Question 270116
It is not possible to know whether your equation is
{{{y = (1/4)^(x-3)}}}
or
{{{y = (1/4)^x-3}}}
Please use parentheses generously to make your problem clear. The first equation should be typed in as
y = (1/4)^(x-3)<br>
I will answer the question for both equations. The key to both solutions is to understand what values a positive number raised to a power can be. Your equations both have a positive number, 1/4, raised to some power. No matter what exponent you put on 1/4, the result will NEVER be:<ul><li>Zero. (Remember {{{(1/4)^0 = 1}}} not zero!)</li><li>Negative. (Remember 1/4 to a negative power is not a negative number. Negative exponents just mean reciprocals. And any reciprocal of 1/4 is still positive.)</li></ul>
While 1/4 to some power can never be zero, it can, however, be an extremely small positive fraction (i.e. a very tiny fraction just above zero). 1/4 to very large positive powers will be very small fractions near zero. The larger the exponent, the closer we get to zero. This is how an asymptote works.<br>
So for
{{{y = (1/4)^(x-3)}}}
where y equals a power of 1/4, y = 0 is a horizontal asymptote.
and for
{{{y = (1/4)^x-3}}}
where y equals a power of 1/4 minus 3, then y = 0-3 = -3 is a horizontal asymptote.