Final answer:
To find the exponential function that passes through two given points, we can use the formula y = abˣ and solve the equations using the coordinates of the points.
So, the correct answer is B) Exponential growth function calculator.
Step-by-step explanation:
The given points (3,4) and (4,1) are used to determine the type of function that passes through these points. Since the function is exponential, we can use the formula y = abˣ, where (x,y) are the coordinates of the points, a is the initial value, and b is the base of the exponential function.
Let's substitute the first point (3,4) into the equation:
4 = ab³
Similarly, substituting the second point (4,1) into the equation:
1 = ab⁴
We can solve this system of equations to find the values of a and b.
Dividing equation 1 by equation 2:
4/1 = (a b³) / (a b⁴)
4 = 1/b
b = 1/4
Now that we have found the value of b, we can substitute it back into one of the original equations to solve for a. Using equation 1:
4 = a (1/4)³
4 = a 1/64
a = 256
Therefore, the exponential function that passes through the points (3,4) and (4,1) is:
y = 256 * (1/4)ˣ
This is an example of an exponential decay function, as the base (1/4) is between 0 and 1.
So, the correct answer is B) Exponential growth function calculator.