Question

Suppose the graph of a function \(f\) is known. Then the graph of \(y=f(x-2)\) is obtained by a __________ shift of the graph of \(f\) to the ___________ a distance of 2 units.

Short answer

Horizontal shift of the graph to the right by 2 units.

Step-by-step solution

Identify the transformation type

When analyzing the function transformation, note that the form of the function is given as \(y = f(x - a)\). This indicates a horizontal shift.

Determine the direction of the shift

In the equation \(y = f(x - 2)\), the \( - 2 \) inside the parentheses suggests that the shift will be to the right. Graph transformations involving \(x - a\) result in a shift to the right.

Specify the distance of the shift

The shift is by a distance of the value inside the parentheses. Since it is \(x - 2\), the horizontal shift will be 2 units to the right.

Key concepts

Horizontal Shift

When working with functions, one common transformation is the horizontal shift. This involves moving the entire graph of the function left or right along the x-axis. The general form that indicates a horizontal shift is given by the function:

y = f(x - a)

Here, the value inside the parentheses (in this case, 'a') determines how far and in what direction the graph shifts. For example, if we have y = f(x - 2), the shift is 2 units to the right. Similarly, if the function were y = f(x + 2), the shift would be 2 units to the left.

Graph of a Function

To better understand a horizontal shift, consider the graph of a function. The graph displays all the possible values of y for each x, based on the given function rule. Moving the graph horizontally does not change the shape or steepness; it only changes the position.

For instance, if we initially have y = f(x), and we apply the horizontal shift (e.g., y = f(x - 2)), every point on the graph of y = f(x) will move 2 units to the right. This process helps in visualizing how the function behaves under different transformations.

Function Transformation Types

Function transformations are modifications made to the basic shape of a function's graph. Some common types are:

• Vertical Shift: Moving the graph up or down (e.g., y = f(x) + C)
• Horizontal Shift: Moving the graph left or right (e.g., y = f(x - a))
• Vertical Stretch/Compression: Changing the steepness by multiplying the function by a factor (e.g., y = k*f(x))
• Horizontal Stretch/Compression: Changing the width by altering the x inside the function (e.g., y = f(b*x))

Each type affects the graph differently. For example, a vertical shift will raise or lower the entire graph, while a horizontal shift moves it left or right.

Distance of Shift

The distance of a shift in the graph of a function is determined by the value of the constant added or subtracted inside the function's argument.
In our example of y = f(x - 2), the constant '-2' tells us the graph will move 2 units to the right. It's important to note that even though it appears as '-2,' the direction of the horizontal shift is to the right.
Understanding the distance of the shift helps in predicting how the graph will look after the transformation. Always remember:

• If y = f(x - a), shift right by 'a' units.
• If y = f(x + a), shift left by 'a' units.

These shifts allow you to control and manipulate the graph's position to better understand the function's behaviour.