8
advanced-math

The graph of y=a(b)xy = a(b)^x has a horizontal asymptote at y=0y = 0. Which must be true?

A

b>0b > 0 and no vertical shift

B

a=0a = 0

C

b=0b = 0

D

The graph is a horizontal line

Correct Answer: A

Choice A is the correct answer. Identify conditions for asymptote at y=0y=0.

  1. Standard Form: y=abxy = ab^x has asymptote at y=0y = 0 (x-axis).
  2. Shifted Form: y=abx+ky = ab^x + k has asymptote at y=ky = k.
  3. Condition: For asymptote at y=0y=0, there must be no vertical shift (k=0k=0) and b>0b > 0.

💡 Strategic Tip: Basic exponential y=abxy = ab^x always approaches y=0y=0 as x→±∞x \to \pm \infty (depending on bb).

Choice B is incorrect because if a=0a=0, the function would be y=0y=0 (constant). Choice C is incorrect becausebb must be positive in exponential functions. Choice D is incorrect because exponential graphs curve, not flat.