Problem 22 - Algebra Form and Function

Question

Refer to the graph of \(y=f(x)\) in Figure 8.21 . Sketch the graph of each function. \(y=f(x-3)-1\)

Step-by-Step Solution

Answer

Answer: The graph of \(y=f(x)\) undergoes a horizontal translation of 3 units to the right and a vertical translation of 1 unit downward to create the graph of \(y=f(x-3)-1\). This causes the key points and features of the original graph, such as x-intercepts and y-intercepts, to shift right by 3 units and down by 1 unit.

Step 1: Identify the original graph

First, examine Figure 8.21 which shows the graph of \(y=f(x)\). Take note of its key points and characteristics, such as its x-intercepts, y-intercepts, and any other noticeable features.

Step 2: Apply horizontal translation

The function \(y=f(x-3)\) implies a horizontal translation to the right by 3 units. Therefore, shift all the key points and features of the original graph to the right by 3 units. For example, if there is an x-intercept at \(x=1\) on the graph of \(y=f(x)\), the x-intercept on the graph of \(y=f(x-3)\) will be at \(x=1+3=4\).

Step 3: Apply vertical translation

The function \(y=f(x-3)-1\) implies a vertical translation downward by 1 unit. After completing the horizontal translation in Step 2, shift all the key points and features resulting from the horizontal translation downward by 1 unit. For example, if there is a y-intercept at \(y=2\) on the graph of \(y=f(x-3)\), the y-intercept on the graph of \(y=f(x-3)-1\) will be at \(y=2-1=1\).

Step 4: Sketch the transformed graph

Finally, with all the translations performed, sketch the new graph of \(y=f(x-3)-1\). Make sure the key points and features match those from Steps 2 and 3, and maintain the overall shape of the original graph. Remember to label the axes and any key points, such as x-intercepts and y-intercepts, to make the graph clear for others to understand.

Key Concepts

Horizontal Translation

Horizontal translation is a crucial aspect of function transformations that involves shifting the graph of a function along the x-axis. When you see a function in the form of \(y = f(x-a)\), this signifies a horizontal translation.

• The graph is moved to the right if \(a\) is positive.
• Conversely, the graph shifts to the left if \(a\) is negative.

This transformation does not affect the shape of the graph; it only changes the position of the graph along the x-axis. So, the whole graph, including x-intercepts and any other significant features, are simply slid horizontally across the plane.
In the example given, \(y = f(x-3)\), the graph is shifted to the right by 3 units. This means if the original graph of \(y = f(x)\) has an important point at \(x = 1\), after the transformation, this point will be at \(x = 4\). This movement is consistent throughout all the points on the graph.

Vertical Translation

Vertical translation involves shifting the graph along the y-axis. For the transformation written as \(y = f(x) + b\), the translation depends on the sign and value of \(b\).

• If \(b\) is positive, move the graph up by \(b\) units on the y-axis.
• If \(b\) is negative, shift the graph down by \(|b|\) units.

This can shift features such as the y-intercepts and peaks, but it does not change the shape of the graph.
In our problem, we observe a function written as \(y = f(x-3) - 1\), where \(-1\) indicates a downward shift by one unit. After shifting horizontally, this transformation moves the entire graph downward. For instance, if a point after the horizontal translation is at \(y = 2\), it will now be at \(y = 1\), maintaining the same x-coordinate.

Graph Sketching

Graph sketching is the final step in understanding function transformations. It involves accurately illustrating the cumulative effect of transformations on the graph of a function.

• Begin by noting the original shape and features of the graph.
• Apply the horizontal and vertical shifts as calculated from translations.
• Ensure the key points, like intercepts and vertices, are clearly labeled.

Approach graph sketching with patience. It's essential to ensure that the proportional relationships and general shape of the graph remain consistent, even after applying the transformations.
For example, with the function \(y = f(x-3) - 1\), one should first shift the graph three units to the right and then move it one unit down. Make sure to replicate the spacing between key points and shape accurately. Clearly label axes and key points, which helps in verifying the transformations and understanding the function's behavior more intuitively.