Question

The function g is related to one of the parent functions

g(x) = −(x − 2)^3
a.) Identify the parent function f.

b.) Use function notation to write g in terms of f.

Answer 1

a. f(x) = x³

b. g(x) = -f(x-2)

Answer 2

a) parent: f(x) = x³

b) g(x) = -f(x -2)

Explanation:

You want the parent function and the given function written in terms of the parent function for g(x) = -(x -2)³.

a) Parent function

The parent function is the function that remains after scale factors and translations are removed. In general, a transformed function will look like ...

g(x) = c·f((x-a)/b) +d

which scales the function f(x) horizontally by a factor of b, vertically by a factor of c, and translates the result by (a, d).

Here, we recognize that ...

g(x) = -(x -2)³

has c = -1, a = 2, b = 1, d = 0

and the parent function is ...

f(x) = x³

b) g in terms of f

Using the values for a, b, c, d that we recognized above, we have ...

g(x) = -f(x -2)

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Additional comment

When scale factors are negative, they reflect the function over the relevant axis. Here, the negative vertical scale factor reflects the function vertically over the x-axis. The translation is 2 units to the right.