Analyzing Functions and Graphs worksheet with questions on interpreting distance-time graphs and matching scenarios to graph representations.

Let's go through each question on the worksheet step by step and solve them with explanations.

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

Graph: A curve that starts at around (0, 10), rises to a peak near (12, 14), then decreases after that. We are only concerned with the interval x from 0 to 10.

- From x = 0 to x ≈ 12, the graph is increasing.
- Between x = 0 and x = 10, the graph is increasing, but it’s not a straight line — it curves upward.

So:
- Not linear → eliminate (a) and (b)
- It's increasing → eliminate (d)

✔ Answer: c. Non-linear and increasing

> ✔ Explanation: The graph curves (so not linear), and as x increases from 0 to 10, the y-values increase, so it's increasing.

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

Graph: A V-shaped graph with a sharp point at (-3, 0), going up to (0, 5), then down to (1, 0), etc.

We’re analyzing the interval x from -2 to 0.

- At x = -2, the graph is at y = 0
- As x goes from -2 to 0, the graph moves upward to y = 5 at x = 0

This segment is a straight line going upward.

So:
- It's linear
- It's increasing

✔ Answer: The graph is linear and increasing from x = -2 to x = 0.

> ✔ Explanation: The portion of the graph between x = -2 and x = 0 is a straight line rising from (−2, 0) to (0, 5). So it's both linear and increasing.

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Matching Graphs A, B, C to Scenarios

We have three graphs showing distance vs. time.

#### Graph A:
- Starts at 0 (home), goes up, then dips back down, then up again, then flat, then steeply up.
- Shows multiple changes in direction: away, back, forward, stop, then fast forward.

#### Graph B:
- Starts high (at bowling alley), drops sharply, then levels off (stopped), then drops to zero (arrived home).
- Then it goes back up (turns around), then drops again (back home).

#### Graph C:
- Starts high (at bowling alley), drops slowly, then flat (stopped), then continues dropping, then faster drop to zero.

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#### 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.

- Starts at home (distance = 0), goes toward bowling alley.
- Stops twice (flat segments), then continues to bowling alley.
- Ends at maximum distance (bowling alley).

That matches Graph A:
- Starts at 0 → increases → flat (stop) → increases → flat (stop) → increases to max.

✔ Answer: A

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#### 4. Paula left the bowling alley to drive home. She realized that 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 (max distance), drives toward home (distance decreases).
- Then turns around (goes back to bowling alley), so distance increases again.
- Then drives home again (distance decreases), with a flat segment (stopped at light).

This is Graph B:
- Starts high → drops → flat (stopped) → goes back up → drops again to zero.

✔ Answer: B

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

Unused graph: C

Graph C shows:
- Starts at high distance (bowling alley)
- Drops gradually
- Then flat (stopped)
- Then continues dropping
- Then drops more quickly to zero

So it's a trip from bowling alley to home with:
- A slow start
- A stop
- Then continues home, possibly speeding up

✔ Answer: C

Scenario:
> Jenna left the bowling alley and walked slowly toward home. She stopped to talk to a friend for a while, then continued walking, picking up speed as she got closer to home.

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6. Who reached the house first? Explain.

Graph: Distance vs. time for Nate and Myla.

- Both start at 40 yards from home.
- They race the last 40 yards home.
- The one whose graph reaches distance = 0 first arrives first.

Look at the graph:
- Nate’s line hits 0 at about 8 seconds
- Myla’s line hits 0 at about 9 seconds

✔ Answer: Nate reached the house first.

> ✔ Explanation: Nate's graph crosses the x-axis (distance = 0) earlier than Myla’s, meaning he arrived home sooner.

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7. Describe the difference in each sibling's pace.

- Nate: His line is steeper than Myla’s, especially in the beginning.
- Steeper slope = faster rate of change = faster speed.

- Myla: Her line is less steep, meaning slower speed overall.

But notice:
- Both start at same point and end at same point.
- Nate gets there faster → he has a higher average speed.

Also, Myla’s graph appears to curve slightly — maybe she started slow, then sped up.

But Nate’s pace is consistently faster than Myla’s.

✔ Answer:
> Nate walked at a faster, more consistent pace than Myla. His graph has a steeper slope, indicating a greater rate of change in distance over time, which means he covered the distance quicker. Myla walked more slowly overall, possibly starting slowly and accelerating later, but still took longer to reach home.

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✔ Final Answers Summary:

1. c. Non-linear and increasing
2. Linear and increasing (from x = -2 to 0)
3. A
4. B
5. C – Scenario: Someone leaves the bowling alley, walks slowly, stops, then speeds up to get home.
6. Nate reached the house first because his graph reaches zero distance earlier.
7. Nate walked faster than Myla; his graph has a steeper slope, indicating a higher speed.

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