Final answer:
To find an exponential equation from two points, set up two equations using the general exponential form y = ab^x, solve for 'a' and 'b', and then write the complete equation. Use logarithms if necessary to determine the growth rate 'b'.
Step-by-step explanation:
Finding the Exponential Equation Given Two Points
To solve the mathematical problem completely, we need to find the exponential equation that passes through two given points. Assuming these points are represented by (X1, Y1) and (X2, Y2), we need to determine the base and the growth rate of our exponential function.
The general form of an exponential function is y = abx, where 'a' is the initial value when x = 0, and 'b' is the growth rate. To find 'a' and 'b', we set up two equations based on our two points.
The steps are as follows:
- Substitute the coordinates of the first point into the exponential equation to create the first equation.
- Do the same with the second point to create a second equation.
- Solve the system of equations to find the values of 'a' and 'b'.
- Once 'a' and 'b' are determined, write the complete exponential equation.
We may need to use logarithms to solve for 'b'. The properties of logarithms, especially the fact that the exponential and natural logarithm are inverse functions, are particularly useful. For instance, knowing that eln(x) = x and ln(ex) = x can be used to transform the equations and make 'b' solvable.
If we're dealing with squaring exponentials, remember to square the digit term as usual and multiply the exponent by 2. This may be necessary in some cases where the exponential equation is part of a larger problem.
Once you have both 'a' and 'b', you can plug them back into the general form to get the specific exponential equation that goes through the two given points.