AP Precalculus · Unit 2: Exponential and Logarithmic Functions · Lesson 2.3

Deep Dive: Exponential Functions

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

2.3.A2.3.A.1Concept

The General Form of an Exponential Function

In Lesson 2.2 you met f(x) = ab^x. Now we pin down exactly what is allowed. The general form of an exponential function is:

f(x) = a·bx  with  a ≠ 0,  b > 0,  b ≠ 1

The number a is the initial value (the output at x = 0) and b is the base. Each restriction has a reason:

Restriction	Why
a ≠ 0	If a = 0, then f(x) = 0 for every x. There is nothing to grow or shrink.
b ≠ 1	If b = 1, then f(x) = a for every x. That is a constant function.
b > 0	A negative base would make the outputs flip sign as x changes, and many inputs (like x = 0.5) would have no real output.

Growth and decay

Conditions	Name	Example
a > 0 and b > 1	exponential growth	f(x) = 3·2^x
a > 0 and 0 < b < 1	exponential decay	f(x) = 100(0.5)^x

Set the initial value a and the base b. The tool classifies f(x) = a·b^x and tells you what each step does to the output. Try a = 0 and b = 1 too.

initial value a2

base b1.5

f(x) = 2(1.5)xa > 0 and b > 1: exponential GROWTH.Each time x goes up by 1, the output is multiplied by 1.5: that is a +50% change in the output.

⚠️Decay does not mean the base is negative. A decaying function has a base between 0 and 1, such as 0.5. The outputs shrink because each step multiplies by a number smaller than 1.

2.3.A2.3.A.2Concept

What the Input Means: Factors of the Base

When the input is a natural number n, the exponent has a very concrete meaning: it tells you how many factors of the base are applied to the initial value.

f(n) = a·b·b·…·b  (n factors of b)

Example. For f(x) = 3·2^x: f(0) = 3 (no factors of 2), f(1) = 3·2 = 6, f(2) = 3·2·2 = 12, f(3) = 3·2·2·2 = 24. This is the geometric sequence from Lesson 2.1.

When the input is a natural number n, it counts how many factors of the base b are applied to the initial value a. Slide n and watch the chain of factors grow.

initial value a3

base b2

input n4

f(4) = 3(2)4= 3 × 2 × 2 × 2 × 2 = 484 factors of 2 have been applied to the initial value 3. The input does not have to be a whole number, though: the domain is all real numbers.

🔑The domain of an exponential function is all real numbers. The natural numbers are a way to see what the exponent means. Other inputs work too: f(−1) = 3·2⁻¹ = 1.5 and f(0.5) = 3·2^0.5 ≈ 4.243.

2.3.A2.3.A.3Concept

Always Increasing or Always Decreasing, and Always Concave

Over equal-length input intervals, the outputs of an exponential function change proportionally. That has two big consequences for the graph:

An exponential function is always increasing or always decreasing.
Its graph is always concave up or always concave down.

a	b	Behavior	Example
a > 0	b > 1	increasing, concave up	2·3^x
a > 0	0 < b < 1	decreasing, concave up	64(0.5)^x
a < 0	b > 1	decreasing, concave down	−3·2^x
a < 0	0 < b < 1	increasing, concave down	−40(0.5)^x

The tool finds the average rate of change over each interval of length 1. If the rates keep the same sign, the function is always increasing or always decreasing. If the rates keep going up, the graph is concave up; if they keep going down, it is concave down.

initial value a2

base b1.5

interval	[0, 1]	[1, 2]	[2, 3]	[3, 4]
average rate of change	1	1.5	2.25	3.375

✓ The function is increasing and its graph is concave up.On the closed interval [0, 4] the extreme values are at the endpoints: f(0) = 2 and f(4) = 10.125. No relative extrema and no point of inflection appear anywhere.

💡Here is the pattern behind the table. The sign of a decides concavity: concave up when a > 0, concave down when a < 0. Whether the function increases or decreases depends on the sign of a and on whether b is above or below 1.

🔑Because the graph never changes direction and never changes concavity, an exponential function has no relative extrema and no points of inflection. The only extreme values appear on a closed interval [c, d], and they sit at the endpoints.

2.3.A2.3.A.4Math

When the Outputs Are Not Proportional Yet

Sometimes a function f looks exponential, but its outputs are not proportional over equal-length intervals. Add or subtract a constant from every output. If the new outputs are proportional, then f can be modeled by an additive transformation of an exponential function:

f(x) = a·bx + k

Example. The outputs 8, 11, 17, 29, 53 have ratios 1.375, 1.545, 1.706, 1.828, which are not constant. Subtract 5 from each output: 3, 6, 12, 24, 48. Those have a constant ratio of 2. So f(x) = 3·2^x + 5.

Each list is the output of a function at x = 0, 1, 2, 3, 4. First check whether the outputs are proportional. If not, slide k to subtract a constant from every output, and look for a value of k that makes the new outputs proportional.

subtract k0

x	0	1	2	3	4
f(x)	8	11	17	29	53
f(x) − 0	8	11	17	29	53
ratios	—	1.375	1.546	1.706	1.828

✗ The outputs of f are not proportional (the ratios above are not constant).

🔑A quick way to find k: look at the differences between successive outputs. For f(x) = ab^x + k the differences are proportional too (3, 6, 12, 24 above). The ratio of the differences is b. Then use f(0) = a + k and f(1) = ab + k to solve for a and k.

💡The shift moves the graph up or down, and the outputs level off toward the line y = k instead of toward 0. Think of k as the value the function settles near.

2.3.A2.3.A.5Concept

End Behavior: Where the Outputs Go

For an exponential function in general form, look at what happens as the input increases or decreases without bound. The outputs either increase without bound, decrease without bound, or get arbitrarily close to zero. In limit notation, for f(x) = ab^x:

limx→±∞ abx = ∞,  −∞,  or  0

Function type	x → ∞	x → −∞
a > 0, b > 1	f(x) → ∞	f(x) → 0
a > 0, 0 < b < 1	f(x) → 0	f(x) → ∞
a < 0, b > 1	f(x) → −∞	f(x) → 0
a < 0, 0 < b < 1	f(x) → 0	f(x) → −∞

Pick a function. The table shows what happens to the outputs as x moves far to the right and far to the left.

x	-20	-10	-5	0	5	10	20
f(x)	2.86 × 10^-6	0.003	0.094	3	96	3,072	3.15 × 10^6

as x → ∞,  f(x) → ∞as x → −∞, f(x) → 0On one side the outputs get arbitrarily close to zero but never reach it. On the other side they increase or decrease without bound.

💡The rule: if b > 1, repeated multiplication blows the outputs up as x increases and shrinks them toward 0 as x decreases. If 0 < b < 1, it is the reverse. The sign of a decides whether "blows up" means ∞ or −∞.

⚠️"Close to zero" is not "equal to zero." No exponential function with a ≠ 0 ever outputs 0. The outputs just get as close as you like, and the graph never crosses the x-axis.

← Back to Lesson 2.3 Try Guided Practice →Ready for 2.4? Exponential Function Manipulation shows how to rewrite these expressions in equivalent forms.