Analyzing Functions and Graphs: A math worksheet featuring questions on interpreting function graphs, matching real-world scenarios to distance-time graphs, and comparing motion patterns.

Let’s solve each question step by step, with clear explanations.

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Question 1:
> Which of the following best describes the graph of the function when x is between 0 and 10?

Looking at the graph:
- The curve rises from x=0 to about x=10 (peaks around x=10), then falls after that.
- But we’re only concerned with x between 0 and 10.
- The graph is curved, not a straight line → so it’s non-linear.
- From x=0 to x=10, the y-values are going up → so it’s increasing.

✔ Correct answer: c. Non-linear and increasing

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Question 2:
> Describe the characteristics of the function’s graph when x is between -2 and 0.

Looking at the second graph (the V-shaped one):
- Between x = -2 and x = 0, the graph goes downward from left to right.
- At x = -2, y ≈ 3; at x = 0, y = 5? Wait — let’s check again.

Actually, looking closely:
- At x = -2, the point is at (–2, 0)
- Then it goes up to (–1, 5) — that’s steeply increasing
- Then from (–1, 5) to (0, 3) — that’s decreasing

Wait — the question says “when x is between –2 and 0”.

So from x = –2 to x = –1: graph increases (from y=0 to y=5)

From x = –1 to x = 0: graph decreases (from y=5 to y=3)

So overall, it’s not monotonic — it increases then decreases.

But since the question asks for “characteristics”, we can say:

> The graph is linear in segments (piecewise linear), and first increases then decreases as x goes from –2 to 0. It reaches a maximum at x = –1.

If forced to pick a single description (though none are given as options here — this is open response), you could say:

✔ “The graph is piecewise linear, increasing from x = –2 to x = –1, then decreasing from x = –1 to x = 0.”

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Matching Graphs A, B, C to Scenarios (Questions 3–5):

We have three graphs showing distance from bowling alley vs. time.

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Graph A: Distance goes up → down → up → flat → up.
This suggests: leaving home → stopping (distance drops? No — wait, if distance from bowling alley, then going *away* from it would increase distance, coming closer decreases it.)

Wait — let’s clarify: Distance from bowling alley.

So:
- If person is at bowling alley → distance = 0
- If person is leaving bowling alley → distance increases
- If person is going toward bowling alley → distance decreases

Now let’s interpret each scenario.

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Question 3:
> Blake left his house to drive to the bowling alley. He stopped to pick up two friends on the way and then ended at the bowling alley.

So:
- Starts at home → distance from bowling alley is high
- Drives toward bowling alley → distance decreases
- Stops to pick up friends → distance stays same (flat line)
- Continues to bowling alley → distance decreases to 0

Look at Graph B:
- Starts high → decreases → flat (stop) → decreases to 0

✔ This matches Graph B

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Question 4:
> Paula left the bowling alley to drive home. She realized she left her purse at the bowling alley so she went back to pick it up. She then drove back home, stopping at one light on the way.

Starts at bowling alley → distance = 0
Drives away → distance increases
Realizes mistake → turns back → distance decreases to 0 (back at alley)
Picks up purse → maybe brief stop? (flat line at 0?)
Then drives home again → distance increases again
Stops at light → flat line somewhere
Then continues → distance increases to final value

Look at Graph A:
- Starts at 0 → goes up (leaving alley)
- Goes down to 0 (returning)
- Flat at 0? Not really — it goes down to 0, then immediately up again — maybe no stop shown
- Then goes up → flat (stop at light) → up again

Yes! Graph A shows:
- Up → Down to 0 → Up → Flat → Up

Perfect match for Paula’s trip.

✔ This matches Graph A

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Question 5:
> Record the letter of the unused graph, and write your own scenario to match the graph.

Unused graph is C

Graph C:
- Starts high → decreases → flat → decreases → flat → decreases to 0

Scenario idea:

> Samantha left her house to go to the bowling alley. She stopped at a store halfway there, then continued driving. She stopped again at a traffic light before finally arriving at the bowling alley.

Breakdown:
- Starts far from alley → distance high
- Drives toward alley → distance decreases
- Stops at store → flat line
- Continues → distance decreases
- Stops at light → flat line
- Arrives → distance = 0

✔ So unused graph is C

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Questions 6 & 7: Nate and Myla walking home

Graph shows distance from house (y-axis) vs. time (x-axis).

At time = 0, both are 40 yards from home (since they’re walking home from school, and last 40 yards are raced).

They start at 40 yds → distance decreases to 0 as they reach home.

Look at the curves:

- Nate’s line: straight line from (0,40) to (8,0) → he takes 8 seconds to cover 40 yards.
- Myla’s curve: starts at (0,40), ends at (9,0) → she takes 9 seconds.

Also, Nate’s line is steeper — meaning he walks faster.

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Question 6: Who reached the house first? Explain.

✔ Nate reached the house first.

Because his distance reaches 0 at 8 seconds, while Myla’s reaches 0 at 9 seconds. So Nate finishes 1 second earlier.

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Question 7: Describe the difference in each sibling’s pace.

✔ Nate walks at a constant pace — his graph is a straight line, meaning he covers equal distance per second (constant speed).

✔ Myla walks at a varying pace — her graph is curved, meaning she slows down or speeds up during the race. Since the curve is concave up (getting less steep over time), she likely started fast and slowed down near the end.

Alternatively, you could say:
> Nate maintained a steady speed throughout the 40-yard race, while Myla’s speed changed — possibly starting quickly and slowing down as she got tired.

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## ✔ Final Answers Summary:

1. c. Non-linear and increasing

2. The graph is piecewise linear. From x = –2 to x = –1, it increases sharply; from x = –1 to x = 0, it decreases. The maximum occurs at x = –1.

3. B

4. A

5. C — Scenario: Samantha left her house to go to the bowling alley. She stopped at a store halfway there, then continued driving. She stopped again at a traffic light before finally arriving at the bowling alley.

6. Nate reached the house first because his distance reached 0 at 8 seconds, while Myla’s reached 0 at 9 seconds.

7. Nate walked at a constant pace (straight line = constant speed). Myla walked at a varying pace (curved line = changing speed), likely starting faster and slowing down.

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