this table represents some points in the graph of an exponential function. What function represents the relationship?
x y
2 6
3 3
4 1.5
5 .75
6 .375
To determine the function that represents the relationship between the values in the table, we can observe the pattern of the y-values as the x-values increase.
We can see that the y-values are decreasing as the x-values increase by a constant factor of 0.5. This indicates an exponential decay.
By dividing each y-value by the previous y-value, we get the following sequence:
6/2 = 3
3/6 = 0.5
1.5/3 = 0.5
0.75/1.5 = 0.5
0.375/0.75 = 0.5
Since the ratio between consecutive y-values is constant, we can conclude that the function representing the relationship is an exponential decay function.
To find the specific function, we can express it in the form:
y = a * b^x
Substituting the given values, we have the following equations:
6 = a * b^2
3 = a * b^3
1.5 = a * b^4
0.75 = a * b^5
0.375 = a * b^6
By taking the ratio of consecutive equations, we can eliminate the constant "a":
3/6 = b^3/b^2 = b
1.5/3 = b^4/b^3 = b
0.75/1.5 = b^5/b^4 = b
0.375/0.75 = b^6/b^5 = b
Simplifying these equations, we have:
0.5 = b
0.5 = b
0.5 = b
0.5 = b
Since all the ratios are equal to 0.5, we can conclude that b = 0.5.
Plugging this value back into any of the original equations, we can solve for "a". Let's use the first equation:
6 = a * (0.5)^2
6 = a * 0.25
a = 6/0.25
a = 24
Therefore, the function representing the relationship in the table is:
y = 24 * 0.5^x