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3.1 — Getting Ready for a Pool Party
Human review completed .
1. In class: A Develop Understanding Task
Turn a real situation into a graph
Learn to label the axes and show changes in water level as you prepare to sketch Sylvia’s pool graph.
A graph can show how one quantity depends on another. In a pool story, the independent variable is elapsed time: it belongs on the horizontal (x) axis. The dependent variable is the height of the water: it belongs on the vertical (y) axis. One point, such as
As you move from left to right along a graph, focus on the direction and steepness:
| Shape of a section | What it says | Pool example |
|---|---|---|
| Rises | Increasing: the dependent quantity becomes larger. | Water is added. |
| Falls | Decreasing: the dependent quantity becomes smaller. | Water is removed. |
| Horizontal | Constant: the dependent quantity does not change. | No water enters or leaves. |
| Steeper than another section | The quantity changes faster in magnitude over the same time interval. | Three buckets can remove water faster than one. |
A complete example: Sylvia’s pool
See one possible graph for all six activities and practice telling the story it represents.
Sylvia must empty the pool, clean it, and refill it. The original packet lists six activities: use one bucket; drain with a hose; use three buckets with friends; clean the empty pool; take a break; and refill with a hose. We can choose a sensible order because the packet does not prescribe one. Cleaning must happen after emptying and before final refilling.
Example assumptions: The MVP task gives no exact times or water depths. Suppose the pool has vertical sides, starts 4 feet deep, and the height changes steadily during each removal or filling interval. The values below describe one possible situation, not measurements supplied by the packet.
| Elapsed time | Activity | Height at start → end |
|---|---|---|
| 0–2 min | One bucket | 4 → 3.5 ft |
| 2–4 min | Three buckets | 3.5 → 2 ft |
| 4–8 min | Drain hose | 2 → 0 ft |
| 8–11 min | Clean empty pool | 0 → 0 ft |
| 11–13 min | Break | 0 → 0 ft |
| 13–21 min | Fill hose | 0 → 4 ft |
Label each part of your graph with its activity. The drain and fill hose pieces should be equally steep in opposite directions.
“Sylvia began with four feet of water. During the first two minutes, the water level decreased slowly while she used one bucket. For the next two minutes the level decreased more quickly because three people used buckets. She then drained the rest with a hose until the pool was empty at eight minutes. The level remained constant at zero while she cleaned and took a break. At thirteen minutes she started refilling; the level increased steadily until the pool reached four feet at twenty-one minutes.”
Why this water-height graph is a function
Explain why each moment in time has exactly one water height.
A function assigns exactly one output to each input. At any single instant, the pool has one water height, so water height is a function of time. The same water height can occur at many different times; that does not break the rule.
Changing the times or the order of the activities can give a different valid graph. It will still represent a function: each time corresponds to exactly one water height.
2. READY — Graph linear and exponential functions
Study the worked examples, then try the packet problems on your own. Use the answer key at the bottom of this section to check your work.
Graph linear and exponential equations, identify three points on each graph, and choose useful axis scales.
A point
Linear functions: equal changes in output
Use a linear equation to calculate points and draw a straight line.
A linear function can be written
Use
| x | Calculation | Point |
|---|---|---|
| 0 | ||
| 1 | ||
| 2 |
On the grid, count right by 1 on the x-axis and use y-values that count by 1 or 2. Plot all three points; draw a straight line through them and continue it in both directions. Each extra 1 in x reduces y by 2.
Exponential functions: equal multiplying factors
Find points on exponential curves and select scales that show their growth.
An exponential function has an input in the exponent, such as
Use
| x | Calculation | Point |
|---|---|---|
| 0 | ||
| 1 | ||
| 2 |
Use equal-length units on both axes, with a square plotting area. The graph below labels every fourth unit on both axes; each small grid square is one unit wide and one unit high. Plot the points and sketch a smooth increasing curve, not a straight line. To the left of x=0, the curve remains positive and approaches zero; it does not cross the x-axis.
Three points meet the packet’s instruction, but extra points can help you sketch an exponential curve. Keep using the same rule, k(x) = 4 · 2x.
For negative inputs, \(2^{-1}=\frac12\) and \(2^{-2}=\frac14\). Thus \(k(-1)=4\cdot\frac12=2\) and \(k(-2)=4\cdot\frac14=1\). Add the points (−1, 2) and (−2, 1). Each step left halves the output: \(4,2,1,\frac12\), and so on. This helps show why the curve approaches zero while staying above the x-axis.
To the right, k(3) = 4 · 8 = 32. Add (3, 32) to see how quickly the curve rises. Draw a smooth curve through the points rather than connecting them with straight segments.
Before you finish: label your axis scales and check that your three points satisfy the equation. For large exponential outputs, smaller or negative inputs can make the graph easier to draw.
Check your work — Ready 1–6 answer key
Sample sets of three points are listed below; other correct points are valid. Plots are not shown.
- (0, 5), (1, 3), (2, 1)
- (0, 4), (1, 1), (2, −2)
- (0, 5), (1, 15), (2, 45)
- (0, 4), (1, 8), (2, 16)
- (0, −4), (2, 1), (4, 6)
- (0, 8), (1, 24), (2, 72)
3. SET — Match situations to graphs and compare features
Try the packet problems on your own. Use the answer key at the bottom of this section to check your work.
Match everyday situations to graphs, label the axes, and compare the characteristics of paired graphs.
Match a real-world story to its visible pattern
Choose a situation for each graph and identify the independent and dependent variables.
(c) Gasoline: Read this as “The amount of gasoline at a gas station after a tanker completes one delivery and before the next delivery.”
(d) Watermelons: Read this as “The number of watermelons remaining for sale throughout a single Thursday, as customers buy them. No new watermelons are delivered that day.”
For most of the given everyday situations, time belongs on the x-axis. The output on the y-axis depends on which item is being measured.
Compare graph characteristics
Compare direction, steepness, and whether each graph is connected or has breaks.
A straight-line graph is linear and has a constant rate of change. A curved graph is nonlinear, and its steepness may change as you move along it. For Problem 12, both graphs decrease, but the straight line maintains the same steepness while the curve does not.
For Problem 13: the horizontal marks are segments containing intervals of inputs. The graph has breaks, so describe it as discontinuous. This differs from a discrete graph, which includes only separate input values, usually shown as isolated points.
The domain is the set of allowed inputs; the range is the set of outputs. Look at the horizontal and vertical values each graph includes when comparing them.
Check your work — Set 7–13 answer key
- b — oven temperature. Horizontal axis: time; vertical axis: oven temperature.
- e — delivery-truck odometer. Horizontal axis: time; vertical axis: odometer reading.
- d — watermelons available on Thursday. Horizontal axis: time on Thursday; vertical axis: number of watermelons available.
- a — savings account. Horizontal axis: time; vertical axis: account balance.
- c — gasoline between deliveries. Horizontal axis: time; vertical axis: amount of gasoline on hand.
- Sample comparison: both graphs are continuous and decreasing. One is a straight line with a constant rate of change; the other is curved, flattening toward height 10 to the left and falling increasingly steeply to the right.
- Sample comparison: both have an overall upward trend. One is a continuous straight line; the other has separate horizontal segments, so it is discontinuous and constant within each segment.
4. Go — Solve linear and basic exponential equations
For linear equations, use the same operation on both sides to isolate the variable. For exponential equations here, isolate the power first, then identify its exponent using familiar powers.
Original problem — Go 20: Find the value of \(x\) that makes \(10=2^x-22\) true.
Worked solution: Add 22 to both sides:
Since \(2^5=32\), the solution is \(x=5\).
Check: \(2^5-22=32-22=10\)
Try the packet problems independently, then use the answer key below to check your work.
Check your work — Go 14–22 answer key
- x=5
- x=14
- x=−5
- x=9
- x=4
- x=5
- x=5
- x=30
- x=3
Additional Learning Resources
| Resource and direct link | What it teaches / why use it | Scope / What to skip |
|---|---|---|
| Khan Academy — Intro to exponential functions (video) | Explains how exponential growth differs from linear growth; especially helpful for Ready’s exponential graphs. | Close match. Use for the difference between adding a fixed amount and multiplying by a factor. |
| The Organic Chemistry Tutor — Relations and Functions | Algebra | One output per input, tables, ordered pairs, and the vertical line test. | Supports Main 3’s question about whether water height is a function of time. |
Teacher’s review of the source packet
Section 3.1, Set, descriptions (c) and (d) — Clarify the time window
The gasoline description should identify the period between completed deliveries. The watermelon description should specify time throughout one Thursday, rather than a count recorded each Thursday. The wording clarification in the Set section makes these intended time windows explicit.
Section 3.1, Set, Problem 13 — Discontinuous versus discrete
The upper graph uses separated horizontal segments. These segments contain intervals of inputs, so they should not be reinterpreted as isolated points. The useful classroom comparison is a connected linear graph versus a discontinuous graph with constant pieces.
This connects to the distinction raised in Module 2: whole-number outputs do not by themselves make a domain discrete. Counts can change in steps over time. Keep the allowed inputs separate from whether the graph has breaks or jumps.
Coverage check — Section 3.1 only
| Original packet component | Content needed | Where it is taught here |
|---|---|---|
| Main task, problems 1–2 | Graph water level against time using all six activities; label segments and narrate increases, decreases, and pauses | In-class section; pool graph, activity table, and worked story |
| Main task, problem 3 | Recognize that height depends on time as a function; explain one output per input | In-class section, “Why this water-height graph is a function” |
| Ready 1–6 | Graph linear and exponential equations; compute 3 points; choose suitable scales | Section 2; worked examples 2–3; two equation graphs and tables |
| Set 7–11 | Match graph shapes to five everyday situations; label independent and dependent axes | Section 3, wording clarification, axis-label guidance, and answer key |
| Set 12–13 | Compare increasing/decreasing, continuous/discontinuous, linear/nonlinear, and domain/range features | Section 3, concise graph-comparison guidance and answer key |
| Go 14–22 | Solve linear equations and exponential equations using equal bases, including isolating exponential terms | Section 4; solving reminder, worked solution to Go 20, and answer key |
Source: Mathematics Vision Project, Secondary Mathematics One, Module 3: Features of Functions, Section 3.1 (the Develop Understanding Task and its Ready, Set, Go pages).
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