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In graph transformations, a horizontal shift changes the position of the graph along the x-axis. To understand this better, let’s look at our function:

The base function is given by: \ f(x) = x^{5}.

When you see a term inside the function like (x-1), it indicates a horizontal shift. Specifically, the function \ (f(x)) = (x-1)^{5} means moving the graph one unit to the right.

This happens because x-1 equals zero when x=1, so the zero of the function (and every other point) moves one step to the right. If it were (x+1), the graph would shift one unit to the left.

The general rule can be summarized as:

• \( x - h \) results in a shift of h units to the right.

• \( x + h \) results in a shift of h units to the left.

A vertical shift moves the graph up or down along the y-axis. This is shown by terms added or subtracted outside the function. Let’s consider our example function again:

f(x) = (x-1)^{5} + 2.

The term +2 outside the function tells us to move the graph up by 2 units.

This means that every point on the graph of the function y = x^{5} is shifted upward by 2 units.

Here are key points to remember about vertical shifts:

• Adding a value outside the function, like (x) + c, shifts the graph c units up.

• Subtracting a value outside the function, like (x) - c, shifts the graph c units down.

Combining both horizontal and vertical shifts, the graph of f(x) = (x-1)^{5} + 2 moves one unit to the right and two units up from the base graph y=x^{5}.

The concept of a base function is foundational in graph transformations. The base function provides the original graph before any transformations. In this exercise, our base function is:

y = x^{5}.

This base function is a simple polynomial curve, and understanding its shape helps us visualize the impacts of transformations.

Important characteristics of base functions include:

• Knowing the basic shape (for y = x^{5}, it’s a curve starting from the bottom-left and extending to the top-right).

• Recognizing critical points like intercepts and symmetry.

In this example, our task was to take the base function y = x^{5} and apply both horizontal and vertical shifts to find and graph the transformed function f(x) = (x-1)^{5} + 2. By understanding and practicing the transformations on the base function, you can handle any complex function transformations confidently.