Math worksheet titled 'Applications of Functions: Interpreting Graphs' featuring three sections of graphs to match with real-world statements.

Let's solve each section of this worksheet step by step, interpreting the graphs and matching them to the correct real-life situations.

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Section A: Distance vs. Time Graphs

We are given three graphs (A, B, C) showing distance over time.

- Graph A: Starts slowly, then curves upward more steeply → Increasing speed (accelerating)
- Graph B: Straight line → Constant speed
- Graph C: Starts steep, then flattens out → Slowing down

Now match:

1. The distance travelled by a motorbike moving at a constant speed.
→ Constant speed means straight line → Graph B

2. The distance travelled by a car accelerating.
→ Speed increases → curve that gets steeper → Graph A

3. The distance travelled by a cyclist who starts fast, but gradually slows down.
→ Starts steep, then flattens → Graph C

✔ Answers for Section A:
1. B
2. A
3. C

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Section B: General Trends (e.g., birth rate, gas price)

Now we have four graphs (A, B, C, D) with different trends.

Let’s analyze each graph:

- Graph A: Starts flat, then sharply rises → Steady then rapid increase
- Graph B: Starts low, then gradually rises → Gradual increase
- Graph C: Rises, then falls → Increases then decreases
- Graph D: Steady upward slope → Steady increase

Now match:

1. The birth rate was steady but now it is rising.
→ Flat then increasing → Graph A

2. The price of gas has been rising steadily.
→ Constant increase → Graph D

3. Unemployment rose slowly for 5 years, but now it is rising dramatically.
→ Slow rise then sharp rise → Graph A (same as #1 — but let's check)

Wait — both #1 and #3 seem to point to Graph A? Let's double-check.

But actually:
- #1: "Steady" then rising → could be Graph A (flat then steep)
- #3: "Rose slowly for 5 years" → gradual increase, then "dramatically" → sudden steep rise → also matches Graph A

But wait — only one graph per question.

Looking again:
- Graph A: starts flat, then very steep rise
- Graph B: starts low, then gently rises (no flat start)
- Graph C: peaks then drops
- Graph D: straight line up

So:
- #1: Birth rate was steady → flat, then rising → Graph A
- #3: Unemployment rose slowly → gentle rise, then dramatic → Graph A again?

But can't assign same graph twice.

Wait — maybe Graph B is better for #3? But Graph B doesn’t have a flat start.

Actually, let's re-express:

- Graph A: Flat at first, then sharp rise → steady then rapid rise
- Graph B: Starts low, then curved up → slow then faster rise, but no flat period
- Graph C: Up then down → decreasing after peak
- Graph D: Steady linear rise

So:
1. Birth rate was steady, now rising → Graph A (flat then up)
2. Gas price rising steadily → Graph D (straight line)
3. Unemployment rose slowly for 5 years, now rising dramatically → This implies gradual rise then sharp rise → Graph A again?

But if both #1 and #3 go to A, that's a problem.

Wait — perhaps Graph B is intended for #3?

But Graph B starts low, not flat.

Alternatively, maybe Graph A is best for #3 because it shows slow initial rise (if you interpret the flat part as slow), but the flat part suggests no change, not slow rise.

Ah! Key difference:

- "Steady" = constant (flat)
- "Rising slowly" = increasing gradually (not flat)

So:
- #1: Was steady → flat, then rising → Graph A
- #3: Rose slowly (so increasing, not flat), then dramatically → so should show gentle curve then steep rise → Graph B?

But Graph B starts low, then curves up — yes, but no flat part.

Wait — look at Graph B: It starts at a certain level, then slowly rises, then accelerates — so it could represent slow rise then dramatic rise → yes, Graph B fits #3.

But Graph A starts flat, then rises sharply — so “was steady” → flat → Graph A fits #1.

Then:
- #2: Gas price rising steadily → straight line → Graph D
- #4: Inflation rising steadily but now falling → rising then falling → Graph C

So final assignment:

1. Birth rate was steady but now rising → Graph A
2. Price of gas rising steadily → Graph D
3. Unemployment rose slowly for 5 years, now rising dramatically → Graph B
4. Inflation had been rising steadily but now falling → Graph C

✔ Answers for Section B:
1. A
2. D
3. B
4. C

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Section C: Real-Life Situations (Various Contexts)

Now four graphs again (A, B, C, D):

- Graph A: Downward sloping straight line → steady decrease
- Graph B: U-shaped curve (up then down) → rises to peak, then falls
- Graph C: Upward sloping straight line → steady increase
- Graph D: Step-like decrease with a smooth dip → decreasing in steps, then continuous drop

Now match:

1. The number of Euros that can be purchased with £'s.
→ This is about exchange rate. If the value of the pound falls, you get fewer euros.
So if the euro strengthens or pound weakens → fewer euros per pound → decreasing
But is it steady? Possibly.
→ Likely steady decrease → Graph A

2. The price of housing has been falling steadily since 2011.
→ Falling steadily → straight downward line → Graph A

Wait — both #1 and #2 seem to go to Graph A?

But let's see: both involve steady decrease.

But Graph A is the only one showing steady decrease.

So both could fit, but we need to pick one per graph.

But there are four questions and four graphs.

Wait — let's check:

- Graph A: Linear decrease
- Graph B: Peak then fall (like temperature)
- Graph C: Linear increase
- Graph D: Stepwise decrease

So:

1. Number of Euros per £ → depends on exchange rate. If pound depreciates, fewer euros → decreasing
But is it steady? Maybe not — but if assumed steady, Graph A

2. Housing price falling steadily since 2011 → steady decrease → Graph A

Again conflict.

But Graph D shows stepwise decrease — like someone drinking tea in sips.

So let's reconsider.

Look at Graph D: It has steps down, then a smooth drop — looks like drinking tea in sips → amount of tea decreases in steps → Graph D

So:
4. Amount of tea being drunk → Graph D

Now:
3. Temperature during 24 hours of summer day →
- Starts low, rises to peak (around noon), then falls → U-shaped curve → Graph B

So:
3. → Graph B

Now:
1. Euros per £ → exchange rate. If the pound loses value, you get fewer euros.
But is it linear? Not necessarily. But if it's steadily declining, then Graph A
But Graph A is linear.

2. Housing price falling steadily → Graph A

But both want Graph A?

Wait — maybe Graph A is for one, and Graph D is for tea.

But Graph D is clearly stepwise — perfect for tea being drunk in sips → Graph D

So:
4. → Graph D

3. → Graph B (temperature: morning rise, afternoon fall)

Now:
1. and 2. both seem to want steady decrease → Graph A

But Graph A is the only one showing linear decrease

So which one?

But wait — Graph A is a straight line down → steady decrease

So:
- Housing price falling steadily → Graph A
- Euros per £: if the exchange rate is falling steadily, also Graph A

But we can’t assign same graph twice.

So perhaps one of them isn't steady?

But the statement says:
> 1) The number of Euros that can be purchased with £'s.

This is value of the pound. If it's weakening, the amount of euros you get decreases.

But is it steady? Not necessarily — but the statement doesn't say it's steady, just that it's changing.

But Graph A is steady.

But Graph D is for tea — that’s clear.

Let’s list:

- Graph A: linear decrease
- Graph B: parabolic (peak then down)
- Graph C: linear increase
- Graph D: stepwise decrease

Now:

1. Euros per £ → if the pound is losing value, fewer euros per pound → decreasing
But is it steady? Maybe not — but without info, assume steady → Graph A

2. Housing price falling steadily → Graph A

But again conflict.

Wait — perhaps Graph A is for housing, and Graph D for tea.

But what about Euros per £?

If the exchange rate changes due to market forces, it might not be linear.

But unless specified, we assume steady.

But both statements suggest steady decrease.

But maybe Graph A is used for one, and the other is different?

Wait — let’s read carefully:

1. The number of Euros that can be purchased with £'s.
→ This is pound depreciation → decrease
But if the pound is weakening, the number of euros per pound decreases → downward trend

2. Housing price falling steadily since 2011 → steady decrease

So both are decreasing, possibly steady

But only Graph A shows steady decrease

So either:
- Both use Graph A — but can’t
- Or one of them is not steady

But Graph D is stepwise — not steady

So likely, Graph A is for housing, and Graph D is for tea

But what about Euros per £?

Maybe it’s not steady — but the graph must match.

Wait — Graph D has steps — like drinking tea in sips — so only for tea

So Graph D → 4

Graph B → 3 (temperature: rises, peaks, falls)

Graph C → ? — only one left: increase

But all others are decreasing?

No — Graph C is increasing → but none of the remaining options are increasing?

Wait — Graph C is increasing → but our statements are mostly decreasing

But let’s list:

1. Euros per £ → decreasing
2. Housing price → decreasing
3. Temperature → rises then falls → Graph B
4. Tea → decreasing in steps → Graph D

So Graph C is increasing — but none of the statements describe an increase?

Wait — Graph C is upward straight line → steady increase

But none of the statements say anything increasing?

Wait — Graph A is decreasing

Graph B: up then down

Graph C: up

Graph D: down

So Graph C is only increasing — but no statement describes an increase?

Wait — Statement 1: "The number of Euros that can be purchased with £'s." — if the pound strengthens, you get more euros — but the statement doesn’t say that.

But if the pound weakens, you get fewer euros — so decreasing

Similarly, housing falling → decreasing

So why is Graph C there?

Unless I misread.

Wait — perhaps Graph C is for something else?

Let’s recheck:

1. Euros per £ → decreasing → Graph A or D
2. Housing price → decreasing → Graph A or D
3. Temperature → Graph B
4. Tea → Graph D

So Graph D is for tea

Graph B for temperature

Then Graph A for one of the decreasing ones

Graph C — only one left — but it's increasing

But no statement is increasing?

Wait — Graph C is from origin, increasing

But none of the statements say something is increasing?

Wait — Statement 1: "The number of Euros that can be purchased with £'s."

Suppose the pound is strengthening — then you get more euros → increasing

But the statement doesn't specify direction — but contextually, "number of Euros" — if it's increasing, you get more — but typically, if the pound is strong, you get more euros.

But the statement doesn't say it's increasing — it just says "the number" — but we need to infer.

But the graph must match the trend.

But all statements except #3 and #4 are decreasing?

Wait — Statement 1: "The number of Euros that can be purchased with £'s."

This is exchange rate: €/£

If the pound appreciates, €/£ increases → increases

If the pound depreciates, €/£ decreases → decreases

But which is it?

The statement doesn't say, but in many contexts, pound has been weakening — so decreasing

But Graph C is increasing — so maybe Graph C is not for this.

Wait — perhaps Graph C is for nothing?

But we have to assign.

Wait — Graph C is straight line from origin — starts at zero, increases

But Euros per £ — if you have £0, you get €0 — so origin makes sense

But if the exchange rate is increasing, meaning pound strengthens, then Graph C could work

But the statement doesn't say it's increasing — it just says "the number"

But Graph A is decreasing

So which one?

But in reality, exchange rates fluctuate, but if we assume steady increase, then Graph C

But the statement says nothing about trend — but the graph must reflect a trend.

But Graph C is increasing — so only if the number of euros per pound is increasing

Similarly, Graph A is decreasing

So which is it?

But the context of the worksheet suggests that Graph C is for increase, so likely:

- Graph C: increasing → Euros per £ if pound is strengthening

But the statement doesn't specify direction.

But Graph A is decreasing — so if pound is weakening, then Graph A

But Graph D is for tea — stepwise

Graph B for temperature

So only Graph C and Graph A are left for #1 and #2

But both are decreasing?

Wait — Graph C is increasing — so cannot be for decreasing

So:

- Graph C: increasing → must be for a situation where value is increasing

But none of the statements say increasing?

Wait — Statement 1: "The number of Euros that can be purchased with £'s."

If the pound is getting stronger, then you can buy more euros → increasing

But is that typical? Not necessarily.

But Graph C is the only one with positive slope from origin

So perhaps it's meant for this — assuming the pound is strengthening

But the statement doesn't say.

Alternatively, maybe Graph C is for something else

Wait — Statement 2: "Housing price falling steadily" → decreasing → Graph A

Statement 1: "Number of Euros per £" — if it's decreasing, then Graph A — but then both #1 and #2 would be Graph A

But can't.

So perhaps Graph A is for housing

Graph D is for tea

Graph B is for temperature

Then Graph C is for Euros per £ — but only if it's increasing

But the statement doesn't say.

Wait — perhaps Graph C is for Euros per £ if the pound is strengthening — but the statement doesn't specify.

But maybe the intended answer is:

- Graph C: increasing → Euros per £ (assuming pound strengthens)
- Graph A: decreasing → housing price falling

But then Graph A is used for #2

Graph D: tea

Graph B: temperature

Then Graph C: #1

But Graph C is from origin — if you have £0, you get €0 — so makes sense

And if the exchange rate is increasing, then Graph C

But the statement doesn't say it's increasing.

But perhaps it's implied.

Alternatively, maybe Graph A is for Euros per £ if it's decreasing.

But then housing would need another graph.

But Graph D is for tea — stepwise

So Graph A is the only one for steady decrease

So likely:

- Graph A: Housing price falling steadily → #2
- Graph D: Tea being drunk → #4
- Graph B: Temperature → #3
- Graph C: Euros per £ → #1 (but only if increasing)

But Graph C is increasing — so if the number of euros per pound is increasing, then yes.

But is that realistic?

Perhaps in some contexts.

Alternatively, maybe Graph C is for something else.

Wait — Graph C is straight line from origin — so it starts at zero and increases

But Euros per £ — if you have £0, you get €0 — so origin is fine

But if the exchange rate is constant, it would be horizontal — but here it's increasing

So if the exchange rate is increasing, then Graph C

So likely, the intended answer is:

1. Graph C — Euros per £ (assuming exchange rate increasing)
2. Graph A — Housing price falling
3. Graph B — Temperature
4. Graph D — Tea

But that seems odd.

Wait — Graph A is decreasing — so for housing falling

Graph D is stepwise — for tea

Graph B is peak then down — for temperature

Graph C is increasing — for Euros per £

So even though the statement doesn't specify, the graph must match.

So Graph C must be for Euros per £ if it's increasing.

But the statement says "the number" — not "increasing"

But perhaps it's implied.

Alternatively, maybe Graph A is for Euros per £ if it's decreasing.

But then housing would need a different graph.

But Graph D is for tea — stepwise

So only Graph A is for steady decrease

So Graph A must be for either #1 or #2

But Graph C is for increase

So perhaps the correct assignment is:

1. Graph C — Euros per £ (increasing)
2. Graph A — Housing price falling
3. Graph B — Temperature
4. Graph D — Tea

Even though #1 doesn't specify direction, Graph C is the only increasing graph, and it starts at origin — so it fits Euros per £ if the exchange rate is increasing.

Alternatively, maybe the intended answer is:

1. Graph A — Euros per £ (decreasing)
2. Graph A — Housing price falling — but can't

So impossible.

Wait — perhaps Graph C is not for #1

Let’s look at Graph C: straight line from origin — increasing

Is there any statement that could be increasing?

Only #1 — "number of Euros per £"

But if it's increasing, then Graph C

If decreasing, then Graph A

But both can't use Graph A

So likely, Graph C is for #1, and Graph A is for #2

So:

1. Graph C — Euros per £ (increasing)
2. Graph A — Housing price falling
3. Graph B — Temperature
4. Graph D — Tea

Yes, that works.

Even if the statement doesn't say "increasing", the graph must match the trend.

So Graph C is increasing — so it must be for a situation where the value is increasing.

Therefore, Euros per £ must be increasing — i.e., pound strengthening.

So it's acceptable.

✔ Final Answers for Section C:

1. C — The number of Euros that can be purchased with £'s. (assuming exchange rate increasing)
2. A — The price of housing has been falling steadily since 2011.
3. B — The temperature during the 24 hours of a summer’s day.
4. D — The amount of tea in a cup that is being drunk.

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✔ Final Answer Summary:

#### Section A:
1. B
2. A
3. C

#### Section B:
1. A
2. D
3. B
4. C

#### Section C:
1. C
2. A
3. B
4. D

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📌 Explanation Recap:

- Straight line (constant slope) → constant rate (speed, price, etc.)
- Curving up → accelerating (increasing rate)
- Curving down → decelerating (slowing down)
- Flat then up → steady then rising
- Up then down → peak then decline
- Stepwise → discrete changes (like sipping tea)
- From origin → starts at zero (e.g., tea in cup, money exchanged)

This logic helps match real-world scenarios to graphical representations.