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3.2 — Floating Down the River
Human review completed .
1. In class: A Solidify Understanding Task
Inputs, outputs, and features of a function
Identify the quantities and use domain, range, direction, extremes, and intercepts to describe them.
A function assigns exactly one output to each allowed input. In a river trip, time is the input. At each time, the depth of the water at the floaters’ location is one possible output; total distance traveled is another. These are two different functions, even though both use time.
Read an ordered pair in input–output order. For a depth function, \((20,8)\) means that at 20 minutes the water was 8 feet deep. On Maria’s distance graph, \((20,2000)\) means that she had traveled about 2,000 feet after 20 minutes.
Domain and range answer different questions
The domain is the set of allowed input values. The range is the set of output values the function actually produces. On a graph, look horizontally for the domain and vertically for the range.
Worked example 1 — A restricted trip
A walker travels steadily at 200 feet per minute for 15 minutes, beginning at distance zero. Here time can be any real number from 0 to 15 minutes, including fractional times.
The domain is \([0,15]\). Distance begins at 0 feet and ends at \(200(15)=3000\) feet. Because the walker moves continuously through all intermediate distances, the range is \([0,3000]\).
The square brackets include both endpoints. The interval \([0,15]\) means \(0\le t\le15\), not just the two values 0 and 15. The function’s formula could produce a number when \(t=20\), but 20 minutes is outside this particular trip’s domain.
Changing output and attained extremes
A function is increasing on an interval when later inputs give larger outputs, decreasing when later inputs give smaller outputs, and constant when the output stays the same. Read these changes from left to right. An increasing function can have negative outputs: rising from −8 to −3 is an increase.
A maximum value is the greatest output actually reached anywhere in the domain; a minimum value is the least output actually reached. Distinguish the value from its location: “the maximum is 12 feet” gives an output; “it occurs at 90 minutes” gives an input. More than one input may attain the same maximum or minimum.
An x-intercept is a point where the graph meets the x-axis; a y-intercept is a point where the graph meets the y-axis. Write intercepts as ordered pairs. A graph need not have either kind of intercept, and the origin \((0,0)\) can be both.
For Main Problem 3(d), in Part II: Interpreting data (p. 7): compare the distance traveled during the last ten minutes with the distance traveled during other ten-minute intervals. On a distance-versus-time graph, height shows the total distance traveled; steepness shows how fast someone is moving. To compare distance traveled during equal time intervals, compare how much the graph rises during each interval.
2. Ready — Solve systems by graphing
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 each pair of lines and identify the point that satisfies both equations.
A system of linear equations is a set of linear equations considered together. A solution makes every equation in the system true. For two lines that intersect at one point, you can find the solution by graphing them on the same axes and reading the coordinates of the intersection point.
To graph a linear equation, find two points that satisfy it, plot them, and draw the straight line through them. Choose values that make the arithmetic easy and put the two points well apart on your grid. A point at \(x=0\) is often convenient; choose another input farther away so the points span a substantial part of the graphing region. Closely spaced points make it harder to draw the line accurately by hand. If there’s a fraction, try a multiple of its denominator.
Worked example 2 — Locate and check an intersection
Solve by graphing \(f(x)=x+1\) and \(g(x)=-2x+7\).
- For \(f(x)=x+1\), choose \(x=0\) and \(x=5\). Then \(f(0)=1\) and \(f(5)=5+1=6\), giving \((0,1)\) and \((5,6)\). These points are well apart on the grid. Plot them and draw the line through them.
- For \(g(x)=-2x+7\), choose \(x=0\) and \(x=4\). Then \(g(0)=7\) and \(g(4)=-2(4)+7=-1\), giving \((0,7)\) and \((4,-1)\). Plot these widely separated points and draw the second line on the same axes.
- Draw both lines. Their intersection is \((2,3)\), so the common input is 2 and the common output is 3.
- Check the point in both formulas: \(f(2)=2+1=3\) and \(g(2)=-2(2)+7=3\). Both equal the proposed y-coordinate.
Report the whole ordered pair \(\boxed{(2,3)}\). Writing only “2” omits the output coordinate. Substitution here checks the solution read from the graph.
Some packet intersections have negative coordinates. Extend your grid below or left of the axes as needed; choose a window that contains both y-intercepts and the crossing. A crossing that falls between grid marks may require an estimate, which you should label and check. All six Ready pairs have different slopes, so their complete lines have exactly one intersection each.
Check your work — Ready 1–6 answer key
Points of intersection are given below. Plots are not shown.
- \((2,-3)\)
- \((-1,3)\)
- \((-4,2)\)
- \((3,-2)\)
- \((-3,2)\)
- \((-1,-1)\)
3. Set — Describe intervals, endpoints, and extremes
Study the worked example, then try the packet problems on your own. Use the answer key at the bottom of this section to check your work.
Use interval notation to describe direction, domain, and range, and identify attained maximum and minimum values.
How interval notation works
An interval includes every real number between its boundaries. A square bracket includes a boundary value; a parenthesis excludes it. A filled endpoint belongs to a graph, while an open circle does not.
| Notation | Meaning |
|---|---|
| \([-3,5]\) | \(-3\le x\le5\): both endpoints included |
| \((-3,5]\) | \(-3<x\le5\): exclude −3, include 5 |
| \((2,\infty)\) | \(x>2\): all real numbers greater than 2 |
| \([-4,-1]\cup[2,6]\) | Two intervals, with a gap between them; \(\cup\) means “union,” or combine the sets |
Infinity is not an endpoint you can reach, so always place a parenthesis beside \(\infty\) or \(-\infty\). It can be confusing that we use the same notation for different things: \((2,5)\) can mean an interval or an ordered pair. You generally tell which is intended from the context. In a domain or range statement, it means all values strictly between 2 and 5; when describing a point, it means input 2 and output 5.
You can also describe intervals using inequalities. For example, \(-1<x<3\) or \(4<x\) describes the same values as \((-1,3)\cup(4,\infty)\). Both notations are valid, and you should become comfortable reading and writing both. These Set questions ask for interval notation, so we use that form in the answers below.
Worked example 3 — A graph with completely specified endpoints
The following original graph has an open circle at \((-3,1)\), then straight segments through \((0,4)\), \((2,4)\), and a filled endpoint at \((5,-2)\). There is no continuation beyond its two ends.
Answer the three questions in the same order as the Set directions:
- Where is the function increasing, decreasing, or constant? Increasing on \((-3,0)\), decreasing on \((2,5)\), and constant on \((0,2)\). These intervals describe the inputs.
- Does it have a minimum or maximum? The minimum output is −2, attained at \(x=5\). The maximum output is 4, attained for every \(x\) in \([0,2]\).
- What are the domain and range? The domain is \(\boxed{(-3,5]}\): the open circle excludes −3, while the filled endpoint includes 5. The range is \(\boxed{[-2,4]}\): the graph reaches every output from −2 through 4, including both. The open point at height 1 does not exclude 1 from the range, because the descending segment reaches that height too.
Disconnected domains and outputs that overlap
If a graph contains separate pieces with gaps between their input intervals, list the domain as a union. For instance, a graph defined only from −4 to −1 and from 2 to 6, with filled endpoints, has domain \([-4,-1]\cup[2,6]\). It has no input values in the gap \((-1,2)\).
The range may nevertheless be one continuous interval if the pieces’ output values overlap. To find it, project all the pieces onto the vertical axis. Do not copy the domain’s gaps into the range. In Set 7, the separate pieces have filled endpoints: read each x-interval separately, then check which y-values the combined pieces actually reach.
Check your work — Set 7–9 answer key
7.
Increasing: \((-10,-7)\cup(-6,-2)\cup(7,12)\). Decreasing: \((-13,-10)\cup(-2,5)\). Constant: none.
Minimum: 1 at \(x=-6\) and \(x=5\). Maximum: 8 at \(x=-7\).
Domain: \([-13,-7]\cup[-6,5]\cup[7,12]\). Range: \([1,8]\).
8.
Increasing: none. Decreasing: \((-\infty,\infty)\). Constant: none.
Minimum and maximum: neither. Domain: \((-\infty,\infty)\). Range: \((3,\infty)\).
9.
Increasing: \((-\infty,-2)\). Decreasing: \((5,\infty)\). Constant: \((-2,5)\).
Maximum: 2 throughout \([-2,5]\). Minimum: none. Domain: \((-\infty,\infty)\). Range: \((-\infty,2]\).
For 8 and 9, these answers follow the intended curves continuing beyond the viewing window.
4. Go — Explicit and recursive sequence rules
Try the packet problems on your own. Use the answer key at the bottom of this section to check your work.
Write arithmetic and geometric sequence rules in explicit and recursive form, including tables with skipped indices.
Reminder: an explicit rule gives a term directly from its index; a recursive rule gives a starting term and a rule for finding each next term.
The exercise is a review of arithmetic and geometric rules, so look for a consistent constant difference or constant ratio per one index step.
In 12–14, the printed indices skip values. In 12 and 15, the first listed index is 0. In 14, you may anchor the recursive rule at the first supplied index, 2, and give its valid later indices; do not silently label that supplied term as \(f(1)\).
Check your work — Go 10–15 answer key
Recursive rules are listed first, followed by explicit rules.
- Recursive: \(f(1)=5,\quad f(n+1)=f(n)-3\), for integers \(n\ge1\). Explicit: \(f(n)=5-3(n-1)\), for integers \(n\ge1\).
- Recursive: \(f(1)=6,\quad f(n+1)=2f(n)\), for integers \(n\ge1\). Explicit: \(f(n)=6\cdot2^{n-1}\), for integers \(n\ge1\).
- Recursive: \(f(0)=-13,\quad f(n+1)=f(n)+4\), for integers \(n\ge0\). Explicit: \(f(n)=-13+4n\), for integers \(n\ge0\).
- Recursive: \(f(1)=5,\quad f(n+1)=f(n)+2\), for integers \(n\ge1\). Explicit: \(f(n)=5+2(n-1)\), for integers \(n\ge1\).
- Recursive: \(f(2)=5,\quad f(n+1)=5f(n)\), for integers \(n\ge2\). Explicit: \(f(n)=5\cdot5^{n-2}\), for integers \(n\ge2\).
- Recursive: \(f(0)=-4,\quad f(n+1)=4f(n)\), for integers \(n\ge0\). Explicit: \(f(n)=-4\cdot4^n\), for integers \(n\ge0\).
Equivalent formulas are also valid.
Additional instructional resources
| Resource | Use it for | Scope |
|---|---|---|
| Khan Academy — Domain and range from graphs (video) | Reading the possible inputs and outputs directly from a graph. | Close match for the in-class discussion and Set domain-and-range questions. |
| Khan Academy — Increasing, decreasing, positive or negative intervals (video) | Identifying intervals where a graph rises or falls, and distinguishing these from intervals where its outputs are positive or negative. | Close match for the increasing-and-decreasing questions in Set. |
| Khan Academy — Intro to arithmetic sequence formulas | Explicit and recursive arithmetic rules. | Close match for Go 10, 12, and 13. Check the starting index in each example. |
| Khan Academy — Explicit & recursive formulas for geometric sequences | Writing the two forms of a geometric rule. | Close match for Go 11, 14, and 15. Distinguish a one-step ratio from a ratio across skipped indices. |
| Khan Academy — Use arithmetic sequence formulas (practice) | Additional interactive practice finding terms from rules. | Useful arithmetic review; the packet also includes geometric sequences and skipped indices. |
| The Organic Chemistry Tutor — Solving Systems of Equations By Graphing | Graph both lines and identify their intersection as the solution. | Close match to Ready 1–6; give both coordinates of the intersection. |
| The Organic Chemistry Tutor — Domain and Range of a Function From a Graph | Read domain and range from graphs using interval notation, open circles, and filled endpoints. | Close match to the in-class definitions and Set 7–9. |
An internet connection is required to typeset the equations with MathJax and to open external resources. The original coordinate diagrams are embedded in this file.
Teacher’s review of the source packet
No serious mathematical issues need special treatment here. The river table and hand-drawn graphs use ordinary classroom simplifications; this lesson follows their intended patterns. The main teaching distinctions are depth versus accumulated distance, input intervals versus output values, and one-step sequence changes versus changes across skipped indices.
Coverage — Section 3.2
| Packet component | Instruction provided | Where it is taught |
|---|---|---|
| Main task 1 · p. 6 | Input/output, domain/range, increasing/decreasing intervals, extrema, intercepts | In-class section; definitions and the restricted-trip example. |
| Main task 2 · p. 6 | Domain, range, extrema, intercepts, and distance versus speed | In-class section; general definitions and the distance-versus-speed explanation. |
| Main task 3(a–m) · p. 7 | Concepts needed to evaluate claims about depth, distance, domain, range, and extrema | In-class section; concepts are grouped by skill. |
| Ready 1–6 · p. 8 | Graph both lines, read common ordered pair, check both equations; negative coordinates and fractional slopes | Section 2, worked example 2. Graphical solving; later algebraic system-solving methods are not introduced. |
| Set 7 · p. 9 | Direction, maximum/minimum, union of domain intervals, combined range | Section 3, worked example 3 and disconnected-domain discussion. Read the original filled endpoints. |
| Set 8–9 · p. 9 | Decreasing curves, approached bounds, plateaus, extrema and interval notation | Section 3, interval and extrema definitions, worked example 3, and answer key; no separate example of an approached bound. |
| Go 10, 12, 13 · p. 10 | Arithmetic explicit/recursive rules; indices beginning at 0 or 1; skipped indices | Section 4; explicit/recursive reminder, index guidance, and answer key. |
| Go 11, 14, 15 · p. 10 | Geometric explicit/recursive rules; negative outputs; zero and nonstandard starting indices; missing terms | Section 4; explicit/recursive reminder, index guidance, and answer key. |
Source: Mathematics Vision Project, Secondary Mathematics One, Module 3: Features of Functions (May 2016), Section 3.2, “Floating Down the River,” printed pp. 6–10, with the corresponding teacher notes. The river-depth discussion uses the student packet’s data; the other numerical worked examples and coordinate diagrams are original.
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