Math 8 · 8.F.B.5

Graphs that tell a story

A graph has no numbers on it, only a shape. What can the shape alone tell you about what happened, and how do you draw that shape from a story?

You need

Reading values off a graphMath 8

Where you are 5 in Graphs of functions

Functions13

What a function is5

Graphs of functions5

Features of a graph10 Graphing a function from a table

Graphs that tell a story

Reading values off a graph Solving f(x) = g(x) with graphs and tables

When you reach for it

Many graphs come with no numbers worth reading: a sketch in a report, a curve on a weather app, a line drawn on a board to make a point. The shape still says a lot. When did the quantity grow, shrink or hold still? Quickly or slowly, steadily or not? When was it highest and lowest? This lesson reads those answers in words, then runs the other way and draws the shape from a story.

Reading exact values off a graph with a scale is Reading values off a graph. The pictures here are the same graphs of functions, usually with time across. In Modeling with linear relationships graphs are read with numbers, rates and units; here only the shape is used, and curves join the straight pieces.

The procedure

Day 1: reading the shape

Read the axes first. “Distance from home” and “speed” can draw the same shape and tell different stories, so nothing can be read until both axes are named. Then read left to right, because moving right means a later time.

A piece is increasing if it rises as you move right, decreasing if it falls and constant if it is flat. A flat piece says only that this quantity did not change. These words are made exact for stretches of input in Increasing, decreasing, constant, positive and negative intervals.
A steeper piece means faster change: over the same width of time it climbs or drops through more height. This is Slope without numbers, and it holds for falls too.
A straight piece means steady change, the same amount in every equal stretch of time, a Constant rate of change and linear tables. A curved piece means the rate itself is changing: a rising curve that gets steeper grows faster and faster, one that flattens grows more and more slowly. See Concavity: when the rate of change itself changes and Linear and nonlinear functions.
The highest point and lowest point are where the quantity is greatest and least. Either may be reached more than once, or held along a flat piece.
A corner, where pieces of different steepness meet, is where something in the story changed.

Matching a graph to a story turns this around: list every feature the story fixes (where it starts, the order of rises, falls and flat stretches, which change is faster, where it ends) and rule out every graph that breaks even one.

Day 2: drawing the shape

Label the axes, saying what any distance is measured from. When the story gives no numbers, the sketch needs no scale.

Go through the story one sentence at a time. Each sentence fixes something about a piece: rising, falling or level; steady (straight) or speeding up or slowing down (curved); steeper or gentler than another piece; where it starts and ends. Draw only what the story says. “Walked back more slowly” fixes that the fall is gentler than the rise, not how much gentler.

Choose between jumps and an unbroken graph. Ask whether the quantity could be caught halfway between two of its values. A distance, a height, the water in a bottle or a speed passes through every value in between, so its graph is unbroken and the pieces join. A count of whole things, such as lengths finished or people in a room, changes all at once, so its graph is flat pieces with jumps between them. At a jump a filled dot marks the value the quantity has at that moment and an open circle the value it no longer has. See Continuous graphs, jumps and holes.

Check the sketch against every sentence, both ways: find each sentence in the sketch, and find the sentence behind each feature of the sketch. A feature with no sentence behind it is a claim the story never made. Many sketches can fit one story; a sketch is wrong only when it breaks a sentence.

Worked examples

Example 1

Day 1: a swim, described

The graph shows Rosalind’s distance from the wall of a pool she starts at. Describe what happened, without numbers.

Figure 1Rosalind’s distance from the wall she started at. There is no scale, and the shape still tells the story.

Solution

The graph starts at 0, so she starts at the wall.

The first piece is increasing and straight: she swims away at a steady pace. It ends at the highest point, where the distance starts to fall, so she turned there. In a pool that is the far wall: one length.
The second piece is decreasing, straight, and less steep than the first: she swims back steadily, more slowly, so it takes longer and the piece is wider.
The third piece is constant at 0: she rests at the starting wall. This and the very start are the lowest points.
The last piece rises and is the gentlest of the moving pieces: a third length, the slowest, ending back at the far wall.

The two highest points are at the same height, as they must be, since each is the length of the pool. The graph gives no number for that length or for any time.

Example 2

Day 1: a roller coaster ride

The graph shows a roller coaster car’s height through one ride. Describe the ride, saying which pieces are straight and which curved, and where the highest and lowest points are.

Figure 2The height of a roller coaster car through one ride.

Solution

The graph starts constant: the car waits at the station.
It rises along a straight piece, so the height increases at a steady rate: the lift hill. Its top is the highest point of the ride.
It then falls along a curve, nearly flat just after the top, steep in the middle, flat again near the bottom. The height falls slowly as the car tips over, then quickly, then slowly as the track levels out.
The bottom of that drop is the lowest point, and it is below the station: the track dips lower than where the ride started.
A smaller hill follows, curved on both sides, its top well below the first.
The graph ends constant at the starting height: back at the station.

The lift and the flat stretches are straight; the drop and the second hill are curved. The graph shows how high the car is, not how fast it moves along the track.

Example 3

Day 1: matching stories to graphs

Graphs A to D show the water in a bottle against time. Match each story to its graph, then tell a story for the graph left over.

Quillon fills his empty bottle at a fountain, leaves it untouched through a lesson, then drinks steadily on the walk home until it is empty.
Pilar’s fountain only trickles, so her empty bottle takes a long time to fill. Then she takes three quick gulps, pausing after each.
Marguerite fills her empty bottle, starts drinking at once, drinks steadily until half is gone, and saves the rest.

Figure 3Four graphs of the water in a bottle, drawn on the same scales.

Solution

Story 1 needs a flat piece between filling and drinking, then a steady fall that ends at zero. B falls the moment it is full, B and C never reach zero, and D ends full. Only A fits, so story 1 is graph A.

Story 2 needs a slow fill, a gentle first piece. A, B and D start steeply; only C starts gently, and its three short drops, each followed by a flat piece, are the gulps and pauses. So story 2 is graph C.

Story 3 needs a straight fall starting right at the top, stopping at half, then flat. B and D both start this way, but D rises again, a refill the story never mentions. So story 3 is graph B.

Graph D: the bottle is filled quickly, its owner drinks steadily straight away until half is gone, tops it up again as steeply as the first fill (the same fountain), then leaves it full. Notice that A, B and D begin alike; only their later pieces tell them apart.

Example 4

Day 2: sketching a walk to the park

Nikolai walks his dog from home to the park at a steady pace. They stay at the park while the dog plays. The dog is tired on the way home, so they walk back along the same path, more slowly, until they reach home. Sketch Nikolai’s distance from home against time.

Solution

The axes are time and distance from home, with no scale, since the story gives no numbers.

“From home to the park at a steady pace”: start at 0 and rise along a straight piece.
“They stay at the park”: the distance does not change, so a flat piece at the highest level.
“Back along the same path, more slowly, until they reach home”: fall to 0, through the same height as the rise, and less steep than the first piece, so wider. Nothing says the pace changes on the way, so a straight piece is the simple choice.

Distance cannot jump, so the pieces join.

Figure 4A sketch of Nikolai’s distance from home. The way back is less steep, because it is slower.

Checking back: start at 0 (home), straight rise (steady pace), flat top (park), gentler fall (more slowly), end at 0 (home). Every feature has a sentence and every sentence is shown. The dog running about the park does not move Nikolai, so it does not appear.

Example 5

Day 2: jumps or an unbroken graph

Olufemi swims four lengths of a pool at one steady pace, turning at each wall without stopping, then rests at the wall he started from. Sketch, against time, (a) his distance from the starting wall and (b) the number of lengths he has finished.

Solution

(a) A distance passes through every value in between, so this graph is unbroken. One steady pace means straight pieces of the same steepness: out to the far wall, back to 0, out, back, then flat at 0 while he rests. The touches at the walls are corners, not gaps.

(b) The number finished is a count. It is 0 until he touches the far wall, and at that instant it is 1; it is never anything in between, so this graph changes only by jumps. It is a staircase of flat steps at 0, 1, 2, 3 and 4, with a jump at each touch, then flat at 4 while he rests. At each jump the filled dot sits on the new level, since at the touch that length counts as finished, and an open circle ends the old level. The story gives the values 0 to 4, so this axis gets a scale, and the 0 step is drawn just above the time axis so it can be seen.

Figure 5One swim, two quantities. The distance passes through every value on the way and never jumps; the count of finished lengths changes only in jumps.

The check: four touches in the story, four corners in (a) and four jumps in (b), at the same moments. One quantity moves through every value; the other counts.

Example 6

Day 2: a skydiver’s speed

Marguerite jumps from a plane. Her downward speed builds quickly at first, then more and more slowly, until it levels off at a steady top speed. She falls at that speed for a while, then opens her parachute: her speed drops sharply, then levels off at a much lower steady speed, which she keeps until she lands. Sketch her speed against time, until she is standing on the ground.

Solution

Her downward speed is 0 as she jumps, so the graph starts there.

“Builds quickly at first, then more and more slowly”: increasing and curved, steep at the start and flattening out. A straight piece would mean the same gain every second.
“Levels off at a steady top speed”: the curve runs into a constant piece, the highest point, marked by the dashed line.
“Drops sharply, then levels off at a much lower steady speed”: a steep fall curving into a long flat piece far below the top speed.
Landing brings the speed to 0 in a very short time: a piece that is almost vertical, but still not a jump, because a speed passes through every value on the way. Then flat at 0.

Figure 6A sketch of Marguerite’s speed from the jump to standing on the ground.

The graph is not a picture of the fall: she is moving downwards whenever the speed is above 0, including along both flat pieces.

Example 7

Day 2: checking a sketch against every sentence

(1) A pelican sits on top of a pier post, then sinks into a small dip as it takes off. (2) It then gains height at a steady rate, up to the highest point of its flight. (3) From there it dives steeply to the water. (4) It floats on the water while it swallows a fish, (5) then flies back up to its post, where it settles.

The left-hand graph is a proposed sketch of the pelican’s height above the water. Say which sentences it breaks, and correct it.

Figure 7A proposed sketch, left, and the sketch after checking it against every sentence of the story, right.

Solution

Height is measured from the water, so the water’s surface is the time axis.

(1) A flat piece, then a small dip: shown.
(2) A steady climb adds the same height every second, so this piece should be straight. The sketch’s curve gets steeper, saying the pelican rose faster and faster: it breaks sentence (2).
(3) The dive is steep but stops above the axis, so the pelican never reaches the water: it breaks sentence (3).
(4) Floating on the water is a height of 0, so this flat stretch must lie on the axis: it also breaks sentence (4).
(5) The pelican settles on the post it left, so the last flat piece must be at the starting height. The sketch ends higher: it breaks sentence (5).

The corrected sketch, right, has a straight climb, a dive to the axis, a flat stretch on the axis and a final flat piece level with the first. Its last climb is drawn straight, but the story does not say the pelican flies back up steadily, so a curved climb ending at the post’s height would be just as correct.

Boundaries

Labeled axes are needed. Without them a rising piece could mean anything.
Steeper means faster only on the same scales: pieces of one graph, or graphs side by side on shared scales as here. Stretching an axis makes any piece look steeper.
A graph shows one quantity. A flat distance-from-home piece says the distance did not change, not that the person stood still; walking in a circle around home draws the same line. A falling piece is not a downhill path, and height is not speed.
Jumps belong only to quantities that change all at once. A vertical segment would give one moment two values, which the graph of a function never does.
A sketch from words is one of many. The story fixes order, direction, which piece is steeper, and where the graph starts and ends; not how long a pause lasts, unless it says.
Exact amounts need a scale: Reading values off a graph and Average rate of change.

Where this goes next

Increasing, decreasing and constant become statements about intervals, written with inequalities, in Increasing, decreasing, constant, positive and negative intervals; highest and lowest points and flat stretches are among the Features of a graph; where a graph’s far ends are heading is End behavior. Graphs built from pieces, each with its own rule over its own stretch, are what Graphing a piecewise function draws from formulas. Numbers on the steepness of a straight piece are Slope and Rate of change; the bending of curves is Concavity: when the rate of change itself changes; unbroken and jumping graphs are Continuous graphs, jumps and holes.

Exercises

Day 1: describing a graph

Exercises 1 to 6 use the graph of a kite’s height above the ground.

Figure 8A kite’s height above the ground, for Exercises 1 to 6.

Over which pieces is the height increasing, decreasing and constant?
Of the two rising pieces, which is steeper, and what does that say about the height there?
Which pieces are straight and which curved? What does a straight piece say about the height?
Where are the highest point and the lowest? Is the lowest point lower than the start, A?
On the drop from B to C, does the height fall at the same rate all the way? Describe it.
Is the kite higher at G than at A? What does the flat piece from F to G say happened, and what can it not tell you?

Day 1: matching graphs and stories

Exercises 7 to 10 use graphs P, Q and R of a swimmer’s distance from the starting wall.

Figure 9Three swims, for Exercises 7 to 10, drawn on the same scales.

Marguerite swims one length, holds the far wall for a rest, then swims back. Which graph?
Quillon swims three lengths at one steady pace without stopping. Which graph?
Pilar stops halfway along her first length to fix her goggles, finishes the length at her starting pace, and swims straight back, more slowly. Which graph?
Rosalind swims one length and climbs out at the far end. Does P, Q or R fit? Describe hers.
A graph of the water in Nikolai’s bottle rises steeply in a straight piece from zero, stays flat, then falls along a curve that is steep at first and gets gentler until it reaches zero. Tell a story that fits.

Day 2: sketching from a story

Olufemi walks his dog toward the park at a steady pace. Halfway there he stops to talk to a neighbor, then walks on to the park faster than before. He sits on a bench while the dog plays, then they jog straight home. Sketch Olufemi’s distance from home against time.
For the same walk, sketch the total distance Olufemi has covered. How does it differ from Exercise 12?
Rosalind’s bottle is half full when she leaves home. She drinks steadily on the way to school until it is empty, refills it at the school fountain, then leaves it full on her desk all morning. Sketch the water in her bottle against time.
Can the graph jump, or must it be unbroken? (a) The water in a bottle as it fills. (b) The number of bottles in a recycling bin through a day. (c) A skydiver’s height. (d) The number of people in a lift. (e) A dog’s distance from home.
Sketch the height above the ground of Marguerite, the skydiver of Example 6, from the jump until she is standing on the ground.
Before a puppet show, children come into the room one at a time until all 24 seats are full, the show runs, then the 24 children leave one at a time. Sketch the number of children in the room against time.

Day 2: checking a sketch

Story: Pilar fills her empty bottle, drinks half of it in one go, then sips the rest slowly through the afternoon until it is empty. A sketch rises steeply from zero, falls steeply to half, then falls just as steeply to zero. Which part does it break, and what is the fix?
A sketch for Example 4 shows the way home as a vertical segment from the park level down to zero. What would that claim, and why can it not be right?

Explain

Two graphs have the same shape: a straight rise from zero, a flat stretch, a straight fall back to zero. One is a walker’s distance from home; the other is the walker’s speed. Describe what someone watching would see in each case.
A sketch from a story has no numbers. Explain how it can still be wrong, and name three features a story can fix.
On a rising curve, how can you tell from the shape whether the quantity grows faster and faster or more and more slowly? Use the first part of the skydiver’s speed graph in Example 6.

Answers

Increasing from A to B and from C to D; decreasing from B to C and from E to F; constant from D to E and from F to G. Up as you move right is increasing, down is decreasing, flat is constant.
C to D is steeper than A to B, so the height rises faster there: more height in the same width of time.
Straight: A to B, C to D, D to E, F to G. Curved: B to C and E to F. A straight piece means the height changes by the same amount in every equal stretch of time; on a flat piece that amount is zero.
The highest point is B and the lowest point is C. Yes, C is lower than A: the kite dips below the height it started at.
No. The curve is nearly flat just after B, steepest halfway and nearly flat near C, so the height falls slowly, then fastest about halfway, then slowly again.
No, A and G are at the same height. From F to G the height does not change; the kite could be hanging in one spot or drifting sideways at that height, and a height graph cannot say which.
Graph P: up to the far wall, flat there (the rest), back down to zero. Q has no flat piece and R’s flat piece is halfway.
Graph Q: out, back, out, equally steep, no flat piece, ending at the far wall.
Graph R: up to half height, flat (the goggles), up again just as steeply to the top, then down less steeply (more slowly) with no rest.
None fits. Hers rises from zero to the top and stays flat there, since out of the pool at the far end she stays at that distance. P has the flat top but comes back down.
One story: Nikolai fills his empty bottle quickly at a steady fountain, leaves it for a while, then drinks, gulping at first and then sipping more and more slowly until it is empty.
From zero, a straight rise to the halfway level; flat (the talk); a steeper straight rise to the park level (faster); flat at the park level, the highest point (the bench); then a straight fall to zero, steeper than either walk since jogging is faster than walking. All pieces join.
The same up to the end of the bench stretch, but then it keeps rising, as steeply as the jog, ending at twice the park level. Total distance never decreases; walking home adds to it.
Starts at half full, not zero; a straight fall to zero, the lowest point; a steep rise to full, the highest point (the refill); then flat at full. The pieces join.
(a) Unbroken. (b) Can jump: bottles arrive whole. (c) Unbroken. (d) Can jump: people get in and out whole. (e) Unbroken. Water, height and distance pass through every value in between.
It starts at the plane’s height and only falls: gently at first, then steeper as her speed builds; a straight steep piece at top speed; a quick bend into a much gentler straight piece when the parachute opens, down to zero at landing; then flat at zero.
A staircase: 24 jumps of 1 up from zero with flat pieces between; flat at 24, the highest level, during the show; then 24 jumps of 1 back down to zero. A count of people changes all at once, so it jumps, though from far away each staircase looks like a slope.
It breaks “sips the rest slowly”. The second fall must be gentler than the first and take longer: a gentle straight piece, or a slow staircase of small sips. The rest is right.
That at one moment Nikolai is at every distance from the park to home at once, getting home in no time. Distance passes through every value between, which takes time, so the way back is a falling piece that is not vertical; here, since he walks back more slowly, it is gentler than the way out.
Distance: the walker leaves home at a steady pace, stays the same distance away for a while, then walks steadily back and ends at home. Speed: the walker speeds up steadily from standing, walks at a constant speed without stopping, then slows steadily to a stop; the speed graph does not say which way they went or where they ended.
It is wrong if it breaks a sentence of the story, which needs no numbers. A story can fix the order of rising, falling and flat pieces; which of two pieces is steeper; and where the graph starts and ends. It can also fix straight or curved, and unbroken or jumping.
Compare the rise over equal stretches of time. If the curve gets steeper, each stretch adds more than the last: faster and faster. If it flattens, each adds less: more and more slowly. The skydiver’s speed curve is steep at the start and flattens toward top speed, so her speed grows quickly at first, then more and more slowly.