To determine which option represents the correct exponential equation for the graph, let's analyze the points given: (1, 14) and (2, 4).
An exponential function can typically be represented in the form:
\[ y = a(b)^x \]
where \( a \) is a constant that affects the vertical stretch and \( b \) is the base of the exponential function which determines the growth or decay rate.
Given the two points:
-
At \( x = 1 \), \( y = 14 \): \[ 14 = a(b)^1 \] \[ 14 = ab \]
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At \( x = 2 \), \( y = 4 \): \[ 4 = a(b)^2 \] \[ 4 = ab^2 \]
Now we have a system of equations:
- \( ab = 14 \)
- \( ab^2 = 4 \)
We can solve for \( b \) by dividing the second equation by the first:
\[ \frac{ab^2}{ab} = \frac{4}{14} \] \[ b = \frac{4}{14} = \frac{2}{7} \]
Now substitute \( b = \frac{2}{7} \) back into the first equation to find \( a \):
\[ a\left(\frac{2}{7}\right) = 14 \]
Thus:
\[ a = 14 \cdot \frac{7}{2} = 49 \]
Now we have \( a = 49 \) and \( b = \frac{2}{7} \).
The exponential equation is:
\[ y = 49\left(\frac{2}{7}\right)^x \]
The base \( b \) can also be expressed as \( \frac{2}{7} = 27^{-1} \) because \( \frac{1}{\frac{7}{2}} = \frac{2}{7}\).
Thus, we can write the equation as:
\[ y = 49 \cdot (27)^{-x} \]
Now, checking the options we have, we see:
- Option #1: \( y = 7(12)^x \)
- Option #2: \( y = 49(27)^x \)
- Option #3: \( y = 7(27)^x \)
- Option #4: \( y = 49(72)^x \)
For the decay described, the equation must have a base less than 1 (typical of decay).
None of the options clearly reflect a decreasing function as derived, but the closest functional form we derived is consistent with the transformation into a format resembling option #2 but in a negated form due to the decay nature.
None of these options directly match, but based on the context of decay nature and reflection of points, there may have been transcription errors in provided exponential forms.
From analysis, if I have to select the closest that represents an incorrect transformation consistent with graph behavior presented, I lean towards Option #2: \( y = 49(27)^x \) due to proximity of base representation.
Hence, please take note that while a direct correct match cannot be selected from options provided, while picking logic suggests selecting a functional base comparisons around decay.
In conclusion, based on breakdown analysis: Option #2 might be the one to select by process of elimination of nearest proximity representation – ensure alignment with growth/decay nature expected in curves when open for further graph review.