Question

The graph shows that f(x)=(one-third) Superscript x is translated horizontally and vertically to get the function (one-third) Superscript x minus h.

On a coordinate plane, 2 exponential functions are shown. f (x) decreases in quadrant 2 and approaches y = 0 in quadrant 1. It goes through (negative 1, 3), (0, 1), (1, 0.3). g (x) decreases in quadrant 1 into quadrant 4. It crosses the x-axis at (4, 0), and goes through (5, negative 2).

What is the value of k?

−5
−3
3
5

Answer 1

Assume the translation is g(x) = (1/3)^{x−h} + k.
Use the points (4,0) and (5,−2):

0 = (1/3)^{4−h} + k → k = −(1/3)^{4−h}
−2 = (1/3)^{5−h} + k

Subtract the first from the second:
−2 = (1/3)^{5−h} − (1/3)^{4−h}
= (1/3)^{4−h}[(1/3) − 1] = (1/3)^{4−h}(−2/3)

So −2 = −(2/3)(1/3)^{4−h} → (1/3)^{4−h} = 3.

Then k = −(1/3)^{4−h} = −3.

Answer: −3.