An exponential function passes through the points (2,3), (9,33), (15,K). What is the value of K?
y = ab^x
a*b^2 = 3
a*b^9 = 33
a = 3/b^2, so
a*b^9 = 3/b^2 * b^9 = 3b^7 = 33
b = 11^(1/7)
a = 3/11^(2/7)
y = 3/11^(2/7) * 11^(x/7)
= 3*11^((x-2)/7)
y(15) = 3*11^(13/7) = k
To find the value of K, we need to use the given points to form an exponential function that represents the pattern of the data. We can start by writing the general form of an exponential function as:
f(x) = a * b^x
where 'a' and 'b' are constants.
Using the first point, (2,3), we can substitute the values of x and f(x) into the equation:
3 = a * b^2
Similarly, using the second point, (9,33), we get:
33 = a * b^9
Now, let's solve these two equations simultaneously to find the values of 'a' and 'b':
Divide the second equation by the first equation:
(33/3) = (a * b^9) / (a * b^2)
11 = b^7
Take the seventh root of both sides:
b = √(11)
Substitute this value of 'b' back into the first equation to find 'a':
3 = a * (√(11))^2
3 = a * 11
a = 3/11
So now we have the values of 'a' and 'b', which are a = 3/11 and b = √(11).
Finally, we can use these values to determine the value of K using the third point, (15,K). Substitute x = 15 into the exponential function:
f(15) = (3/11) * (√(11))^15
Calculate this expression to find the value of K.