To determine which function best models the number of game systems sold, we can analyze the points given: (0, 30.5), (1, 21.35), and (2, 14.95).
From the point (0, 30.5), we see that when \( x = 0 \), \( y \) is 30.5, which suggests that \( y \) starts from 30.5 millions when \( x = 0 \).
We also see that the values are decreasing over time, which suggests an exponential decay function.
Now let's examine the given options:
a. \( y = 21.35(0.7)^{x} \) - This starts at 21.35 when \( x = 0 \), which does not match the initial point.
b. \( y = 30.5(21.35)^{x} \) - This is not an exponential decay since \( 21.35 > 1 \), which means it increases.
c. \( y = 30.5(1.3)^{x} \) - This suggests growth, not decay, since \( 1.3 > 1 \).
d. \( y = 30.5(0.7)^{x} \) - This starts at 30.5 when \( x = 0 \) and indicates decay with a base less than 1.
Now, let's quickly confirm that option d fits with the subsequent points. If \( x=1 \): \[ y = 30.5(0.7)^{1} = 30.5 \cdot 0.7 = 21.35 \] For \( x=2 \): \[ y = 30.5(0.7)^{2} = 30.5 \cdot 0.49 \approx 14.95 \]
Thus, the function that best models the data provided is:
d. \( y = 30.5(0.7)^{x} \)