(b) Read how many hours it takes to reach $22$ km.
(c) Write one sentence explaining what the point $(2,14)$ means.
find x on the bottom axis, go up to the line, read y → for part (b) start from y and go across → explain the pair with units
Lesson 34.1 · Interpreting Graphs
Travel Graphs
Read the start, the rate, and the story in every line.
A graph is a trip told in one picture
Mia's family travels a lot, so graphs of journeys are everywhere: bike routes, trains, buses, hiking trails. Before you calculate anything, read the story of the graph in three steps.
1 · AxesWhat does $x$ measure? What does $y$ measure? Which units?
2 · StartWhere does the line cross the $y$-axis? That is the value when $x=0$.
3 · ChangeHow much does $y$ change for each $+1$ in $x$? That is the rate, with units.
Worked example · A bike ride along the Valencia coast
Mia cycles at a steady speed away from her campsite. Her father starts the stopwatch when they are already $12$ km out.
Axes. $x$ is hours riding. $y$ is kilometres from the campsite.
Start. The line meets the $y$-axis at $(0,12)$: at time $0$ they were already $12$ km from the campsite.
Change. From $(2,48)$ to $(3,66)$: $\Delta y=66-48=18$ km and $\Delta x=3-2=1$ h, so the rate is $\frac{18\text{ km}}{1\text{ h}}=18$ km/h.
Sentence: “They start $12$ km from the campsite and move away at $18$ km each hour.”
Practice · Read the graph
Read points and meanings
(b) Read how many hours it takes to reach $22$ km.
(c) Write one sentence explaining what the point $(2,14)$ means.
choose two exact points → signed coordinate changes → divide → kilometres per hour
(a) Read the $y$-intercept.
(b) Explain what the $y$-intercept means for the bus. Use the frame: “At time $0$ the bus was already ___ .”
(c) Predict the distance at $x=5$ hours by continuing the pattern of the line.
the intercept is the start, not the rate → use the sentence frame → extend the line by one more hour of 45 km
From here, you decide what to read and what to calculate.
Compare, decrease, and scale
(b) Decide who is faster, and give both speeds in km/h.
(c) Estimate from the graph how many hours pass before they are side by side.
compare starts first, then rates, then the crossing point
(b) Explain what the point $(8,0)$ means in the barn.
(c) Decide whether the point $(10,-12)$ is sensible for the barn. Give a reason.
decreasing line → negative rate → x-intercept means the stock is gone → stay inside the real situation
(a) Calculate the correct rate using $(1,60)$ and $(2,100)$.
(b) Explain Sofia's mistake in one sentence.
(c) Predict the distance from Munich at $x=3$ hours.
read the numbers on the axes, never the number of squares → rate with units → continue the pattern
(b) Calculate her rate on the way out (from $x=0$ to $x=2$), with its sign.
(c) Calculate her rate on the way back (from $x=3$ to $x=5$), with its sign.
(d) Decide whether one constant rate describes the whole trip. Give a reason.
(e) Calculate the total distance she rode, and her distance from camp at $x=5$.
each straight piece has its own rate → a flat piece has rate 0 → the graph shows position, not distance travelled
Spaced review
Determine the missing coordinate $(8,y)$ and show how the known slope connects it to $(2,5)$.
coordinate change → use slope to connect the points → verify
(a) Calculate the rate from $1$ g to $3$ g.
(b) Calculate the rate from $3$ g to $6$ g.
(c) Decide whether the rate is constant and state it with units.
the x-steps are uneven, so divide each Δy by its own Δx
(a) Subtract $2$ from both sides and write the new equation.
(b) Solve for $x$ by multiplying both sides by $\frac43$.
undo the +2 first → multiply every term by the reciprocal
fence follows the edge, sand covers the surface
Show answers
Problem 1a
$y=18$ km.
Problem 1b
$x=4$ hours.
Problem 1c
After $2$ hours of walking, Mia is $14$ km from the village.
Problem 2a
$\Delta y=200-20=180$ km.
Problem 2b
$\Delta x=4-0=4$ hours.
Problem 2c
$\frac{180}{4}=45$ km/h.
Problem 3a
$y$-intercept $=20$, the point $(0,20)$.
Problem 3b
At time $0$ the bus was already $20$ km from the depot.
Problem 3c
$y=45(5)+20=245$ km (check: $200+45=245$).
Problem 4a
Marco starts $9$ km further along (his line begins at $9$, Lena's at $0$).
Problem 4b · Lena
Lena walks $5$ km/h.
Problem 4b · Marco
Marco walks $3$ km/h, so Lena is faster.
Problem 4c
About $4.5$ hours (they meet near $22.5$ km). Check: $5x=3x+9$ gives $x=4.5$.
Problem 5a
$\frac{24-48}{4-0}=-6$ bags per day. The stock falls by $6$ bags each day.
Problem 5b
After $8$ days the barn has $0$ bags: the hay has run out.
Problem 5c
Not sensible: $-12$ bags cannot exist. The graph is only meaningful from $0$ to $8$ days.
Problem 6a
$\frac{100-60}{2-1}=40$ km/h.
Problem 6b
She counted gridlines, but each gridline is worth $20$ km, so $2$ squares are $40$ km.
Problem 6c
$y=40(3)+20=140$ km.
Problem 7a
She rests for $3-2=1$ hour (the flat part, where the distance stays $16$ km).
Problem 7b
$\frac{16-0}{2-0}=8$ km/h away from camp.
Problem 7c
$\frac{0-16}{5-3}=-8$ km/h: the distance from camp falls by $8$ km each hour.
Problem 7d
No. The rate is $8$, then $0$ during the rest, then $-8$, so the graph is not one straight line.
Problem 7e · total ridden
$16+16=32$ km (the graph shows distance from camp, not distance travelled).
Problem 7e · from camp
$0$ km: she is back at camp.
Problem 8a
$\Delta x=8-2=6$.
Problem 8b
$\Delta y=\frac32\times6=9$.
Problem 8c
$y=5+9=14$.
Problem 9a
$\frac{13-7}{3-1}=\frac62=3$ € per gram.
Problem 9b
$\frac{22-13}{6-3}=\frac93=3$ € per gram.
Problem 9c
Yes, constant: both intervals give $3$ € per gram.
Problem 10a
$\frac34x=18$.
Problem 10b
$$\frac34x=18\quad\mid\times\frac43$$$$x=24$$So the trail is $24$ km long.
Problem 11a
$P=2(18+12)=60$ m.
Problem 11b
$A=18\times12=216$ m².
Next upThe next lesson turns this around: instead of reading a story from an equation, you will read two points off a graph and write the equation $y=mx+b$ yourself.