and the amount left in Dayne’s is \(d(x)\), measured in gallons.
The packet’s graph has time on the horizontal axis and water remaining on the vertical axis. Both lines slope downward: as time passes, less water remains. Each line is straight, so each pump removes water at a constant rate.
Even before numbering the axes, you can read four things: Aly starts with more water; her line falls more steeply, so her pump drains faster; her pool empties first; and the crossing shows a time when the pools have equal amounts.
Worked example 1 — Write Dayne’s draining rule
Dayne’s pump removes 1,000 gallons per minute and empties his pool in 24 minutes. That tells us how much water was there at the start:
After \(x\) minutes, the pump has removed \(1000x\) gallons. Subtract that amount from the starting amount:
The constant 24,000 is the initial amount. The coefficient −1,000 is the change in water amount per minute. At \(x=0\), the graph is at \((0,24{,}000)\); at \(x=24\), it reaches \((24,0)\).
These two points let us number the original graph. Dayne’s time intercept is 24 minutes, and his water intercept is 24,000 gallons. Count the equal spaces from zero to each intercept to establish the scale on that axis.
Worked example 2 — Write Aly’s draining rule
On the newly labeled packet graph, Aly starts with 28,000 gallons and empties her pool in 20 minutes. Her pumping rate is
After \(x\) minutes, 1,400 gallons have been removed for each minute elapsed. Her rule is therefore
Check the endpoints: \(a(0)=28{,}000\), and \(a(20)=28{,}000-1400(20)=0\).
Use function notation to find an amount or a time
Distinguish a known input from a known output, then interpret the result in the pool situation.
Read \(a(x)\) as “a of x.” The letter \(a\) names Aly’s function; the number in parentheses tells you which input to use. The whole expression gives the output, in gallons. Thus \(a(5)\) is the amount in Aly’s pool after 5 minutes.
Worked example 3 — Find the amount in Aly’s pool after 5 minutes
To find \(a(5)\), substitute 5 for \(x\):
After 5 minutes, Aly’s pool contains 21,000 gallons. The corresponding point on her graph is \((5,21{,}000)\).
In \(d(x)=2000\), the output is known: Dayne’s pool contains 2,000 gallons. The input—the time—is what we need to find.
Worked example 4 — Find when Dayne’s pool has 2,000 gallons remaining
Replace \(d(x)\) with its rule and solve for \(x\):
Dayne’s pool contains 2,000 gallons after 22 minutes. Check: \(d(22)=24{,}000-22{,}000=2000\). On the graph, you could find the same time by starting at 2,000 gallons on the vertical axis, moving across to Dayne’s line, then down to the time axis.
Limit the equations to the draining situation
State the domain and range, accounting for the beginning and end of draining.
The domain describes the allowed times; the range describes the amounts of water reached. Time begins at 0, and draining ends when each pool is empty. Fractional minutes are allowed.
- Dayne: domain \([0,24]\) minutes; range \([0,24{,}000]\) gallons.
- Aly: domain \([0,20]\) minutes; range \([0,28{,}000]\) gallons.
The brackets include both endpoints: the starting amount and the empty pool are part of the process. Write each range from the smaller amount to the larger, even though the pool is draining.
As an unrestricted linear function, \(a(x)=28{,}000-1400x\) has domain and range \((-\infty,\infty)\). The pool situation restricts those values: negative times are before the start, and times beyond 20 would give negative water amounts. For example, the formula gives \(a(22)=-2800\), but 22 is outside the draining model’s domain.
Compare the pools at the same time
Interpret the intersection and the times when Aly’s pool contains more water.
At a given time, the higher line represents the pool with more water. The statement \(a(x)=d(x)\) means their amounts are equal at the same time; \(a(x)>d(x)\) means Aly’s amount is greater.
Worked example 5 — Read the intersection and compare the amounts
The two lines meet at \((10,14{,}000)\). After 10 minutes, both pools contain 14,000 gallons. Check that time in both equations:
Before the crossing, Aly’s line is above Dayne’s. Therefore \(a(x)>d(x)\) for \(0\le x<10\), written \([0,10)\) in interval notation. Include 0 because Aly starts with more water; exclude 10 because the amounts are equal then.
After the crossing, Aly’s line is below Dayne’s while both pools are draining. For example, at 12 minutes, \(a(12)=11{,}200\) and \(d(12)=12{,}000\): Dayne has more water left.
2. Ready — Compare linear and exponential functions
Describe similarities and differences between linear and exponential functions.
This problem asks for a comparison in words: how the functions change, what their graphs look like, and which inputs and outputs they allow. Write your own comparison first, then use the sample answer below to check that you have included both similarities and differences.
Check your work — Ready 1 sample answer
Similarities: Both kinds of functions assign one output to each input. Both can describe an increasing or decreasing quantity, and both have a y-intercept. Their usual unrestricted forms allow all real inputs; a real situation may restrict that domain.
Differences in change and shape: A linear function has a straight graph and adds the same amount to its output for each one-unit increase in the input. An exponential function has a curved graph and multiplies its output by the same factor for each one-unit increase. For example, \(L(x)=6+2x\) adds 2 each step, while \(E(x)=6\cdot2^x\) doubles each step. Both begin at \((0,6)\).
Differences in outputs: An unrestricted nonconstant linear function has all real numbers in its range. For an exponential example such as \(y=6\cdot2^x\), the range is \((0,\infty)\): its outputs remain positive. Its graph approaches the x-axis without reaching it. A line can cross the x-axis; this exponential curve cannot.
3. Set — Read domain, range, and direction from a graph
Give each graph’s domain, range, and increasing or decreasing behavior.
These are similar to the graph-reading problems in Sections 3.2 and 3.3. Use the interval explanation and endpoint example in Section 3.2 if you need a reminder about endpoints and interval notation. Try the problems yourself before opening the answer key.
Check your work — Set 2–9 answer key
The details matter: write each interval from the smaller number to the larger. Use a square bracket when an endpoint is included and a parenthesis when it is excluded. Always use parentheses with \(-\infty\) and \(\infty\).
- Domain: \([-5,2)\). Range: \((-6,5]\). Direction: decreasing.
- Domain: \((-\infty,4)\). Range: \((-6,-1)\). Direction: decreasing.
- Domain: \((-\infty,\infty)\). Range: \((-\infty,4)\). Direction: increasing.
- Domain: \([1,6]\). Range: \([-2,-1]\). Direction: increasing.
- Domain: \((-\infty,\infty)\). Range: \((-\infty,\infty)\). Direction: increasing.
- Domain: \([-5,\infty)\). Range: \([0.4,\infty)\), with the starting height estimated from the graph. Direction: increasing.
- Domain: \((-\infty,5]\). Range: \((-\infty,-1]\). Direction: increasing.
- Domain: \((-\infty,\infty)\). Range: \((0,\infty)\). Direction: decreasing.
4. Go — Write explicit and recursive sequence rules
Write explicit and recursive rules for the sequence tables.
These revisit the sequence rules from Section 3.2. Refer to its Go reminders if needed. Count steps in the index, not just rows in the table. Try each problem before checking the answers.
Check your work — Go 10–18 answer key
Recursive rules are given first. These answers start at each table’s first supplied index; equivalent formulas or valid earlier starting indices are also acceptable.
- Recursive: \(f(1)=3,\quad f(n+1)=f(n)+2\)Explicit: \(f(n)=3+2(n-1)\)For integer indices \(n\ge1\).
- Recursive: \(f(2)=4,\quad f(n+1)=2f(n)\)Explicit: \(f(n)=4\cdot2^{n-2}\)For integer indices \(n\ge2\).
- Recursive: \(f(6)=23,\quad f(n+1)=f(n)-4\)Explicit: \(f(n)=23-4(n-6)\)For integer indices \(n\ge6\).
- Recursive: \(f(1)=1,\quad f(n+1)=3f(n)\)Explicit: \(f(n)=3^{n-1}\)For integer indices \(n\ge1\).
- Recursive: \(f(3)=8,\quad f(n+1)=\frac12f(n)\)Explicit: \(f(n)=8\left(\frac12\right)^{n-3}\)For integer indices \(n\ge3\).
- Recursive: \(f(6)=7,\quad f(n+1)=f(n)+2\)Explicit: \(f(n)=7+2(n-6)\)For integer indices \(n\ge6\).
- Recursive: \(f(2)=40,\quad f(n+1)=f(n)-4\)Explicit: \(f(n)=40-4(n-2)\)For integer indices \(n\ge2\).
- Recursive: \(f(2)=16,\quad f(n+1)=\frac14f(n)\)Explicit: \(f(n)=16\left(\frac14\right)^{n-2}\)For integer indices \(n\ge2\).
- Recursive: \(f(17)=5,\quad f(n+1)=f(n)+\frac53\)Explicit: \(f(n)=5+\frac53(n-17)\)For integer indices \(n\ge17\).
Additional learning resources
| Resource and direct link | What it supports | Scope |
|---|---|---|
| The Organic Chemistry Tutor — Domain and Range of a Function From a Graph | Worked graph-reading examples with interval notation and endpoint inclusion. | A refresher for Set 2–9; it does not use the water-park context. |
Teacher’s review of the source packet
No serious mathematical or instructional errors identified in Section 3.4. The initially unnumbered pool graph is intentional; Part II supplies the information needed to establish its scales. Go 18 is an arithmetic pattern with difference \(\frac53\) per index step; the answer key reflects the skipped indices.
Coverage — Section 3.4
| Packet component | Skills supported | Where taught |
|---|---|---|
| Main 1 · p. 16 | Interpret starting amounts, rates, emptying times, and a crossing | In class; numbered water-park graph |
| Main 2–3 and 6 · p. 17 | Write a linear draining rule; interpret coefficients; label graph scales | In class; Dayne’s and Aly’s worked models |
| Main 4–6 · p. 17 | Algebraic versus contextual domain/range, units, endpoint inclusion | In class |
| Main 7–10 · p. 17 | Known input versus known output; equal amounts; compare graph heights | In class; evaluation, solving, and intersection examples |
| Ready 1 · p. 18 | Linear/exponential changes, graph shapes, intercepts, domain and range | Section 2; sample comparison answer |
| Set 2–9 · pp. 18–19 | Segments, rays, lines, curves, excluded endpoints and horizontal bounds | Section 3; answer key and link to Section 3.2 reminders |
| Go 10–14 and 17 · p. 20 | Arithmetic/geometric explicit and recursive rules; non-1 starting indices | Section 4; answer key and link to Section 3.2 reminders |
| Go 15–16 and 18 · p. 20 | Skipped indices, per-step differences, fractional difference | Section 4; answer key with correct index steps |
Source: MVP, Secondary Mathematics One, Module 3: Features of Functions (May 2016), Section 3.4, printed pp. 16–20, and its teacher notes. The in-class lesson uses the packet’s Aly-and-Dayne example; the pool diagrams recreate its starting graph and then add numbered axes. The Ready comparison is a sample answer, and the homework keys refer to the packet’s exercises. Mathematical notation needs an internet connection to load its renderer.
Last updated .