MATH 130 · Section 4.2

Exponential Functions

Growth, decay, compound interest, and modeling

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    recognize and graph exponential functions

  2. 2

    evaluate exponential expressions and use the natural base e

  3. 3

    select and apply simple, periodic, and continuous interest formulas

  4. 4

    interpret exponential growth and decay models

Activate · 0.5 minSource: 2

Predict: recognize and graph exponential functions

Predict first

Which grows faster for large x: x² or 2ˣ? Explain before calculating.

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

recognize and graph exponential functions

Concept
  • An exponential function has the variable in the exponent: $f(x)=b^x$, where $b>0$ and $b\ne1$.
  • All parent exponential graphs pass through $(0,1)$, have domain $(-\infty,\infty)$, range $(0,\infty)$, and asymptote $y=0$.

Explain · 2.5 minSource: 3, 4

Model the Skill: recognize and graph exponential functions

Annotated diagram
  • Build and graph a five-point table for $f(x)=2^x$.
  • For $x=-2,-1,0,1,2$, the values are $1/4,1/2,1,2,4$.
  • Plot the five points and draw a smooth increasing curve through $(0,1)$.
  • Domain: $(-\infty,\infty)$; range: $(0,\infty)$; horizontal asymptote: $y=0$.
Graph of the exponential function two to the x
Graph of the exponential function two to the x

Model · 3.0 minSource: 5, 6

You Try: recognize and graph exponential functions

Student practice

Graph $g(x)=3^x$ using five points. Label the intercept, domain, range, and asymptote.

Practice · 3.0 minSource: 7, 8

Check and Diagnose: recognize and graph exponential functions

Error analysis
  • Do not confuse $x^2$ with $2^x$.
  • A larger growth base is steeper to the right; a base between 0 and 1 produces decay.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 9, 10

Pacing checkpoint

Make It Stick: recognize and graph exponential functions

Explain how the base determines growth or decay and name the four invariant graph features.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 11

Predict: evaluate exponential expressions and use the natural base e

Predict first

What value should $e^0$ have, and why must that be true?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

evaluate exponential expressions and use the natural base e

Formula focus
  • $e\approx2.71828$ is the natural base.
  • Calculator entry requires parentheses around negative or fractional exponents.

Explain · 2.5 minSource: 12

Model the Skill: evaluate exponential expressions and use the natural base e

1

$3.5^{1.6}\approx7.422$; a base above 1 with a positive exponent should exceed 3.5.

2

$(3/4)^{-0.95}\approx1.314$; a negative exponent takes a reciprocal, so the result exceeds 1.

3

Both results are positive, as required for positive bases.

Model · 3.0 minSource: 13

You Try: evaluate exponential expressions and use the natural base e

Student practice

Evaluate $e^{-0.15}$ and $175e^{0.5}$ to the nearest thousandth.

Practice · 3.0 minSource: 14

Check and Diagnose: evaluate exponential expressions and use the natural base e

Error analysis
  • A negative exponent does not make the result negative.
  • Estimate first: $e^{0.5}$ lies between 1 and e.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 15

Pacing checkpoint

Make It Stick: evaluate exponential expressions and use the natural base e

Describe one calculator check that catches an exponent-entry error.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 16

Predict: select and apply simple, periodic, and continuous interest formulas

Predict first

Which wording tells you whether an interest problem needs n or e?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

select and apply simple, periodic, and continuous interest formulas

1

Simple: $I=Prt$.

2

Periodic compound: $A=P(1+r/n)^{nt}$.

3

Continuous compound: $A=Pe^{rt}$.

Explain · 2.5 minSource: 17, 18

Model the Skill: select and apply simple, periodic, and continuous interest formulas

1

Annual compounding gives $n=1$ in $A=P(1+r/n)^{nt}$.

2

$A=44000(1.0325)^{13}\approx66684.28$.

3

The balance exceeds the principal and retains currency units.

Model · 3.0 minSource: 19, 20

You Try: select and apply simple, periodic, and continuous interest formulas

Student practice

$2000 is invested at 3.1% compounded semiannually for 3 years. Find A.

Practice · 3.0 minSource: 21, 22

Check and Diagnose: select and apply simple, periodic, and continuous interest formulas

  • Convert percentages to decimals.
  • n is compounds per year, not the number of years.

Continuous compounding has no n.

Feedback · 2.0 minSource: 23, 24

Pacing checkpoint

Make It Stick: select and apply simple, periodic, and continuous interest formulas

Give a decision rule that selects the correct interest formula from the wording.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 25, 26

Predict: interpret exponential growth and decay models

Predict first

In $A(t)=3200(1/2)^{t/14}$, what does each number mean?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

interpret exponential growth and decay models

Concept
  • A coefficient gives the initial amount.
  • A factor above 1 models growth; a factor between 0 and 1 models decay.
  • The exponent $t/14$ counts the number of half-lives.

Explain · 2.5 minSource: 27

Model the Skill: interpret exponential growth and decay models

For $A(t)=3200(1/2)^{t/14}$, evaluate the model at $t=0$ and $t=40$ hours.

  1. $A(0)=3200(1/2)^0=3200$ units.
  2. $A(40)=3200(1/2)^{40/14}\approx441.64$ units.
  3. The decrease is reasonable because 40 hours is almost three half-lives.

Model · 3.0 minSource: 28

You Try: interpret exponential growth and decay models

Student practice

A culture begins with 500 cells and doubles every 3 hours. Write a model and estimate the population after 10 hours.

Practice · 3.0 minSource: Authored

Check and Diagnose: interpret exponential growth and decay models

Error analysis
  • The initial amount is found by setting time equal to zero.
  • Keep units attached to time and rate quantities.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 29

Pacing checkpoint

Make It Stick: interpret exponential growth and decay models

Explain how to read initial value, growth or decay factor, and time scale from an exponential model.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Session Summary and Exit Ticket

  1. State the most important idea from today without looking at your notes.
  2. Complete one representative setup and identify the step most likely to cause an error.
  3. Write one question that should be answered before the next assessment.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: 30, 31