g10-06-04 · Core topics
Crossing the base 1 reverses growth
What changes in aˣ when a passes through 1?
1 MIN · PREDICT ON PAPER
Compare outputs at x=−1,0,1 for bases 0.5,1,2.
Discuss or write first. No typed answer is required to continue.
a=2: increasing; a^1=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
For a>1, aˣ is increasing; for 0<a<1 it is decreasing. Every positive base gives a⁰=1. At a=1 the rule is constant, not an exponential function in this course.
Conditions: The real exponential family requires a>0 and a≠1. Its domain is all real numbers and its range is (0,∞).
Mathematical justification: For x₂>x₁, a^x₂/a^x₁=a^(x₂−x₁). This ratio exceeds 1 if a>1 and is below 1 if 0<a<1.
1 MIN · EXIT CHECK
Is (1/2)ˣ increasing or decreasing, and can its value be zero?
Check after answering
Decreasing; every finite real input has positive output.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Keep the point (0,1) fixed while comparing a positive and a negative input.
Watch for: A larger base always gives a larger output for every x.
Challenge: Compare 2ˣ and 3ˣ for x<0; explain why their ordering reverses.
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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