Maths › Exponentials and logarithms › Exponential functions and e

Exponential functions and e

Put x in the exponent and everything changes: equal steps in x now multiply y instead of adding to it. Among all possible bases, one is special. The curve of e to the x climbs at a rate equal to its own height, which is why the same number turns up in banking, biology and radioactive decay.

Year 12EDEXCEL 9MA0 6.1, 6.2

Builds on Indices and surds and Graphs, proportion and transformations.

IN THIS TOPIC

  • Sketch y = ax for any positive base, growth and decay alike, with the right asymptote.
  • Use the gradient property of ekx, and transform e-curves with the standard moves.
  • Recognise when a situation calls for an exponential model, and read its constants.

WHAT YOU PROBABLY THINK

Exponential growth means growing very fast.

The family a to the x

The function y = ax (with a positive) puts the variable in the exponent, so each unit step in x multiplies y by a rather than adding anything. Every member of the family passes through (0, 1), because a0 = 1 from the index laws, and the x-axis is a horizontal asymptote on one side. Which side depends on the base: a > 1 grows to the right, while 0 < a < 1 decays, and the two are mirror images since (1/2)x = 2−x.

Growth and decay: y equals 2 to the x climbs to the right while y equals a half to the x falls, and the two curves are mirror images in the y-axis, meeting at 0 comma 1growth: base 2decay: base ½mirror images in the y-axis
FIG. 1The growth curve of base 2 and the decay curve of base ½, reflections of each other in the y-axis, meeting where every exponential starts, at (0, 1).

Every exponential is positive everywhere. The curve hugs its asymptote without touching, so ax = 0 has no solution, a fact exam questions poke at from several directions.

The special base e

Compare gradients where the curves cross (0, 1). Base 2 leaves that point with gradient about 0.69, base 3 with about 1.10, so somewhere between them lives a base leaving at gradient exactly 1. That base is e = 2.71828…, and the property extends along the whole curve and to every multiple of x,

gradient of ekx = kekxNOT IN THE BOOKLET — LEARN IT

so the gradient of ekx at any point is just k times the height there.

The curves of 2 to the x and e to the x both pass through the point 0 comma 1, but only e to the x leaves it with gradient exactly 1, shown by the dashed tangent linebase ebase 2tangent slope 1 at (0, 1)every base passes through (0, 1); only e leaves at slope 1
FIG. 2Bases 2 and e through their shared point (0, 1). The dashed tangent has slope exactly 1, and only the e-curve leaves along it.

WORKED EXAMPLE

A gradient with no calculus in sight

Find the gradient of y = e2x at the point where x = 1.

The gradient function is 2e2x, by the boxed property with k = 2.

At x = 1 the gradient is 2e2 = 14.8 to 3 significant figures.

The height there is e2 ≈ 7.39, and the gradient is twice the height, which is what k = 2 promises everywhere on this curve.

YOUR TURN

A transformed exponential

Sketch y = e−x − 2, stating the equation of its asymptote and its y-intercept, before opening the working.

Show the working

Build it in two moves from y = ex: the −x reflects in the y-axis, giving decay, and the −2 translates down 2.

The asymptote moves with the curve, to y = −2; the y-intercept is e0 − 2 = −1.

The curve falls from the top left, cuts the y-axis at −1, and flattens onto y = −2 without reaching it. The transformations lesson runs the whole show; nothing about e changes the rules.

Why nature keeps choosing e

The deep reason exponentials model so much is the gradient property read in reverse. Whenever a quantity's rate of change is proportional to its current value, more bacteria breed more bacteria, more atoms decay more often, more money earns more interest, the quantity must follow P = Aekt. Growth like this can start slowly, which retires the opening lie; what defines it is not speed but the proportionality.

TRY IT UNSEEN

Reading a growth model

A population is modelled by P = 500e0.2t, with t in days. Write down the initial population, find P after 5 days, and explain what 0.2 says about the growth.

Show the working

At t = 0 the exponential is 1, so the initial population is 500.

After 5 days, P = 500e1 = 500e ≈ 1360 to 3 significant figures.

The 0.2 is the proportionality constant: at every moment the population grows at 0.2 × its current size per day. Early on that is only 100 a day; by day 5 it is about 272 a day, from the same rule.

Questions probing \u201cinitial\u201d always mean t = 0, and the answer is the constant in front, A, whenever the model is written as Aekt.

THE EXAM BIT

  • Every ax sketch needs three features labelled: the point (0, 1), the asymptote, and the correct growth or decay direction for the base.
  • Quote the gradient property, gradient of ekx is kekx, before using it; the quoted line is a mark.
  • Transformations of e-curves follow the standard rules, and the asymptote transforms with the curve. State its new equation explicitly.
  • In models written P = Aekt, A is the value at t = 0 and k is the per-unit-time proportionality constant; interpret both with units.
  • ax is never zero and never negative, and \u201cexplain why the model predicts P never reaches 0\u201d is asking for the asymptote.

CHECK YOURSELF

For the curve y = e3x, write down the gradient function, the gradient at x = 0, and the y-intercept.

Show a hint

One boxed property answers all three parts.

Show the answer

The gradient function is 3e3x, by the property with k = 3.

At x = 0 the gradient is 3e0 = 3, three times the height there.

The y-intercept is e0 = 1, shared by the whole exponential family.

Every base's curve passes through (0, 1) and hugs an asymptote it never touches.

e is the base whose gradient equals its height; e to the kx grows at k times itself.

CHECK YOUR PROGRESS

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  • Sketch y = ax for any positive base, growth and decay alike, with the right asymptote.
  • Use the gradient property of ekx, and transform e-curves with the standard moves.
  • Recognise when a situation calls for an exponential model, and read its constants.

No animated video for this topic yet; these notes stand alone.