Question

use a transformation of the graph of \\(y = x^5\\) to graph the function.

\\f(x) = (x - 1)^5 + 2\\

Explanation:

Identify the parent function

The parent function is:

$$ y = x^5 $$

Analyze horizontal shift

The term \((x - 1)\) shifts the graph horizontally.
This represents a shift right by 1 unit.

Analyze vertical shift

The term \(+ 2\) shifts the graph vertically.
This represents a shift up by 2 units.

Determine key point transformations

The reference point \((0,0)\) on \(y = x^5\) shifts to:

$$ (0 + 1, 0 + 2) = (1, 2) $$

The reference point \((1,1)\) shifts to:

$$ (1 + 1, 1 + 2) = (2, 3) $$

The reference point \((-1,-1)\) shifts to:

$$ (-1 + 1, -1 + 2) = (0, 1) $$

Answer:

To graph the function \(f(x) = (x - 1)^5 + 2\), shift the graph of the parent function \(y = x^5\) horizontally to the right by 1 unit and vertically upward by 2 units.