Rewind, Replay, Fast-Forward

Stand Up & Math · Functions · SL 2.11
Teacher: unlock codes (screen only, not printed)
Phase 1: HEIGHT
Phase 2: SPEED
Phase 3: REWIND
Phase 4: PARENT
🎬 The Scenario

One drone, one flight, a lot of editing

The drone club filmed an eight-second test flight. The drone's altitude after \(t\) seconds is \(h(t)\) metres, shown in the graph.

The club's video editor speeds the footage up, slows it down, plays it backwards and compares it with other drones. Every edit creates a new altitude graph. Predict each one and explain why before anyone tells you a rule.

Copy the graph onto your board first. Watch the clips on the class screen or on the activity page when each phase starts.

⏱ ~45 min  ·  👥 Groups of 3–4  ·  SL & HL
-112345678924681012ht(0, 0)(2, 6)(4, 6)(6, 10)(8, 0)h(t): altitude of the drone in metres
1
Phase 1 · Same Clock, Different Heights
Vertical stretches and reflections
Task 1a
Fill in the heights
Watch the clip: Drone A (original), Drone B (always twice as high), Drone C (always half as high) and A's reflection in a lake. Make a table for \(t = 0, 2, 4, 6, 8\). Heights below the water are negative.
Task 1b
Write each altitude using \(h\)
Write B, C and the reflection in terms of \(h(t)\). Sketch all four on one set of axes and label every corner point.
Task 1c
Where does a point go?
\((a, b)\) is on \(y = h(t)\). Where does it end up on \(y = k\,h(t)\)? Which points never move, whatever \(k\) is? Why?
2
Phase 2 · Playback Speed
Horizontal stretches
Task 2a · predict first
Two questions, two speeds
At 2× speed, when is the drone highest? When does the clip end? Same for 0.5× speed. Write your predictions, then watch the clip.
Task 2b
What does \(h(2t)\) mean?
Why does the 2 go inside? Find \(F(t) = h(2t)\) for \(t = 0, 1, 2, 3, 4\) and sketch it with \(h\). What happens at \(F(5)\)?
Task 2c
Slow motion: \(h\!\left(\tfrac t2\right)\)
Find \(S(t)\) for \(t = 0, 4, 8, 12, 16\). Add it to your sketch.
Task 2d
"There's a 2, so it stretches by 2"
Convince that student they are wrong, using the video. Then complete: \((a, b)\) on \(y = h(t)\) goes to \((\ \_\_\ ,\ \_\_\ )\) on \(y = h(kt)\).
3
Phase 3 · Rewind and Remix
Reflections and combined edits
Task 3a
What is \(h(-t)\)?
The rewound clip is \(R(t) = h(-t)\) for \(-8 \le t \le 0\). Find \(R(-8), R(-6), R(-4), R(-2), R(0)\) and sketch it. Where is the mirror? Compare with Phase 1.
Task 3b
Three mystery clips
Dashed: the original flight. Describe each edit in video words, then write it as a function using \(h\).
-112345678948121620hth(t)Clip A
-1123456789-12-10-8-6-4-224681012hth(t)Clip B
-9-8-7-6-5-4-3-2-112345678924681012hth(t)Clip C
4
Phase 4 · Back to the Parents
When two stretches agree
Task 4a
Two edits of \(y = x^2\)
Using \(x = -2, -1, 0, 1, 2\), sketch \(y = 4f(x)\) and \(y = f(2x)\) with \(f(x) = x^2\). Describe each. Something strange happens: what is it? Prove it.
Task 4b
The square root
Compare \(y = 2\sqrt x\) with \(y = \sqrt{4x}\) using \(x = 0, 1, 4, 9\). What do you notice? Prove it.
Task 4c
Find the value of \(k\)
\(9x^2 = (kx)^2\)   \(3\sqrt x = \sqrt{kx}\)   \(5|x| = |kx|\)
Task 4d · Challenge
Can the editor fake Drone B?
Could Drone B's footage be made just by changing the playback speed of Drone A's clip? Why does the 4a trick work for \(x^2\) but not for the drone?