Kenneth ChengMathematical modeling
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Exponential functions

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.

What you will be able to explain

  • Interpret and use exponential functions with domain checks.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Base

A real exponential base is positive and different from one.

y=axy=a^x

Growth and decay

Bases above one increase; bases between zero and one decrease.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in exponential functions changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the positive exponential base.

f(x)=ax,f(2)=4f(x)=a^x,\quad f(2)=4
  • Use the domain, units and sampling assumptions stated in the question.
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 real exponential base is positive.
Worked solution

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

  2. Calculate or simplify this relation.

    a2=4,a>0⇒a=2a^2=4,\quad a>0\Rightarrow a=2

  3. The negative algebraic square root is not an admissible exponential base.

The requested value is 2.

Checks and common pitfalls: The negative algebraic square root is not an admissible exponential base.

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

02 / Standard#Worked example

Solve the exponential equation.

2x=242^x=2^{4}
  • Use the domain, units and sampling assumptions stated in the question.
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
Equal outputs of a strictly monotone exponential have equal inputs.
Worked solution

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

  2. Calculate or simplify this relation.

    x=4x=4

  3. The base is positive and not one.

The requested value is 4.

Checks and common pitfalls: The base is positive and not one.

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

03 / Transfer#Worked example

Explain why an exponential model with base one is excluded from the usual increasing/decreasing classification.

f(x)=1xf(x)=1^x
  • Use the domain, units and sampling assumptions stated in the question.
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
Check two different inputs.
Worked solution

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

  2. Calculate or simplify this relation.

    1x=1(∀x∈R)1^x=1\quad(\forall x\in\mathbb R)

  3. It is not strictly monotone and has no inverse on the full real line.

It is the constant function 1.

Checks and common pitfalls: It is not strictly monotone and has no inverse on the full real line.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

Find the positive exponential base.

f(x)=ax,f(2)=9f(x)=a^x,\quad f(2)=9
  • Use the domain, units and sampling assumptions stated in the question.
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 real exponential base is positive.
Worked solution

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

  2. Calculate or simplify this relation.

    a2=9,a>0⇒a=3a^2=9,\quad a>0\Rightarrow a=3

  3. The negative algebraic square root is not an admissible exponential base.

The requested value is 3.

Checks and common pitfalls: The negative algebraic square root is not an admissible exponential base.

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

05 / Foundation#Your turn

Solve the exponential equation.

2x=252^x=2^{5}
  • Use the domain, units and sampling assumptions stated in the question.
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
Equal outputs of a strictly monotone exponential have equal inputs.
Worked solution

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

  2. Calculate or simplify this relation.

    x=5x=5

  3. The base is positive and not one.

The requested value is 5.

Checks and common pitfalls: The base is positive and not one.

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

06 / Foundation#Your turn

Compare the two values for x<y.

f(t)=(1/3)tf(t)=(1/3)^t
  • Use the domain, units and sampling assumptions stated in the question.
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

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

  2. Calculate or simplify this relation.

    0<1/3<1;x<y⇒(1/3)x>(1/3)y0<1/3<1;\quad x<y\Rightarrow (1/3)^x>(1/3)^y

  3. 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
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

07 / Foundation#Your turn

State the range and horizontal asymptote.

y=2x+3y=2^x+3
  • Use the domain, units and sampling assumptions stated in the question.
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
Shift the positive exponential graph upward.
Worked solution

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

  2. Calculate or simplify this relation.

    2x>0⇒y>3;2x→0 (x→−∞)2^x>0\Rightarrow y>3;\quad 2^x\to0\ (x\to-\infty)

  3. The asymptote is approached but never reached.

Range: y>3; asymptote: y=3.

Checks and common pitfalls: The asymptote is approached but never reached.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

08 / Standard#Your turn

Find the y-intercept.

y=3⋅3xy=3\cdot3^x
  • Use the domain, units and sampling assumptions stated in the question.
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
At a y-intercept, x=0.
Worked solution

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

  2. Calculate or simplify this relation.

    y(0)=3⋅30=3y(0)=3\cdot3^0=3

  3. The outside multiplier scales the graph vertically.

The requested value is 3.

Checks and common pitfalls: The outside multiplier scales the graph vertically.

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

09 / Standard#Your turn

A quantity starts at 3 and doubles each hour. Find it after 3 hours.

  • Use the domain, units and sampling assumptions stated in the question.
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
Apply the same multiplicative factor three times.
Worked solution

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

  2. Calculate or simplify this relation.

    N(3)=3⋅23=24N(3)=3\cdot2^3=24

  3. Doubling is multiplicative growth, not addition of a fixed amount.

The requested value is 24.

Checks and common pitfalls: Doubling is multiplicative growth, not addition of a fixed amount.

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

10 / Standard#Your turn

Prove that the difference of these two exponentials is odd.

g(x)=2x−2−xg(x)=2^x-2^{-x}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Replace x by −x in both terms.
Hint 2
The two terms exchange positions.
Worked solution

  1. Replace x by −x in both terms.

  2. Calculate or simplify this relation.

    g(−x)=2−x−2x=−g(x)g(-x)=2^{-x}-2^x=-g(x)

  3. A difference of non-odd functions can still be odd.

g is odd on R.

Checks and common pitfalls: A difference of non-odd functions can still be odd.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

Find the positive exponential base.

f(x)=ax,f(2)=16f(x)=a^x,\quad f(2)=16
  • Use the domain, units and sampling assumptions stated in the question.
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 real exponential base is positive.
Worked solution

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

  2. Calculate or simplify this relation.

    a2=16,a>0⇒a=4a^2=16,\quad a>0\Rightarrow a=4

  3. The negative algebraic square root is not an admissible exponential base.

The requested value is 4.

Checks and common pitfalls: The negative algebraic square root is not an admissible exponential base.

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

12 / Transfer#Your turn

Solve the exponential equation.

2x=262^x=2^{6}
  • Use the domain, units and sampling assumptions stated in the question.
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
Equal outputs of a strictly monotone exponential have equal inputs.
Worked solution

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

  2. Calculate or simplify this relation.

    x=6x=6

  3. The base is positive and not one.

The requested value is 6.

Checks and common pitfalls: The base is positive and not one.

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

13 / Transfer#Your turn

Compare the two values for x<y.

f(t)=(1/4)tf(t)=(1/4)^t
  • Use the domain, units and sampling assumptions stated in the question.
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

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

  2. Calculate or simplify this relation.

    0<1/4<1;x<y⇒(1/4)x>(1/4)y0<1/4<1;\quad x<y\Rightarrow (1/4)^x>(1/4)^y

  3. 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
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for exponential functions?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

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

    Anticipated thinking

    • 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.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗