For the function g(x)=(-2/3^4x+8)-5

Question

a) State the transformations from the parent function.

b) Rewrite in the form y=abx+c

c) Determine the horizontal asymptote

d) State whether the function is increasing or decreasing

e) State the Domain

f) State the Range

Answer 1

(a) Assume g(x)=-(⅔^4x+8)-5.

p(x)=e^x could be regarded as the parent function, because it has the same basic shape.

If e^xln(b)≡b^x, (16/81)^x=e^xln(16/81)=e^-1.622x. A negative coefficient on the exponent reflects the parent in the y (vertical) axis. In other words, e^-1.622x is a reflection of e^1.622x. So this is the early step in the transformation process. The magnitude of the coefficient produces a horizontal dilation. The transformation is e^x→e^px, where p=ln(16/81). The next part of the transformation is e^px→ae^px (dilation, or amplification, vertically), and the final part is a downward shift (relocation) caused by the negative value of c.

(b) g(x)=-(⅔)⁸(2⁴/3⁴)^x-5=ab^x+c, so a=-(⅔)⁸=-256/6561, b=16/81, c=-5.

(c) The horizontal asymptote is y=-5, because the exponential term shrinks to zero as x increases, that is, b=16/81<1 so increasing powers make it smaller. e^-1.622x→0 as x→∞.

(d) g(x) increases towards the asymptote.

(e) The domain is (-∞,∞), that is, it's unrestricted.

(f) The range is (-∞,-5). The asymptote is the upper limit to g(x).