You can write an exponential function from two points on the function's graph. For example, write an exponential function y = ab^{x} for a graph that includes (1,1) and (2, 4)
The goal is to use the two given points to find a and b. Then, we can replace a and b in the equation y = ab^{x} with the values we found.
Use the general form of the exponential function y = ab^{x} and substitute for x and y using (1, 1)
1 = ab^{1}
1 = ab
Divide both sides by b to solve for a
Use the general form of the exponential function y = ab^{x} again and substitute for x and y using (2, 4)
4 = ab^{2}
4 = b.
The exponential function is y = (1/4)(4)^{x}
Example #2
Find y = ab^{x} for a graph that includes (1, 2) and (-1, 8)
Use the general form of the exponential function y = ab^{x} and substitute for x and y using (1, 2)
2 = ab^{1}
2 = ab
Divide both sides by b to solve for a
Use the general form of the exponential function y = ab^{x} again and substitute for x and y using (-1, 8)
8 = ab^{-1}
b^{2} × 8 = 2
Divide both sides of the equation by 8
(b^{2} × 8) / 8 = 2 / 8
b^{2} = 2 / 8
b^{2} = 1 / 4
b = square root of 1 / 4
b = 1 / 2
Now, let us find a using the equation a = 2 / b
a = 2 ÷ 1 / 2 = 2 / 1 ÷ 1 / 2 = 2 / 1 × 2 / 1 = 4 / 1 = 4
The exponential function is y = 4(1/2)^{x}
Aug 16, 22 04:10 AM