Question

The parent function f(x) = log3x has been transformed by reflecting it over the x-axis, stretching it vertically by a factor of two and shifting it up three units. Which function is representative of this transformation? g(x) = log3(2x) - 3 g(x) =log3(-2x) + 3 g(x) = 2log3(x) - 3 g(x) = -2log3(x) + 3

Answer 1

The function that represents the transformation is g(x) = 2log3x - 3.

Step-by-step explanation:

The function that represents the transformation of the parent function f(x) = log3x, after reflecting it over the x-axis, stretching it vertically by a factor of two, and shifting it up three units, is g(x) = 2log3x - 3.

To reflect a function over the x-axis, we change the sign of the function's output or y-coordinate. To stretch a function vertically by a factor of two, we multiply the function by 2. To shift the function up three units, we subtract 3 from the function's output or y-coordinate.

Therefore, g(x) = 2log3x - 3 is the correct function that represents the given transformation.

Answer 2

The transformation from f(x) to g(x) using

g(x)=a*f(x-h)+k

will stretch f(x) by a scale factor of a (>0), translates to the right by h, and translates upwards of k.

Similarly, the transformation from f(x) to g(x) using

g(x)=-a*f(x-h)+k

will stretch f(x) by a scale factor of a (>0) AND reflects over the x-axis, translates to the right by h, and finally translates upwards of k.

For the given problem,

f(x)=log_3(x)

a=-2 (dilates with scale factor of 2, AND with reflection over the x-axis)

h=0 (no horizontal translation)

k=3 (translates UPwards by 3 units)

Put into the above formula,

g(x)=-2f(x)+3=-2log_3(x)+3.

Note: in case you are not familiar with it, _ means subscript, and ^ means superscript, applicable to some text processing software.