To determine the correct exponential equation for the graph described, we can utilize the points provided: (1, 14) and (2, 4).
In general, an exponential equation can be represented in the form \( y = ab^x \). We can use the two points to set up a system of equations.
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Using the point (1, 14): \[ 14 = ab^1 \implies ab = 14 \quad \text{(Equation 1)} \]
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Using the point (2, 4): \[ 4 = ab^2 \quad \text{(Equation 2)} \]
Now, we can express \( ab^2 \) in terms of Equation 1:
- From Equation 1, we know \( a = \frac{14}{b} \).
- Substitute \( a \) in Equation 2: \[ 4 = \left(\frac{14}{b}\right)b^2 \implies 4 = 14b \implies b = \frac{4}{14} = \frac{2}{7} \]
Now, substituting \( b \) back into Equation 1 to find \( a \):
- \( ab = 14 \) becomes: \[ a \left(\frac{2}{7}\right) = 14 \implies a = 14 \cdot \frac{7}{2} = 49 \]
Putting it all together, we have: \[ y = 49\left(\frac{2}{7}\right)^x \]
Now, we can look at the options provided to find an equivalent expression. Rewriting \(\frac{2}{7}\) as \( \frac{27}{49} \) gives: \[ y = 49 \cdot \left(\frac{2}{7}\right)^x = 49 \cdot \left(\frac{1}{\frac{7}{2}}\right)^x = 49(27)^{-x} \]
After some consideration, we realize that since the curve descends but does not cross the x-axis, it indicates a decay (which the fraction represents). Looking at the options:
- None of the options provide a clearly correct representation since \(27^{-x}\) wasn't listed; however, based on the decay,
- The most logical option that aligns with our value of \(49\) for \(a\) is:
Option #2: \(y=49(27^{-x})\), which matches closely the form derived.
Since the exact negative exponent was absent, we rely on the shape and derived values.
So, the answer is Option #2: y=49(27)^x is the closest representation of the trend represented in the question.