Thursday
Thursday: Shift, Stretch and Flip
Comet sets the launcher on a wooden stand one meter tall. "What does this do to the graph?"
"Every height goes up by one," Wren says. "f(t) + 1. Same shape, slid up. What do you notice about the vertex?"
"(2, 21) instead of (2, 20)," Comet says. She writes four cards: f(t) + 1, 0.5 f(t), f(t - 1), f(2t).
"Half the water, half the height at every moment," Wren says. "0.5 f(t) squashes the arc. Vertex (2, 10)."
"Launch one second late and the whole arc slides right. f(t - 1), vertex (3, 20)."
Nova projects all four arcs over the original. "Would you like a hint? Compare the vertices first."
"And f(2t) is a flight twice as quick," Comet says. "Same top, squeezed to two seconds."
Four changes to one rule
Start with f(t) = -5t² + 20t, vertex (2, 20). Each card is a transformation of f.
Adding k outside, f(t) + k, slides the graph up by k. Multiplying outside, k f(t), stretches or squashes it.
A negative multiplier flips the graph upside down.
Subtracting inside, f(t - k), slides the graph right by k. Adding inside slides it left.
Multiplying inside, f(kt), squeezes the graph toward the axis when k is bigger than 1.
| Card | New rule | Vertex | What changed in the launch |
|---|---|---|---|
| f(t) + 1 | -5t² + 20t + 1 | (2, 21) | launched from a 1 m stand |
| 0.5 f(t) | -2.5t² + 10t | (2, 10) | half the height at every moment |
| f(t - 1) | -5t² + 30t - 25 | (3, 20) | launched one second late |
| f(2t) | -20t² + 40t | (1, 20) | the same flight in half the time |
Every row is the crew's own Loft thought experiment, written as a rule. The vertex tells the story fastest.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| f(t) + 1 has the same shape as f(t), moved up 1. | ? |
| f(t - 1) moves the graph to the left. | ? |
| 0.5 f(t) has a vertex at (2, 10). | ? |
| Multiplying f(t) by -1 flips the graph upside down. | ? |
| f(2t) lands at the same time as f(t). | ? |
For each move, say the rule out loud: f(t) + 2, f(t - 1), -f(t).
Sharp thinking. Tomorrow you spot parabolas in everyday life and review the week.