To determine the correct exponential equation for the curve that descends in the first quadrant and passes through the points (1, 14) and (2, 4), we can analyze the points given.
For an exponential function, we can express it as:
\[ y = a \cdot b^x \]
where \( a \) is a constant, and \( b \) is the base of the exponential, which must be less than 1 for a descending curve.
We'll check each option to see if it satisfies the points (1, 14) and (2, 4):
Option #1: \( y = 7 \cdot (12)^x \)
- For \( x = 1: y = 7 \cdot 12^1 = 7 \cdot 12 = 84 \) (not 14)
- For \( x = 2: y = 7 \cdot 12^2 = 7 \cdot 144 = 1008 \) (not 4)
Option #2: \( y = 49 \cdot (27)^x \)
- For \( x = 1: y = 49 \cdot 27 = 1323 \) (not 14)
- For \( x = 2: y = 49 \cdot 27^2 = 49 \cdot 729 = 35721 \) (not 4)
Option #3: \( y = 7 \cdot (27)^x \)
- For \( x = 1: y = 7 \cdot 27 = 189 \) (not 14)
- For \( x = 2: y = 7 \cdot 27^2 = 7 \cdot 729 = 5123 \) (not 4)
Option #4: \( y = 49 \cdot (72)^x \)
- For \( x = 1: y = 49 \cdot 72 = 3528 \) (not 14)
- For \( x = 2: y = 49 \cdot 72^2 = 49 \cdot 5184 = 254016 \) (not 4)
None of the options seem to match both points. It is likely that there's a misunderstanding or the options may not represent the given points accurately. However, based on the points provided, we would expect that an equation of the form \( y = ab^x \) with \( b < 1 \) would be necessary to create a descending curve that passes through those points.
Since none of the options provided work accurately with both points, I can't select one as the correct equation based on the information given.
You may want to recheck the provided options or clarify the details further.