if f is an exponential function, and f(5)=20 and f(9) = 14, use the add-multiply property to calculate f(13),f(17), and f(21). Show your method
there is a difference of x=+4 between 5, 9
There is a multiply difference of 14/20=.7 between the two.
f(13)=14*.7
f(17)=14*.7*.7
f(21)=14(.7*.7*.7)
let f(x) = a(b^x) or
y = a(b^x)
given: when x=5, y=20
20 = a b^5
given: when x=9, y=14
14 = a b^9
divide the 2nd by the first:
b^4 = 14/20 = 0.7
b = (0.7)^.25 = appr .91469
sub into 20 = a b^5
20 = a(.91469)^5
a = 31.236
f(x) = 31.236 (.91469)^x
f(13) = 31.236(.91469)^13 = 9.8
you do the button-pushing for the rest
Note, I used the memory locations of my calculator to store all the supplied decimal values for both a and b
To calculate the values of f(13), f(17), and f(21) using the add-multiply property, we need to determine the exponential function's base and exponent.
Step 1: Identify the base of the exponential function
The add-multiply property states that if f(x) is an exponential function of the form f(x) = a * b^x, where 'a' is a constant and 'b' is the base, then f(x + k) = a * b^x * b^k. Therefore, to apply this property, we need to find the common ratio 'b'.
Given that f(5) = 20 and f(9) = 14, we can set up a system of equations to solve for the base 'b'.
f(5) = a * b^5 = 20 -- Eq. (1)
f(9) = a * b^9 = 14 -- Eq. (2)
Step 2: Solve the system of equations
Divide Eq. (2) by Eq. (1):
[a * b^9] / [a * b^5] = 14 / 20
b^4 = 14/20 = 7/10
Take the fourth root of both sides to solve for b:
b = (7/10)^(1/4)
b ≈ 0.872
Step 3: Calculate f(13), f(17), and f(21)
Now that we know the base 'b' is approximately 0.872, we can apply the add-multiply property to calculate f(13), f(17), and f(21).
f(13) = f(5 + 8) = f(5) * b^8
= 20 * (0.872^8)
f(17) = f(9 + 8) = f(9) * b^8
= 14 * (0.872^8)
f(21) = f(9 + 12) = f(9) * b^12
= 14 * (0.872^12)
You can now substitute the calculated value of 'b' into the equations above to find the values of f(13), f(17), and f(21).