Question

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

Answer 1

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