To find the exponential function given two points, we need to use the general form of an exponential function, which is:
y = a * b^x
where: - y is the dependent variable (output) - x is the independent variable (input) - a is the initial value or y-intercept - b is the base or growth/decay factor
Let’s say we have two points (x1, y1) and (x2, y2). We can substitute these points into the equation to form a system of equations:
y1 = a * b^x1 y2 = a * b^x2
To solve for a and b, we can divide the second equation by the first equation:
y2 / y1 = (a * b^x2) / (a * b^x1)
Simplifying the equation:
y2 / y1 = b^(x2 - x1)
Taking the logarithm of both sides:
log(y2 / y1) = (x2 - x1) * log(b)
Now, we can solve for log(b):
log(b) = (log(y2 / y1)) / (x2 - x1)
Finally, we can substitute log(b) back into the equation to solve for a:
a = y1 / (b^x1)
Therefore, the exponential function can be written as:
y = (y1 / (b^x1)) * b^x