Thursday
Thursday: The Mirror and the Moves
Wren pins two sheets of graph paper to the Boathouse wall. On the left, the winch rule climbs gently.
"Now swap every point," he says. "(0, 1) becomes (1, 0). (10, 6) becomes (6, 10). What do you notice?"
Comet plots the swapped points. "A steeper line. It crosses the first one."
Nova draws a dashed line from corner to corner. "Would you like a hint? Fold along y = x."
"They match," Comet says, folding the paper. "A mirror. So the inverse is the reflection."
"Now lift the winch rule up 2," Wren says. "Say the drum held 3 meters to start. Where does the inverse go?"
Comet redraws. "It slid right 2. Up on one side of the mirror is right on the other."
Why the inverse graph is a mirror
Here is the reason, in six steps. Read it first, then put it in order below.
- Take any point (a, b) on the graph of f. That means f(a) = b.
- Then f⁻¹(b) = a, because the inverse sends b back to a.
- So the point (b, a) is on the graph of f⁻¹. The coordinates have swapped.
- The midpoint of (a, b) and (b, a) is ((a + b)/2, (a + b)/2). It lies on the line y = x.
- The segment from (a, b) to (b, a) has slope -1, so it is perpendicular to y = x.
- A point with its mirror image across a line has exactly those two properties. So (b, a) is the reflection of (a, b).
The two lines cross at x = 2, because f(2) = 2 there. A point that f sends to itself sits on the mirror.
The same mirror turns y = 2ˣ into y = log₂ x. Every exponential rule and its logarithm are reflections.
Shift, stretch and flip
The moves from Algebra 2 still work. The rule f(x) + 2 lifts every point up 2. The rule f(x - 2) slides it right 2.
The rule 2f(x) stretches it to twice the height. The rule -f(x) flips it over the x-axis.
For the winch: f(x) + 2 = 0.5x + 3, and its inverse is 2x - 6. Compare with f⁻¹(x) = 2x - 2.
Put x - 2 into f⁻¹: 2(x - 2) - 2 = 2x - 6. The inverse of "up 2" is "right 2," exactly as the mirror predicts.
| Rule | Inverse | Move on the graph | Move on the inverse |
|---|---|---|---|
| f(x) = 0.5x + 1 | 2x - 2 | the original | the original |
| f(x) + 2 = 0.5x + 3 | 2x - 6 | up 2 | right 2 |
| 2f(x) = 1x + 2 | x - 2 | stretched 2 times taller | squeezed to half the width |
- ?The segment between them has slope -1, perpendicular to y = x
- ?The midpoint ((a + b)/2, (a + b)/2) lies on y = x
- ?Then f⁻¹(b) = a
- ?So (b, a) is on the graph of f⁻¹
- ?So (b, a) is the reflection of (a, b) across y = x
- ?(a, b) is on the graph of f, so f(a) = b
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| The graph of f⁻¹ is the graph of f reflected across y = x. | ? |
| The graph of f⁻¹ is the graph of f reflected across the x-axis. | ? |
| If f(2) = 2, then f⁻¹(2) = 2. | ? |
| Shifting a graph up 2 shifts its inverse up 2 as well. | ? |
| Reflecting y = 2ˣ across y = x gives y = log₂ x. | ? |
Clear reasoning. Tomorrow the whole week comes back in a mixed set from the Boathouse log.