The Exponential Function and the Number e · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Growth that multiplies every step

Mathematics · Algebra · ages 16-17
Name ______________________   Date ____________

  1. For y = 2^x, what is y when x = 3?

    Answer: ______________

  2. What stays true for every y equals a to the x curve?

    • It crosses below zero often
    • It stays positive with asymptote y equals 0
    • It has no asymptote at all

  3. An exponential curve touches the line y equals 0.

    Circle one:   True   False

  4. How does y equals 2 to the minus x differ from 2 to the x?

    • It decays to the right, mirrored in x
    • It grows twice as fast
    • It shifts upward by 2

  5. Priya says e is special because the gradient of y = e^x at x = 0 equals 1, and e^x is its own derivative. Is Priya right?

    Circle one:   True   False

  6. In A e to the k t, a negative k means decay toward zero.

    Circle one:   True   False

  7. A culture doubles each hour from 1 cell. How many cells after 4 hours?

    Answer: ______________

  8. For y = 2^-x, what is y when x = 1? Give your answer as a decimal.

    Answer: ______________

  9. A story starts at 50 and shrinks in proportion to size. Which model fits?

    • 50 e to the k t with positive k
    • 50 plus k t with negative k
    • 50 e to the k t with negative k

  10. In a model of the form A e^kt, what does the sign of k tell you?

    • It grows when k is positive and decays when k is negative
    • It always grows whatever k is
    • It always decays whatever k is
    • It stays flat whatever k is
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Growth that multiplies every step W1-mt_2hyjN4JfJJ-s1

  1. 8 · 2^3 = 2 x 2 x 2 = 8.
  2. It stays positive with asymptote y equals 0 · Repeated multiplication by a positive base never reaches zero or below.
  3. False · Zero is an asymptote the curve approaches but never touches.
  4. It decays to the right, mirrored in x · Negating x mirrors the climb into a decay.
  5. True · Priya is right: e is the base whose exponential has gradient 1 at x = 0 and equals its own derivative.
  6. True · The sign of k picks the direction, and negative falls.
  7. 16 · Doubling four times gives 1, 2, 4, 8, 16.
  8. 0.5 · 2^-1 = 1/2 = 0.5.
  9. 50 e to the k t with negative k · Proportional shrinking needs the exponential form with a falling k.
  10. It grows when k is positive and decays when k is negative · A positive k makes the power grow with t, a negative k makes it shrink.
Worksheet · LightMySky