Twelfth Grade · Math · week 12

Exponential functions: growth, decay, and the natural base e

Graph exponential functions, model growth and decay, and evaluate continuous growth using the natural base e.

An exponential function f(x) = a*b^x grows (if b>1) or decays (if 0<b<1) by multiplying by the same factor b every time x increases by 1 -- the opposite of a polynomial, which ADDS the same amount every step. Its graph always passes through (0,a), always stays on one side of the x-axis (b^x is never 0 or negative), and has a horizontal asymptote at y=0.

The number e (approximately 2.71828) is the natural base of continuous growth -- growth that compounds constantly, at every instant, rather than at fixed intervals like monthly or yearly. Continuous growth or decay follows A = A0 * e^(kt), where A0 is the starting amount, k is the continuous rate (positive for growth, negative for decay), and t is time. This is the limiting case of compounding more and more often, forever.

📋 For the teacher
Teach it (10–15 min):
  • Point out the horizontal asymptote (y=0) and the fixed point (0,a) on every exponential graph before discussing growth rate -- the shape's two anchors never move.
  • Contrast ordinary percent growth (multiply by (1+rate) once per period) against continuous growth (e raised to rate times time) explicitly, side by side, using the same starting numbers.
  • Compute the exponent (rate times time) as its own separate mini-step before touching e -- rushing this is where most errors happen.
  • Say "e is just a number, like pi" as often as needed -- treating it as some abstract mystery instead of a specific approximately-2.718 value slows students down.
Talk about it:
  • Why does an exponential function's graph never touch or cross the x-axis?
  • What is the practical difference between compounding once a year and compounding continuously?
  • Could two different exponential functions have the same horizontal asymptote but completely different shapes? What would have to differ?

Mastery looks like: They correctly identify growth versus decay from the base, and evaluate both ordinary and continuous exponential models.

Common stumbles: Forgetting to multiply rate and time before exponentiating e; confusing the continuous-rate formula with the once-per-period growth formula.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

ExponentsRepeated multiplication and how fast powers grow.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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