Question

The graph shows the number of game systems sold since 2015. Based on this information, which function best models the number of game systems sold in millions `x` years since 2015? The graph is a slope that shows three points: (0, 30.5) (1, 21.35) (2, 14.95)
a. y=21.35\left(0.7\right)^{x}
b. y=30.5\left(21.35\right)^{x}
c. =30.5\left(1.3\right)^{x}
d. y=30.5\left(0.7\right)^{x}
answer with one of these responses

Answer 1

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. \( y = 30.5(0.7)^{x} \)**