The table represents some points on the graph of an exponential function. Which function represents the relationship?
The table:
x: 2, 3, 4, 5, 6. f(x): 6, 13, 1.5. 0.75, 0.375
To determine the exponential function that represents the relationship between the x-values and the corresponding f(x)-values in the table, we can look for patterns and calculate the common ratio.
Using the given table, we can see that as x increases by 1, f(x) undergoes a certain multiplication by a constant factor.
To find this constant factor, we can divide each f(x) by the preceding f(x). Let's perform the calculations:
𝑓(3)/𝑓(2) = 13/6 ≈ 2.167
𝑓(4)/𝑓(3) = 1.5/13 ≈ 0.115
𝑓(5)/𝑓(4) = 0.75/1.5 = 0.5
𝑓(6)/𝑓(5) = 0.375/0.75 = 0.5
We can observe that the division results in a common ratio of 0.5.
Now, we can use this common ratio to determine the function.
The general form of an exponential function is: 𝑓(𝑥) = 𝑎 * 𝑏^𝑥, where 𝑎 is the initial value and 𝑏 is the common ratio.
Substituting the first point (2, 6) into the function, we get the equation:
6 = 𝑎 * (0.5)^2
Simplifying the equation, we have:
6 = 0.25 * 𝑎
Dividing both sides of the equation by 0.25, we find:
24 = 𝑎
Therefore, the function that represents the relationship is:
𝑓(𝑥) = 24 * (0.5)^𝑥