(0,-2)

Question

y = 5 (.3)^x - 7

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. Exponential Decay Function: The term $(0.3{)}^x$ represents an exponential decay function because the base $0.3$ is between 0 and 1. As $x$ increases, $(0.3{)}^x$ approaches 0.

2. Finding the Asymptote: As $x$ approaches infinity, $(0.3{)}^x$ will approach 0, thus:

   $$y\approx 5(0)-7=-7$$

   Therefore, the horizontal asymptote of the graph of this function is where the function approaches as $x$ grows larger.

3. Conclusion: The horizontal asymptote is at $y=-7$.

Response: The horizontal asymptote is at $y=-7$.