Final answer:
To find an exponential function from two points, use the coordinates to create two equations in the form y = ab^x. Then solve for the base b by dividing one equation by another, and finally solve for a using one of the original equations.
Step-by-step explanation:
Finding an exponential function from two points involves using the fact that the exponential and natural logarithm functions are inverses of each other. To determine the equation of an exponential function, you need the coordinates of two points. Suppose the two points are (x1, y1) and (x2, y2). The general form of an exponential function is y = abx, where a is the initial value and b is the base of the exponent.
First, you derive two equations based on the two points by substituting their coordinates into the general form:
y1 = abx1
y2 = abx2.
Then, you divide the second equation by the first to eliminate a and solve for b. After finding b, substitute it back into one of the equations to solve for a.
For example, if the given points are (1, e0.488) and (2, e0.976), then the equations are:
e0.488 = ab1
e0.976 = ab2.
Upon division, you get b and finally a, which fully determine your exponential function.