Since it has a negative exponent, the equation [tex]y=4^-^2^x[/tex] is showing exponential decay. The graph will decrease.
In order to find the percent of change, we need to review our exponent properties. The product property tells us that when we raise a power to a power, we multiply the exponents. This means we can view our exponent of -2x as the product of two exponents, -2 and x. This would look like
[tex]y=(4^-^2)^x[/tex]. We know that negative number exponents tell us to "flip" sides of the fraction, so that tells us that 4⁻² = 1/4² = 1/16. We can then rewrite our equation as
[tex]y=(\frac{1}{16})^x[/tex]
Exponential decay is in the form y = a(1-b)ˣ. From our equation, a = 1. We know that 1-b = 1/16; this means that b must be 15/16. Thus our percent of decrease would be 15/16 = 0.9375 = 93.75%.
