Interpret, Sketch and Critique Story Graphs | Grade 8 Math | LevelKite

A story graph shows how a quantity changes over time. Here, each story graph follows a trip along one path, such as a straight street or a train track. Time is the input, on the horizontal axis. The traveler's distance from a starting point along the path is the output, on the vertical axis. If time is in minutes and distance is in city blocks, the point $(3, 3)$ means that 3 minutes after leaving, the traveler is 3 blocks from the start.

Read the graph from left to right, the way time moves, one piece at a time. A piece is one straight (linear) or curved (nonlinear) part; you already know how to tell those apart. On a graph of distance from a starting point along one path, plotted against time:

a rising (increasing) piece means the distance is growing, so the traveler is moving away;
a flat (constant) piece means the distance stays the same, so the traveler is resting;
a falling (decreasing) piece means the distance is shrinking, so the traveler is coming back.

The graph is not a picture of the road; it shows only how far the traveler is from the start. The graph below shows one such trip.

Priya's distance from home: rising from

(0, 0)

to

(6, 6)

, flat to

(12, 6)

, falling to

(16, 0)

, with marked points

(3, 3)

and

(14, 3)

.

Worked example

Reading Priya's trip

Priya walks along her straight street to the library and back. The graph above shows her distance from home, $d$ city blocks, $t$ minutes after she leaves, for $0 \le t \le 16$ . Its pieces join the points $(0, 0)$ , $(6, 6)$ , $(12, 6)$ and $(16, 0)$ , and the marked points are $(3, 3)$ and $(14, 3)$ . What does Priya do on each piece, how fast does she walk each way, and when is she 3 blocks from home?

The first piece rises in a straight line from (0, 0) to (6, 6), so her distance from home grows at a steady rate: she walks away from home. Her speed is the change in distance divided by the change in time. The change in time is 6 minutes, which is not zero, so the division makes sense.
$\frac{6 - 0}{6 - 0} = \frac{6}{6} = 1 \text{ block per minute}$
The second piece is flat from (6, 6) to (12, 6). Her distance stays at 6 blocks, so she is not moving: she is at the library. Subtract the start time from the end time to find how long she stays there.
$12 - 6 = 6 \text{ minutes}$
The third piece falls in a straight line from (12, 6) to (16, 0), so her distance shrinks to 0 at a steady rate: she walks back home. The result is negative because the distance is shrinking, and its size, 1.5 blocks per minute, is her speed. That is faster than her 1 block per minute on the way out.
$\frac{0 - 6}{16 - 12} = \frac{-6}{4} = -1.5 \text{ blocks per minute}$
Both marked points, (3, 3) and (14, 3), have a distance of 3. So she is 3 blocks from home twice: at 3 minutes, on the way to the library, and at 14 minutes, on the way home.

Answer

Priya walks away from home at 1 block per minute, stays at the library 6 blocks from home from $t = 6$ to $t = 12$ , and walks home faster, at 1.5 blocks per minute, arriving at $t = 16$ . She is 3 blocks from home at 3 minutes and again at 14 minutes.

Key concept

Direction, steepness and shape

On a graph of a traveler's distance from a starting point along one path, plotted against time, read each piece in three ways.

Direction: rising means moving away, flat means resting, and falling means coming back.
Steepness: how sharply a piece rises or falls shows the speed. On the same graph, a steeper piece covers more distance in each unit of time, so it means faster motion.
Shape: a straight (linear) piece keeps the same steepness, so the speed is steady. A curved (nonlinear) piece changes steepness, so the speed is changing: getting steeper means speeding up, and getting flatter means slowing down.

Now go from a story to a graph. Story: Eli lives on a straight street. He starts at his front door and jogs away, speeding up gradually for 4 minutes until he reaches a park 4 blocks from home. He rests at the park for 2 minutes. Then he walks back home at a steady speed and arrives 4 minutes later. The trip lasts $4 + 2 + 4 = 10$ minutes, so the graph uses time $t$ in minutes for $0 \le t \le 10$ and Eli's distance from home $d$ in blocks. It is built one phase of the story at a time.

Step 1: the jog away. From $t = 0$ to $t = 4$ , Eli's distance grows from 0 to 4 blocks, so the piece rises from $(0, 0)$ to $(4, 4)$ . Because he is speeding up, he covers more distance in each minute than in the minute before, so the piece must get steeper as time passes: its slope changes. On this graph a straight piece would mean one steady speed, so this piece is a curve that bends upward, as shown below.

Step 1: a curve from

(0, 0)

to

(4, 4)

that gets steeper, because Eli is speeding up.

Step 2: the rest. Eli rests for 2 minutes, from $t = 4$ to $t = 6$ , so his distance from home stays at 4 blocks. The next piece is flat, from $(4, 4)$ to $(6, 4)$ . It starts exactly where the curve ends, because Eli does not jump from place to place: at $t = 4$ he is exactly 4 blocks from home. The graph below shows both pieces.

Step 2: the flat piece from

(4, 4)

to

(6, 4)

shows the 2-minute rest at the park.

Step 3: the walk home. From $t = 6$ to $t = 10$ , Eli comes back, so his distance shrinks from 4 blocks to 0. The last piece falls from $(6, 4)$ to $(10, 0)$ . It is straight because his speed is steady: he covers 4 blocks in $10 - 6 = 4$ minutes, which is $4 \div 4 = 1$ block per minute. The finished graph below tells the whole story, with one piece for each phase.

Step 3: the straight falling piece from

(6, 4)

to

(10, 0)

shows the steady walk home.

You can also go the other way, from a graph to a story. To describe a graph of distance from a starting point along one path, plotted against time, take its pieces in order from left to right. For each piece, say:

what changes: moving away, resting or coming back, from whether the piece rises, stays flat or falls;
how fast: a steady speed, speeding up or slowing down, from its steepness and whether it is straight or curved;
over which times: from the piece's endpoints.

A curve may look like it gets steeper or flatter, but coordinates are stronger evidence than looks. The train graph below has marked points that give you those coordinates.

The train's distance from Station A: pieces meet at

(2, 1)

,

(4, 3)

and

(6, 4)

; marked points are

(1, 0.25)

,

(3, 2)

and

(5, 3.75)

.

Worked example

Describing a train trip

A train leaves Station A at $t = 0$ . Station B is 4 kilometers from Station A along the track. The graph above shows the train's distance from Station A along the track, $d$ kilometers, $t$ minutes after it leaves, for $0 \le t \le 8$ . It starts at $(0, 0)$ , its pieces meet at $(2, 1)$ , $(4, 3)$ and $(6, 4)$ , and it ends at $(8, 4)$ . The marked points are $(1, 0.25)$ , $(3, 2)$ and $(5, 3.75)$ . Describe the trip.

From t = 0 to t = 2 the graph is increasing and curved, rising from (0, 0) to (2, 1), so the train moves away from Station A. The curve looks steeper as time passes, and the marked point (1, 0.25) confirms it: the second minute adds more distance than the first. More distance in each minute means the train is speeding up.
$0.25 - 0 = 0.25 \text{ km}, \quad 1 - 0.25 = 0.75 \text{ km}$
From t = 2 to t = 4 the graph is increasing and straight, from (2, 1) to (4, 3). The marked point (3, 2) shows each minute adding the same 1 kilometer, so the train moves away at a steady speed of 1 kilometer per minute.
$2 - 1 = 1 \text{ km}, \quad 3 - 2 = 1 \text{ km}$
From t = 4 to t = 6 the graph is still increasing, from (4, 3) to (6, 4), but it is curved and gets flatter. The marked point (5, 3.75) shows the first minute adding 0.75 kilometer and the second only 0.25 kilometer, so the train keeps moving away but slows down.
$3.75 - 3 = 0.75 \text{ km}, \quad 4 - 3.75 = 0.25 \text{ km}$
From t = 6 to t = 8 the graph is constant at 4 kilometers, so the train is not moving. Station B is 4 kilometers from Station A, so the train is stopped at Station B, and the stop lasts 2 minutes.
$8 - 6 = 2 \text{ minutes}$

Answer

The train speeds up as it leaves Station A from $t = 0$ to $t = 2$ , runs at a steady 1 kilometer per minute from $t = 2$ to $t = 4$ , slows down from $t = 4$ to $t = 6$ as it reaches Station B, and waits at Station B from $t = 6$ to $t = 8$ .

Worked example

Fixing a wrong reading

Jalen describes the same train graph: “From $t = 2$ to $t = 4$ the train is speeding up, because that is the steepest part of the graph. From $t = 6$ to $t = 8$ the train is going fastest, because the graph is highest there.” Find each error and correct it.

Check the first claim against the piece from (2, 1) to (4, 3). Jalen is right that it is steep: it adds 1 kilometer each minute, while each one-minute step on the curved pieces adds at most 0.75 kilometer. But the piece is straight, so its steepness never changes. Dividing the change in distance by the change in time (2 minutes, not zero) gives one steady speed.
$\frac{3 - 1}{4 - 2} = \frac{2}{2} = 1 \text{ km per minute}$
Name the error: Jalen read a steep straight piece as speeding up. Steepness shows how fast the train is going, not whether its speed is changing. Speeding up shows as a curve that gets steeper, which happens from t = 0 to t = 2, where the two minutes add 0.25 kilometer and then 0.75 kilometer.
Check the second claim against the piece from (6, 4) to (8, 4). It is flat, so the distance does not change at all, and the train is not moving.
$4 - 4 = 0 \text{ km}$
Name the error: Jalen read a flat piece as motion because it sits high on the graph. Height shows how far the train is from Station A, not how fast it is going. The train is at its farthest point here, but it is stopped at Station B.

Answer

Jalen should say that the train moves at a steady 1 kilometer per minute from $t = 2$ to $t = 4$ , after speeding up from $t = 0$ to $t = 2$ , and that it is stopped at Station B from $t = 6$ to $t = 8$ .

Takeaway: On a graph of distance from a starting point along one path, plotted against time, each piece tells one part of the story. Whether a piece rises, stays flat or falls tells you away, resting or back; its steepness tells you how fast; and its shape tells you whether the speed is steady (straight) or changing (curved). Read the pieces in order, take times and distances from stated coordinates, and remember that the graph shows how far away the traveler is, not a picture of the road.