Question

Suppose that the graph of a function \(f\) is known. Explain how the graph of \(y=f(x)-2\) differs from the graph of \(y=f(x-2)\).

Short answer

The graph of \(y=f(x)-2\) moves down by 2 units, while the graph of \(y=f(x-2)\) shifts right by 2 units.

Step-by-step solution

Understand the function translation

The expression \(y = f(x) - 2\) represents a vertical translation. Subtracting 2 from the function moves the graph down by 2 units.

Graph of \(y = f(x) - 2\)

To obtain the graph of \(y = f(x) - 2\), take each point on the graph of \(y = f(x)\) and move it 2 units downward. This shifts the entire graph vertically downward.

Horizontal shift understanding

The expression \(y = f(x-2)\) represents a horizontal translation. Replacing \(x\) by \(x-2\) shifts the graph to the right by 2 units.

Graph of \(y = f(x-2)\)

To obtain the graph of \(y = f(x-2)\), take each point on the graph of \(y = f(x)\) and move it 2 units to the right. This shifts the entire graph horizontally to the right.

Conclusion: Comparing the transformations

In summary, \(y = f(x) - 2\) shifts the graph of \(f\) downward by 2 units, while \(y = f(x-2)\) shifts the graph of \(f\) to the right by 2 units.

Key concepts

Vertical Translation

Vertical translation is a type of function transformation that shifts the graph of a function up or down. This is done by adding or subtracting a constant value to the function. For example, when you have the original function represented by the graph of \(y = f(x)\), applying a vertical translation using \(y = f(x) - 2\) shifts the entire graph down by 2 units.

To understand it better:

• Imagine the graph is pasted on a transparent sheet.
• If you move the sheet 2 units down, every single point on the graph moves 2 units down as well.

For each point \((x, f(x))\) on the original graph, you will now have a corresponding point \((x, f(x)-2)\) on the translated graph.

Vertical translations do not affect the shape of the graph; they only change its position vertically. This means the peaks, troughs, and intercepts of the graph will all drop by the same amount.

Horizontal Translation

Horizontal translation shifts the graph of a function left or right. This transformation is done by changing the input value \(x\) by a constant. For the function \(y = f(x)\), applying a horizontal translation using \(y = f(x-2)\) moves the graph 2 units to the right.

Here's a detailed look at it:

• As if you were sliding the transparent sheet horizontally.
• When you replace \(x\) with \(x-2\), each point on the original graph \((x, f(x))\) shifts to the right, becoming \((x+2, f(x))\).

This movement affects every point on the graph equally, shifting the entire graph to the right without changing its shape.

Therefore, like vertical translations, horizontal translations also do not alter the overall shape or curvature of the graph, just its horizontal positioning.

Graph Shifting

Graph shifting is a general term that includes both vertical and horizontal translations. It helps to change the position of the graph without altering its shape.

To recap:

• Vertical shifting moves the graph up or down by adding or subtracting a value from the entire function.
• Horizontal shifting moves the graph left or right by modifying the input \(x\) in the function.

These transformations are crucial for understanding how functions behave under translations. They show how each point on the graph is systematically repositioned, making them a fundamental concept in graphing functions.

In mathematical notation:

• Vertical translation of \(a\) units: \(y = f(x) + a\)
• Horizontal translation of \(b\) units: \(y = f(x-b)\)

More complex transformations can occur when both types of shifts are combined, allowing for even more dynamic graphing possibilities.