g10-06-06 · Core topics
Swap coordinates for inverse graphs
If (2,4) lies on y=2ˣ, which point lies on y=log₂x?
1 MIN · PREDICT ON PAPER
Reflect the point across y=x before plotting its inverse partner.
Discuss or write first. No typed answer is required to continue.
(2,4) reflects to (4,2).
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
The relation y=aˣ is equivalent to x=logₐy. Swapping coordinates reflects the graph in y=x. Domain and range exchange: real inputs/positive outputs become positive inputs/real outputs.
Conditions: Require a>0,a≠1. The inverse graph has no point with input zero, even when its output is negative.
Mathematical justification: Applying logₐ to aˣ gives x and exponentiating logₐu gives u for u>0. These identities establish inversion beyond the displayed points.
1 MIN · EXIT CHECK
Reflect (−2,1/4) from y=2ˣ into its inverse graph.
Check after answering
(1/4,−2), on y=log₂x.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Have students label each coordinate as input or output before swapping it.
Watch for: An inverse graph is obtained by changing y to −y.
Challenge: Repeat for a=1/2 and input −1. Explain why both inverse functions decrease.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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