To determine the horizontal asymptote of the exponential decay function based on the provided information, we can analyze the given equation and properties of exponential functions.
The equation given is:
\[ y = 5(0.03)^x - 7. \]
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Understanding the function: The term \( (0.03)^x \) represents an exponential decay function since the base (0.03) is between 0 and 1. As \( x \) increases, \( (0.03)^x \) approaches 0.
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Finding the horizontal asymptote: The horizontal asymptote of a function describes the value that the function approaches as \( x \) approaches infinity. In this case, as \( x \) increases, \( (0.03)^x \) approaches 0, which leads us to analyze the entire function:
\[ y = 5(0.03)^x - 7. \]
As \( x \) approaches infinity:
\[ (0.03)^x \to 0 \quad \text{so} \quad y \to 5(0) - 7 = -7. \]
Thus, the horizontal asymptote of this function is given by:
\[ y = -7. \]
- Summary of properties: The key properties of the graph are:
- It decays towards the horizontal asymptote at \( y = -7 \) as \( x \) increases.
- The function will never actually reach this horizontal asymptote but will get infinitely close as \( x \) increases.
Based on this analysis, the correct response regarding the horizontal asymptote is:
The horizontal asymptote is at \( y = -7 \).
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