y = 5 (.3)^x -7

Question

Use the graph of the exponential decay function that models an exponential decay sequence to discuss the properties and determine the equation for the horizontal asymptote of the graph.

(1 point)
Responses

The horizontal asymptote is at y=5
.
The horizontal asymptote is at y equals 5 .

The horizontal asymptote is at y=0
.
The horizontal asymptote is at y equals 0 .

The horizontal asymptote is at y=0.3
.
The horizontal asymptote is at y equals 0.3 .

The horizontal asymptote is at y=−7
.

Answer 1

To determine the horizontal asymptote of the function \( y = 5(0.3)^x - 7 \), we need to analyze the behavior of the function as \( x \) approaches infinity.

1. As \( x \) becomes very large (i.e., \( x \to \infty \)), the term \( (0.3)^x \) approaches 0 because it is a decay function (since \( 0 < 0.3 < 1 \)).

2. So, as \( x \to \infty \): \[ y = 5(0.3)^x - 7 \to 5(0) - 7 = -7 \]

This indicates that the function \( y \) will get closer and closer to \(-7\) as \( x \) increases.

Thus, the horizontal asymptote of the graph is \( y = -7 \).

Therefore, the correct response is:

The horizontal asymptote is at y = -7.