To determine the correct exponential equation for the graph based on the points given, we can analyze the two points plotted on the curve: (1, 14) and (2, 4).
The general form of an exponential equation is \( y = a \cdot b^x \), where:
- \( a \) is the initial value (when \( x = 0 \))
- \( b \) is the base (the growth or decay factor)
Using the point (1, 14), we can plug it into the equation: \[ 14 = a \cdot b^1 \] \[ 14 = a \cdot b \] (1)
Using the point (2, 4): \[ 4 = a \cdot b^2 \] (2)
Now, we can express \( a \) in terms of \( b \) from equation (1): \[ a = \frac{14}{b} \]
Substituting this into equation (2): \[ 4 = \left(\frac{14}{b}\right) \cdot b^2 \] \[ 4 = 14b \] \[ b = \frac{4}{14} = \frac{2}{7} \]
Now substitute \( b \) back to find \( a \): \[ a = \frac{14}{\frac{2}{7}} = 14 \cdot \frac{7}{2} = 49 \]
Thus, the exponential equation is: \[ y = 49 \cdot \left(\frac{2}{7}\right)^x \]
Expressing \(\frac{2}{7}\) as \( \left(27\right)^{-1} \): \[ y = 49 \cdot (7^{-1})^x = 49 \cdot 72^{-x} \]
Hence, in standard exponential form the equation looks like \( y = 49 \cdot \left(\frac{1}{27}\right)^x \).
Comparing this with the options provided, the form of the equation matches:
Option #4: \( y = 49(72)^{-x} \)
Therefore, the correct answer is:
Option #4 is the correct equation for the graph.