LEARN · EXPLAIN · REVISE

Compare the two values for x<y.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 4.2 · PDF 118 / printed page 111

Revisit first: Exponents

TOPIC 01

Exponential functions

Build understanding of exponential functions through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

f(t)=(1/3)^t

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the base and whether the exponential graph grows or decays.

Hint 2

A base between zero and one gives decay.

Worked solution

Identify the base and whether the exponential graph grows or decays.
Calculate or simplify this relation.
$0<1/3<1;\quad x<y\Rightarrow (1/3)^x>(1/3)^y$
The order reverses because the function decreases.

f(x)>f(y).

Checks and common pitfalls: The order reverses because the function decreases.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Interpret and use exponential functions with domain checks.
Base: A real exponential base is positive and different from one.
$y=a^x$
Growth and decay: Bases above one increase; bases between zero and one decrease.
Close with: conditions → representation → reasoning → check.

Expected reasoning: A real exponential base is positive and different from one.
Expected reasoning: Bases above one increase; bases between zero and one decrease.
Expected correction: An exponential never reaches its zero asymptote.