DWG NO. 3.1 — Lesson 1 of 6

Exponential Functions and Their Graphs

Unit 3: Exponential and Logarithmic Functions · ~20 min

Objective Graph an exponential function of the form f(x) = a·\(b^{x}\) and describe its domain, range, intercept, and asymptote from the values of a and b.

An exponential function has the form f(x) = a·\(b^{x}\), where a ≠ 0, b > 0, and b ≠ 1. The variable is in the exponent instead of the base — the opposite arrangement from every function in Units 1 and 2. That one change is enough to produce a completely different kind of growth: each time x increases by 1, the output doesn't add a fixed amount, it multiplies by b.

Growth vs. decay

The base b decides the shape:

b > 1 — exponential growth. Output increases as x increases, slowly at first and then steeply. The larger b is, the faster it climbs.
0 < b < 1 — exponential decay. Output decreases as x increases, approaching (but never reaching) 0. The graph is a mirror image of a growth curve across the y-axis when b and \(\frac{1}{b}\) are reciprocals.

Key features

Feature	b > 1 (growth)	0 < b < 1 (decay)
Domain	(−∞, ∞)	(−∞, ∞)
Range (a > 0)	(0, ∞)	(0, ∞)
y-intercept	(0, a)	(0, a)
Horizontal asymptote	y = 0	y = 0
As x → ∞	f(x) → ∞	f(x) → 0
As x → −∞	f(x) → 0	f(x) → ∞

Notice that an exponential function never crosses the x-axis — a·\(b^{x}\) is always positive when a > 0, so y = 0 is a horizontal asymptote the graph approaches but never touches. That asymptote, not a zero, is the defining boundary of an exponential graph.

y = 2ˣ (growth) and y = (½)ˣ (decay) — both pass through (0, 1) with horizontal asymptote y = 0

Worked Example 1 · Growth, a = 1

ProblemDescribe the key features of f(x) = \(3^{x}\).

1Base b = 3 > 1, so this is growth. y-intercept: f(0) = 3⁰ = 1, so (0, 1).

2Horizontal asymptote y = 0 as x → −∞; f(x) → ∞ as x → ∞. Domain: all reals. Range: (0, ∞).

Growth curve through (0, 1), asymptote y = 0, range (0, ∞)

Worked Example 2 · Decay, a ≠ 1

ProblemDescribe the key features of f(x) = 4(0.5\()^{x}\).

1Base b = 0.5, with 0 < b < 1, so this is decay. y-intercept: f(0) = 4(0.5)⁰ = 4(1) = 4, so (0, 4).

2Each time x increases by 1, the output is cut in half: f(1) = 2, f(2) = 1, f(3) = 0.5, approaching 0 but never reaching it.

Decay curve through (0, 4), asymptote y = 0, range (0, ∞)

Worked Example 3 · Transformation

ProblemDescribe g(x) = \(2^{x-1}\) + 3 as a transformation of f(x) = \(2^{x}\) (recall the shift rules from 1.4).

1Subtracting 1 inside the exponent shifts the graph right 1 unit. Adding 3 outside shifts it up 3 units.

2The horizontal asymptote shifts along with the vertical translation: y = 0 becomes y = 3. New y-intercept: g(0) = 2⁻¹ + 3 = 0.5 + 3 = 3.5.

Shift of 2ˣ: right 1, up 3 · asymptote y = 3 · y-intercept (0, 3.5)

Guided practice

1.Is f(x) = 5(1.2\()^{x}\) growth or decay? State its y-intercept.
Show answer

b = 1.2 > 1, so growth. y-intercept: f(0) = 5(1.2)⁰ = (0, 5).
2.Is f(x) = 7(\(\frac{2}{3} )^{x}\) growth or decay? State its horizontal asymptote.
Show answer

b = \(\frac{2}{3}\), and 0 < b < 1, so decay. Horizontal asymptote: y = 0.
3.State the domain and range of any exponential function f(x) = a·\(b^{x}\) with a > 0.
Show answer

Domain: (−∞, ∞). Range: (0, ∞). The graph is defined for every real x but never reaches or crosses y = 0.
4.Find the y-intercept and horizontal asymptote of g(x) = \(3^{x}\) − 2.
Show answer

g(0) = 3⁰ − 2 = 1 − 2 = −1, so y-intercept (0, −1). The whole graph of 3ˣ shifts down 2, so the asymptote y = 0 becomes y = −2.
5.Explain why an exponential function a·\(b^{x}\) (with a > 0) can never have an x-intercept.
Show answer

\(b^{x}\) is positive for every real x (a positive base raised to any power stays positive), so a·\(b^{x}\) is always positive when a > 0. The graph can get arbitrarily close to y = 0 but can never equal it, so there is no x-intercept.

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