There are different ways to approach this problem, but one common method is to use the general form of an exponential function:
y = a ⋅ b^x
where a is the initial value (when x=0), b is the base of the exponential (a constant greater than 0), and x is the independent variable (usually representing time or another variable that changes continuously).
To find the value of a, we can see from the table that y=0 when x=0, so a=0. To find the value of b, we can notice that each y-value is half of the previous y-value, meaning that the ratio between consecutive y-values is the same:
1/0.005 = 200
0.005/0.0025 = 2
0.0025/0.00125 = 2
This common ratio is equal to b^1 (since it is the ratio between consecutive outputs for a one-unit increase in the input). Therefore:
b^1 = 2
b = 2^(1/1) = 2
Substituting a=0 and b=2 into the general form of an exponential function, we get:
y = 0 ⋅ 2^x = 0
This means that the exponential function represented by the table is y = 0 for all values of x. Alternatively, we can notice that the values in the table are decreasing exponentially as x increases, which means that the function could be expressed as the inverse of an exponential function, namely:
y = 1/(2^x)
Find the equation of the exponential function represented by the table below:
x
x
y
y
0
0
0.01
0.01
1
1
0.005
0.005
2
2
0.0025
0.0025
3
3
0.00125
0.00125
1 answer