To determine which of the given options is the correct exponential equation for the graph based on the points plotted and the information provided, we can evaluate each option with the coordinates provided in the points (1, 14) and (2, 4).
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Option #1: \( y = 7\left(\frac{1}{2}\right)^x \)
- For \( x = 1 \): \( y = 7\left(\frac{1}{2}\right)^1 = 7 \times \frac{1}{2} = 3.5 \) (not 14)
- For \( x = 2 \): \( y = 7\left(\frac{1}{2}\right)^2 = 7 \times \frac{1}{4} = 1.75 \) (not 4)
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Option #2: \( y = 49\left(\frac{2}{7}\right)^x \)
- For \( x = 1 \): \( y = 49\left(\frac{2}{7}\right)^1 = 49 \times \frac{2}{7} = 14 \) (correct)
- For \( x = 2 \): \( y = 49\left(\frac{2}{7}\right)^2 = 49 \times \frac{4}{49} = 4 \) (correct)
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Option #3: \( y = 7\left(\frac{2}{7}\right)^x \)
- For \( x = 1 \): \( y = 7\left(\frac{2}{7}\right)^1 = 7 \times \frac{2}{7} = 2 \) (not 14)
- For \( x = 2 \): \( y = 7\left(\frac{2}{7}\right)^2 = 7 \times \frac{4}{49} = \frac{28}{49} = \frac{4}{7} \) (not 4)
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Option #4: \( y = 49\left(\frac{7}{2}\right)^x \)
- For \( x = 1 \): \( y = 49\left(\frac{7}{2}\right)^1 = 49 \times \frac{7}{2} = 171.5 \) (not 14)
- For \( x = 2 \): \( y = 49\left(\frac{7}{2}\right)^2 = 49 \times \frac{49}{4} = 600.25 \) (not 4)
Based on the evaluations, Option #2 is the only one that correctly matches the points (1, 14) and (2, 4).
Therefore, the answer is: Option #2 is the correct equation for the graph.