Question

which of the following describes exponential decay?
a. the graph is symmetrical around the y - axis.
b. the base of the exponent is between 0 and 1.
c. the graph has no horizontal asymptote.
d. the base of the exponent is greater than 1.

Explanation:

To determine which option describes exponential decay, we analyze each choice:

• Option a: Symmetry around the y - axis is a property of even functions (like \(y = x^{2}\) or \(y=\cos(x)\)), not a defining feature of exponential decay. Exponential functions of the form \(y = a^{x}\) are not generally symmetric about the y - axis (except when \(a = 1\), but \(y = 1^{x}=1\) is a constant function, not a decaying exponential).
• Option b: The general form of an exponential function is \(y=a^{x}\), where \(a>0,a

eq1\). For exponential decay, as \(x\) increases, \(y\) decreases. If the base \(a\) is between 0 and 1 (i.e., \(0 < a<1\)), then as \(x\) increases, \(a^{x}\) gets smaller (e.g., \(y=(0.5)^{x}\), when \(x = 1,y = 0.5\); \(x = 2,y=0.25\), etc.). This is the definition of exponential decay.

• Option c: Exponential functions of the form \(y=a^{x}\) (for \(a>0,a

eq1\)) have a horizontal asymptote at \(y = 0\). So the statement that the graph has no horizontal asymptote is false.

• Option d: If the base of the exponent \(a>1\), then as \(x\) increases, \(a^{x}\) increases (e.g., \(y = 2^{x}\), when \(x=1,y = 2\); \(x = 2,y = 4\)), which is exponential growth, not decay.

Answer:

b. The base of the exponent is between 0 and 1.