Rewind, Replay, Fast-Forward

The editor plays the clip at 2× speed. In the 2× version, at what time does the drone reach its highest point? At what time does the clip end?

Answer the same two questions for 0.5× speed (slow motion). Write your predictions on the board, then watch the clip to check.

In the 2× clip, the altitude at real time \(t\) is \(F(t) = h(2t)\). On your board, explain in one sentence why the 2 goes inside the brackets: after \(t\) real seconds, how much of the original flight has been shown?

Use the scenario graph to find \(F(0),\ F(1),\ F(2),\ F(3),\ F(4)\). Sketch \(y = h(t)\) and \(y = F(t)\) on the same axes. What happens when you try to find \(F(5)\)?

The 0.5× clip has altitude \(S(t) = h\!\left(\tfrac{t}{2}\right)\). Find \(S(0),\ S(4),\ S(8),\ S(12),\ S(16)\) and add \(y = S(t)\) to your sketch in a third colour.

A student in another group says: "In \(h(2t)\) there is a 2, so the graph gets stretched by a factor of 2." Write a reply on your board that would convince them they are wrong. You must use the video in your argument.

Then complete the rule: the point \((a, b)\) on \(y = h(t)\) ends up at \(\big(\ \_\_\_\ ,\ \_\_\_\ \big)\) on \(y = h(kt)\).

The rewound clip has altitude \(R(t) = h(-t)\) for \(-8 \le t \le 0\). Find \(R(-8),\ R(-6),\ R(-4),\ R(-2),\ R(0)\) and sketch \(y = R(t)\) together with \(y = h(t)\).

The two graphs are mirror images. Where is the mirror? Compare it with the mirror in Phase 1.

The editor exported three more clips (solid lines). The dashed line is the original flight. For each clip, describe the edit in video words (speed, height, direction) and then write its altitude as a function using \(h\).

Let \(f(x) = x^2\). Using \(x = -2, -1, 0, 1, 2\), sketch \(y = 4f(x)\) and \(y = f(2x)\) on the same axes as \(y = f(x)\). Describe each one using the words from Phases 1 and 2.

Something strange happens. What is it? Prove it with algebra.

Let \(g(x) = \sqrt{x}\). Compare \(y = 2g(x)\) with \(y = g(4x)\), using \(x = 0, 1, 4, 9\). What do you notice? Prove it.

Find a value of \(k\) that makes each statement true for every \(x\) where both sides are defined:

Could the editor create Drone B's footage from Phase 1 (always twice as high) just by changing the playback speed of Drone A's clip? Use your sketches to decide.

Then explain: why does the trick in 4a work for \(y = x^2\) but not for the drone's flight?