Construct exponential functiom of points (0,200) and (6,380).
The exponential function is
y(x)=____
let the exponential equation be y = a e^(kx)
for (0,200)
200 = a e^0
a = 200
so you have y = 200 e^kx
for (6,380)
380 = 200 e^(6k)
1.9 = e^(6k)
take ln of both sides, and use rules of logs
ln 1.9 = 6k lne, but lne = 1
6k = ln 1.9
k = ln 1.9/6
the exponential equation is
y = 200 e^(ln1.9/6 x)
To construct an exponential function using two points, we can use the general form of an exponential function:
y(x) = ab^x
Where 'a' is the initial value or y-intercept, 'b' is the base or growth/decay factor, and 'x' is the independent variable.
Given the points (0, 200) and (6, 380), we can substitute the coordinates of the first point (0, 200) into the equation to solve for 'a':
200 = ab^0
200 = a * 1
a = 200
Hence, the value of 'a' is 200.
Now, substitute the coordinates of the second point (6, 380) into the equation and use the value of 'a' to solve for 'b':
380 = 200 * b^6
Divide both sides by 200:
380/200 = b^6
1.9 = b^6
To isolate 'b', we need to take the 6th root of both sides:
b = (1.9)^(1/6)
Using a calculator, we find that b is approximately 1.136.
Therefore, the exponential function is:
y(x) = 200 * (1.136)^x