Question

an exponential growth function has an asymptote of \\(y = -3\\). which might have occurred in the original function to permit the range to include negative numbers?

a whole number constant could have been added to the exponential expression.
a whole number constant could have been subtracted from the exponential expression.
a whole number constant could have been added to the exponent.
a whole number constant could have been subtracted from the exponent.

Explanation:

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<reasoning>

Analyze the parent exponential function

An exponential growth function in its simplest parent form is \(y = b^x\), where \(b > 1\). The range of this parent function is \((0, \infty)\), which contains only positive numbers, and its horizontal asymptote is \(y = 0\).

Determine the effect of transformations

To permit the range to include negative numbers, the graph must be shifted vertically downward. A vertical shift is achieved by adding or subtracting a constant to or from the entire exponential expression.

Using the Translation Rule concept, shifting a function \(f(x)\) vertically downward by \(k\) units is represented by \(f(x) - k\).

Evaluate the horizontal asymptote

The problem states that the transformed function has a horizontal asymptote of \(y = -3\). For a function of the form \(y = b^x + k\), the horizontal asymptote is \(y = k\). Therefore, \(k = -3\), which means a constant of \(3\) was subtracted from the exponential expression.

Match with the given options

Subtracting a whole number constant from the exponential expression shifts the graph down, moving the asymptote to \(y = -3\) and allowing the range to be \((-3, \infty)\), which includes negative numbers. This matches the second option.
</reasoning>

<answer>
<mcq-option>A whole number constant could have been added to the exponential expression.</mcq-option>
<mcq-correct>A whole number constant could have been subtracted from the exponential expression.</mcq-correct>
<mcq-option>A whole number constant could have been added to the exponent.</mcq-option>
<mcq-option>A whole number constant could have been subtracted from the exponent.</mcq-option>
</answer>

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Answer:

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<reasoning>

Analyze the parent exponential function

An exponential growth function in its simplest parent form is \(y = b^x\), where \(b > 1\). The range of this parent function is \((0, \infty)\), which contains only positive numbers, and its horizontal asymptote is \(y = 0\).

Determine the effect of transformations

To permit the range to include negative numbers, the graph must be shifted vertically downward. A vertical shift is achieved by adding or subtracting a constant to or from the entire exponential expression.

Using the Translation Rule concept, shifting a function \(f(x)\) vertically downward by \(k\) units is represented by \(f(x) - k\).

Evaluate the horizontal asymptote

The problem states that the transformed function has a horizontal asymptote of \(y = -3\). For a function of the form \(y = b^x + k\), the horizontal asymptote is \(y = k\). Therefore, \(k = -3\), which means a constant of \(3\) was subtracted from the exponential expression.

Match with the given options

Subtracting a whole number constant from the exponential expression shifts the graph down, moving the asymptote to \(y = -3\) and allowing the range to be \((-3, \infty)\), which includes negative numbers. This matches the second option.
</reasoning>

<answer>
<mcq-option>A whole number constant could have been added to the exponential expression.</mcq-option>
<mcq-correct>A whole number constant could have been subtracted from the exponential expression.</mcq-correct>
<mcq-option>A whole number constant could have been added to the exponent.</mcq-option>
<mcq-option>A whole number constant could have been subtracted from the exponent.</mcq-option>
</answer>

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