Question: Indicate which of the following four graphs is the correct graph of this function. \[ p(x) =…

Indicate which of the following four graphs is the correct graph of this function.

\[ p(x) = \frac{1}{\left(\frac{15}{4}\right)^{2-x}} \]

Answer

[Graphs are shown, but not transcribed as they are visual representations.]

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Solution

To determine the correct graph of the function \[ p(x) = \frac{1}{\left( \frac{15}{4} \right)^{2-x}} \] start by rewriting it for easier analysis. Rewrite the expression inside the denominator using exponent rules: \[ p(x) = \left( \frac{15}{4} \right)^{x-2} \] Recognize that this is an exponential function. The general form of an exponential function is \( a^{x-c} \), where \( a \) is the base. Here, \( a = \frac{15}{4} > 1 \), which means the function will be increasing as \( x \) increases. Now, analyze the behavior: - As \( x \to \infty \), \( p(x) \to \infty \). - As \( x \to -\infty \), \( p(x) \to 0 \). The graph will approach the x-axis to the left and rise steeply to the right. The correct graph is the one in the bottom right quadrant, which reflects an increasing exponential function.