identify axes and units → read height → interpret a flat section → read endpoint
Lesson 34.2 · Graphs tell journeys
Read What Happened
Turn rises, flats, and falls into a real-world story.
On a travel day, a graph can show when a train moved, waited, or returned toward a station. Before calculating anything, name both axes and read the journey from left to right. The line is not a road drawing: its height records the quantity on the vertical axis.
See the story before the arithmetic
Read left to right. From $0$ to $30$ minutes, distance increases. From $30$ to $50$ minutes, time passes but distance stays at $30$ km, so the car is stopped. Then distance increases again.
Boundary idea: a falling line does not mean time moves backwards. It means the vertical quantity decreases as time moves forward.
Worked example · A horse-trail pause
A trail graph passes through $(0,0)$, $(1,4)$, $(2,4)$, and $(3,7)$, where $x$ is time in hours and $y$ is distance from the stable in kilometres.
Name the axes. Horizontal: time in hours. Vertical: distance from the stable in kilometres.
Locate the flat section. From hour $1$ to hour $2$, distance remains $4$ km.
Interpret, do not just describe. The rider stops for $1$ hour, perhaps to let the horse drink.
Check the ending. At hour $3$, the rider is $7$ km from the stable. That is a location relative to the stable, not necessarily $7$ km travelled altogether.
Practice · Read each graph as evidence
Guided: axes → shape → meaning
(b) Explain what the flat section from hour $2$ to hour $4$ means, using the frame “The amount of water ___ because ___.”
(c) Calculate the signed rate from hour $4$ to hour $7$, in litres per hour.
(d) Choose REFILLING or USING for the final section and justify with the sign of the rate.
signed changes → flat means zero change → signed rate → connect sign to action
(b) Calculate the average rate of change from minute $15$ to minute $35$, in degrees Celsius per minute.
(c) Identify the time interval when the temperature stays constant.
(d) Explain why “the tray cooled for all $50$ minutes” is not accurate.
change is ending minus starting → divide by elapsed time → find zero-rate interval → challenge the claim
From here, choose the important graph evidence yourself.
(b) Identify the interval when Mia stays in one place.
(c) Calculate the signed rate from hour $3$ to hour $5$, in kilometres per hour.
(d) Explain what the falling line means without saying that time or total distance travelled decreases.
maximum height → flat interval → negative rate → distinguish position from total travel
(b) Calculate Lake's rate in kilometres per hour.
(c) Choose the faster route and justify using both rates.
(d) Decide which route better fits a relaxed ride no faster than $3.5$ km/h.
same distance, different time → calculate each slope → compare → apply a real constraint
Choose one story: A Mia begins at camp, rides steadily for $4$ hours, and stops $3$ km away. B Mia begins $3$ km from camp, rides farther away, rests for $1$ hour, then returns to her starting distance from camp. C Mia begins $7$ km from camp and rides back continuously.
(a) Choose A, B, or C.(b) State one coordinate that supports your choice.
(c) State one interval whose shape supports your choice.
starting point → rises/flats/falls → endpoint → match all evidence, not one clue
Targeted review
same point order → signed changes → divide → attach kg/box
substitute into m = change in y / change in x → solve the resulting equation
unequal time steps → divide each temperature change by its own time change → compare
Retention check
percent to decimal → multiply principal, rate, time → add principal
Show answers
Problem 1a
The horizontal axis measures time in hours.
Problem 1b
The vertical axis measures distance from the station in kilometres.
Problem 1c
At $1.5$ hours, the train is $60$ km from the station.
Problem 1d
$1.5-1=0.5$ hour, or $30$ minutes.
Problem 1e
The train ends $120$ km from the station.
Problem 2a
$\frac{56-40}{2-0}=8$ L/hour.
Problem 2b
The graph is flat, so the net change in water is $0$ litres: the amount stays constant. (Either nothing flows in or out, or inflow and outflow exactly balance.)
Problem 2c
$\frac{32-56}{7-4}=\frac{-24}{3}=-8$ L/hour.
Problem 2d
USING; the rate is negative, so the amount in the tank decreases.
Problem 3a
$60-90=-30^\circ$C.
Problem 3b
$\frac{40-60}{35-15}=\frac{-20}{20}=-1^\circ$C/min.
Problem 3c
From minute $35$ to minute $50$.
Problem 3d
The temperature stays at $40^\circ$C during the final $15$ minutes, so it is not cooling then.
Problem 4a
The farthest distance is $6$ km.
Problem 4b
From hour $2$ to hour $3$.
Problem 4c
$\frac{2-6}{5-3}=\frac{-4}{2}=-2$ km/hour.
Problem 4d
As time moves forward, Mia gets $2$ km closer to the guesthouse each hour. Her distance from the guesthouse decreases.
Problem 5a
Pine: $\frac{12-0}{3-0}=4$ km/hour.
Problem 5b
Lake: $\frac{12-0}{4-0}=3$ km/hour.
Problem 5c
Pine is faster because $4$ km/hour is greater than $3$ km/hour.
Problem 5d
Choose Lake: $3$ km/hour is no faster than the $3.5$ km/hour limit.
Problem 6a
Story B.
Problem 6b
For example, $(0,3)$ shows that Mia begins $3$ km from camp; $(4,3)$ shows that she returns to that distance.
Problem 6c
For example, the graph is flat from hour $1$ to hour $2$, showing a $1$-hour rest.
Problem 7c
$m=\frac{25}{5}=5$ kg/box.
Problem 8a
$-3=\frac{y-14}{5-2}$.
Problem 8b
$$-3=\frac{y-14}{3}\quad\mid\times3$$$$-9=y-14\quad\mid+14$$$$5=y$$
Problem 9a
$\frac{76-88}{3-0}=\frac{-12}{3}=-4^\circ$C/min.
Problem 9b
$\frac{56-76}{8-3}=\frac{-20}{5}=-4^\circ$C/min.
Problem 9c
Yes. Both interval rates are $-4^\circ$C/min.
Problem 10a
$I=Prt=450(0.04)(3)=€54$.
Problem 10b
$450+54=504$, so the total is €504.
Next · Make the graph speak algebraYou can now read a graph as a sequence of real events. Next, you will lift exact points from a straight graph, organise them in a table, and compress the whole pattern into one equation.