To identify which exponential function properly represents the points depicted in the table, we can consider the general form of an exponential function:
\[ f(x) = a \cdot b^x \]
Where:
- \( a \) is the initial value (the value of \( f(x) \) when \( x = 0 \)),
- \( b \) is the base of the exponential function which determines the growth factor.
From the given options, we can analyze the points at specific \( x \) values.
Let’s evaluate those functions:
A. \( f(x) = 15(2)^{-x} \)
At \( x=0 \): \( f(0) = 15(2)^0 = 15 \)
At \( x=1 \): \( f(1) = 15(2)^{-1} = 15/2 = 7.5 \)
(This does not match with our exponential growth so this might be incorrect)
B. \( f(x) = 18(1)^{-x} \)
At \( x=0 \): \( f(0) = 18(1)^0 = 18 \)
(This also doesn't make sense as it is a constant function)
C. \( f(x) = 15(1.2)^{x} \)
At \( x=0 \): \( f(0) = 15(1.2)^0 = 15 \)
At \( x=1 \): \( f(1) = 15(1.2)^1 = 18 \)
At \( x=2 \): \( f(2) = 15(1.2)^2 = 21.6 \)
(This sequence of points indicates that it is exponentially growing. Hence, it seems appropriate)
D. \( f(x) = 18(1.2)^{x} \)
At \( x=0 \): \( f(0) = 18(1.2)^0 = 18 \)
At \( x=1 \): \( f(1) = 18(1.2)^1 = 21.6 \)
At \( x=2 \): \( f(2) = 18(1.2)^2 = 25.92 \)
(This shows a constant growth rate. Thus also matching)
Comparing to the points presented:
- If the values you’re looking at have an initial value of f(0) matching closely with 18, then options C and D don’t match perfectly as they lead us away from the actual relationships.
Based on the points, if you clarify whether they grow (i.e. listing the points of the table), we can pinpoint the exact exponential function. However, looking purely at function increments, option D is more accurate to exhibit an exponential growth pattern.
Therefore: The correct answer depends on the initial values; D is appropriate given your numerical values nearing 18.