We want to write an exponential function for the graph that passes through the given points. Let's consider the general form for this type of function.
y=ab^x
Such functions have horizontal asymptote at y=0. However, we are told that the function has a horizontal asymptote at y=20. This means that the given function is translated 20 units up.
y=ab^x+20Now, since we want the points to lie on the graph they must satisfy this equation. Let's substitute (1,22) into the above formula.
We received two equations, so to find the values of a and b we need to solve the system of equations.
ab=2 & (I) ab^3=0.125 & (II)
The first equation says that the product of a and b is 2. Notice that the second equation contain the same expression multiplied by b^2.
ab=2 ab^3=0.125 ⇒ ab=2 ab* b^2=0.125
This allows us to substitute the value of ab to the second equation and solve for b. Keep in mind that b is the base of the exponential function, so it has to be positive.
