Which statement is true about the graph of an exponential decay function in the form y=a(b)x

Question

, where 0<b<1
?(1 point)
Responses

The function is always increasing, the y-intercept is at (0,a), and the horizontal asymptote is at y = 0.
The function is always increasing, the y- intercept is at (0,a) , and the horizontal asymptote is at y = 0 .

The domain includes all real numbers, the y-intercept is at (0,a), and the horizontal asymptote is at y = 0.
The domain includes all real numbers, the y- intercept is at (0,a) , and the horizontal asymptote is at y = 0 .

The domain includes all real numbers, the y-intercept is at (0,a), and the range includes all real numbers.
The domain includes all real numbers, the y -intercept is at (0,a) , and the range includes all real numbers.

The y-intercept is at (0, a), the range includes all real numbers, and the horizontal asymptote is at y = 0.
The y -intercept is at (0, a) , the range includes all real numbers, and the horizontal asymptote is at y = 0 .

Answer 1

The correct statement about the graph of an exponential decay function in the form \( y = a(b)^x \) where \( 0 < b < 1 \) is:

The domain includes all real numbers, the y-intercept is at (0,a), and the horizontal asymptote is at y = 0.

Here's why:

1. Domain: The domain of an exponential function is all real numbers, as you can input any real value for \( x \).

2. Y-intercept: The y-intercept occurs at \( x = 0 \), where \( y = a(b)^0 = a \), which gives the point \( (0, a) \).

3. Horizontal asymptote: As \( x \) approaches infinity, \( y \) approaches 0 but never actually reaches it, giving a horizontal asymptote at \( y = 0 \).

So the correct response is: "The domain includes all real numbers, the y-intercept is at (0,a), and the horizontal asymptote is at y = 0."