Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x-2)+3\)

Step-by-Step Solution

Answer

Shift right by 2 units and up by 3 units.

Step 1: Identify Original Function

The original function is given by the generic form \(y = f(x)\). This is our starting point for analyzing transformations.

Step 2: Analyze Horizontal Shift

The function \(y = f(x-2)\) indicates a horizontal shift. The \(x-2\) means that the graph of \(f(x)\) is shifted to the right by 2 units. This transformation changes the input directly, which moves the entire graph horizontally.

Step 3: Analyze Vertical Shift

The function has an additional component, \(+3\), outside of the function \(f(x-2)\). This indicates a vertical shift. Specifically, it moves the entire graph of \(f(x-2)\) upwards by 3 units.

Step 4: Combine Transformations

Combining the transformations, the graph of \(f(x)\) undergoes a shift to the right by 2 units, and then it is shifted upwards by 3 units. The final graph of \(y = f(x-2) + 3\) reflects these two transformations together.

Key Concepts

Horizontal Shift

A horizontal shift in a function occurs when the graph of the function is moved left or right. This movement is achieved by altering the input variable of the function.
For example, in the function transformation expressed as \( y = f(x-2) \), the term "\( x-2 \)" signals a horizontal shift. Specifically, the "\(-2\)" inside the function indicates a shift to the right.

• The minus sign (\(-\)) in \(x-2\) is a signal to move right.
• The number 2 indicates the distance of the shift, so the graph moves right by 2 units.

This transformation affects where each \(x\)-value is found on the graph, pulling every point on the graph 2 units right from where it originally was.

Vertical Shift

A vertical shift, on the other hand, adjusts the graph of a function up or down. Unlike the horizontal shift, this transformation does not change the input variable. Instead, it alters the output value directly by adding or subtracting a number from the function result.
In the transformation \( y = f(x-2) + 3 \), the "+3" means a vertical shift. This number sits outside the function and impacts the whole function's outcome.

• The plus (\(+\)) sign in "+3" tells us to move the graph upwards.
• The number 3 indicates the shift is by 3 units.

So, every point on the graph moves up by 3 units, making the whole function rise higher on the graph. It's like taking the graph as is and lifting it by 3 units.

Graph Transformation

Graph transformations are changes made to the position, size, or shape of a graph. The transformation might involve shifting, stretching, compressing, or reflecting the graph. In this case, we are looking at a combination of transformations through horizontal and vertical shifts. The function \( y = f(x-2) + 3 \) combines these shifts:

• First, we apply a horizontal shift to the right by 2 units.
• Next, we apply a vertical shift upwards by 3 units.

Each transformation applies sequentially, adjusting the graph's layout in the coordinate plane. Together, they result in a translated graph that maintains the same general shape as the original \( f(x) \) but appears in a different position on the graph.