Question 232438
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ \left(\frac{1}{4}\right)^{x-3}].


Evaluate:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(3),\,f(2),\,f(1),\,f(0),\,f(-1),\,f(-2),\,f(-3)]


I'll do three of these for you:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(3)\ =\ \left(\frac{1}{4}\right)^{(3)-3}\ =\ \left(\frac{1}{4}\right)^{0}\ =\ 1].


So the ordered pair is (3, 1)


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(1)\ =\ \left(\frac{1}{4}\right)^{(1)-3}\ =\ \left(\frac{1}{4}\right)^{-2}\ =\ 4^2 = 16].


So the ordered pair is (1, 16)



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(-2)\ =\ \left(\frac{1}{4}\right)^{(-2)-3}\ =\ \left(\frac{1}{4}\right)^{-5}\ =\ 4^5 = 1024].


So the ordered pair is (-2,1024)


I presume you know how to plot points.  Once you have done all the required arithmetic and plotted your set of points, draw a smooth curve.  It should look something like this:


{{{drawing(
500, 500, -2, 8, -2, 18,
grid(1),
blue(circle(1,16,.1)),
blue(circle(3,1,.1)),
blue(circle(2,4,.1)),
blue(circle(4,.25,.1)),
graph(
500, 500, -2, 8, -2, 18,
(1/4)^(x-3))
)}}}


Notice that no matter how large you make *[tex \Large x], the value of the function will never be exactly zero.  It gets very close, very fast, but never quite makes it.  Therefore asymptotic to *[tex \Large y = 0], in other words, to the *[tex \Large x]-axis.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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