Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 1 to 6 and its y-axis ranges from negative 10 to 10, both by 1-unit increments. A solid curve and a dotted line with arrows at both ends are plotted. The solid curve intersects a marked point on the y-axis.

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=−7
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The horizontal asymptote is at y equals negative 7 .

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

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

The horizontal asymptote is at y=5

1 answer

To determine the correct horizontal asymptote of the exponential decay function based on the properties of such graphs, we typically note that exponential decay functions approach a constant value but never actually reach it as the x-values increase or decrease towards infinity.

In general, for an exponential decay function of the form \( y = ae^{bx} + c \), where \( a \) is a positive constant, \( b \) is a negative constant (indicative of decay), and \( c \) represents a vertical shift, the horizontal asymptote is normally at \( y = c \).

Since you mentioned that the graph intersects a marked point on the y-axis and may involve a horizontal asymptote, without the exact graph or additional context like the equation parameters, we need to make some assumptions based on standard properties.

If the horizontal asymptote is at a y-value of 0 or a small positive value, it often indicates that the graph does not fall below this line as x approaches either positive or negative infinity.

Given the typical behaviors of exponential decay functions, the choice of the horizontal asymptote at y = 0 is frequently observed. Exponential decay functions approach this value as they decline.

Thus, based on the common properties of exponential decay functions, it is appropriate to conclude that:

The horizontal asymptote is at y = 0.