Source: Calculus (Texas A&M University), Section 4.1 | Exploration worksheet
Tags: exponential function, exponential growth, exponential decay, base, exponent, exponent rules, laws of exponents, domain, range, asymptote, y-intercept, calculus, TAMU
An exponential function has the form f(x) = k · aˣ, where the variable sits in the exponent rather than the base. All basic exponential functions pass through (0, 1), share a domain of (−∞, ∞) and a range of (0, ∞), and approach zero (without reaching it) as x heads toward negative infinity. Rewriting bases using exponent rules shows that expressions like (1/2)ˣ and 2⁻ˣ are the same function.
Exponential function
A function of the form f(x) = k · aˣ, where a and k are constants and x is the variable. The defining feature is that the variable is the exponent, not the base.
Base (of an exponential function)
The constant a in f(x) = k · aˣ. In the general form, the base must be positive and cannot equal 1.
Exponential growth
Occurs when the base a is greater than 1 (e.g. 2ˣ, 3ˣ, 5ˣ). As x increases, y increases rapidly. Larger bases produce steeper growth.
Exponential decay
Occurs when the base is between 0 and 1 (e.g. (1/2)ˣ). As x increases, y shrinks toward zero. This is the mirror behaviour of exponential growth.
Horizontal asymptote
The line y = 0 for basic exponential functions. The curve approaches it as x → −∞ (for growth) or x → +∞ (for decay) but never touches or crosses it.
The general form is f(x) = k · aˣ.
The variable x appears as the exponent, which is what distinguishes exponential functions from power functions (where x is the base).
Constraints on the base: a > 0 and a ≠ 1. If a = 1, the function collapses to a constant (1ˣ = 1 for all x).
Common point: all four pass through (0, 1), because any positive number raised to the zero power equals 1.
Effect of increasing the base: as a increases from 2 to 5, the y-values grow more steeply for positive x. The curve "climbs faster" with a larger base.
Domain: (−∞, ∞) for all four. You can raise a positive base to any real exponent.
Range: (0, ∞) for all four. The output is always positive, never zero, never negative.
End behaviour: as x → −∞, y → 0. The curve flattens toward the x-axis on the left but never reaches it.
Graphing y = (1/2)ˣ and y = 2⁻ˣ produces identical curves.
This follows directly from exponent rules: (1/2)ˣ = (2⁻¹)ˣ = 2⁻ˣ.
These decay functions share the same domain (−∞, ∞) and range (0, ∞) as the growth functions above.
They also pass through (0, 1), just like the growth versions.
The difference is direction: decay functions decrease as x increases and approach zero as x → +∞ rather than x → −∞.
Rules of exponents (reference set from the worksheet):
aⁿ · aᵐ = aⁿ⁺ᵐ (same base, add exponents)
(aⁿ)ᵐ = aⁿᵐ (power to a power, multiply exponents)
aⁿ / aᵐ = aⁿ⁻ᵐ (same base, subtract exponents)
a⁻ⁿ = 1 / aⁿ (negative exponent flips to a reciprocal)
a⁰ = 1 (any positive base to the zero power is 1)
Equivalence to remember:
(1/a)ˣ = a⁻ˣ
This is the key identity connecting growth and decay forms. It comes from applying the "power to a power" rule to (a⁻¹)ˣ.
⚠️ The point (0, 1) is shared by every basic exponential function (where k = 1). Exam questions love asking for the y-intercept.
⚠️ Domain is all reals; range is (0, ∞). A common mistake is writing the range as [0, ∞) or (−∞, ∞). The function never actually reaches zero.
⚠️ Know the difference between exponential functions and power functions. In xⁿ the variable is the base (power function). In aˣ the variable is the exponent (exponential function).
⚠️ Be ready to show algebraically that (1/a)ˣ = a⁻ˣ. This equivalence is a frequent short-answer or explanation question.
⚠️ End behaviour: as x → −∞, y → 0 for growth functions. As x → +∞, y → 0 for decay functions. The curve approaches zero from above, never crossing the x-axis.
Q: What is the y-intercept of f(x) = aˣ for any valid base a?
A: The y-intercept is (0, 1), because a⁰ = 1 for any a > 0.
Q: What are the domain and range of f(x) = 3ˣ?
A: Domain: (−∞, ∞). Range: (0, ∞).
Q: Prove algebraically that (1/2)ˣ and 2⁻ˣ are the same function.
A: Rewrite (1/2)ˣ as (2⁻¹)ˣ. Applying the power-to-a-power rule, (2⁻¹)ˣ = 2⁻¹·ˣ = 2⁻ˣ.
Q: For y = 5ˣ, what happens to y as x → −∞?
A: y → 0. The function approaches the horizontal asymptote y = 0 from above but never reaches it.
Q: Why must the base of an exponential function be positive and not equal to 1?
A: If the base were negative, non-integer exponents would produce undefined or complex outputs. If the base were 1, the function would just be y = 1 for all x, which is a constant, not exponential behaviour.
Q: As you increase the base from 2 to 5, what happens to the graph of y = aˣ for positive x-values?
A: The curve climbs more steeply. Larger bases produce faster growth for x > 0, while all curves still pass through (0, 1).
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