Problem 17 - 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 \(f\). \(y=f(x)-7\)

Step-by-Step Solution

Answer

The graph of \(y = f(x) - 7\) is a downward shift of \(f(x)\) by 7 units.

Step 1: Identify the Transformation

We start by identifying the transformation applied to the original function. The given function is \(y = f(x) - 7\). Here, we subtract 7 from the function \(f(x)\). This indicates a vertical transformation.

Step 2: Determine the Type of Vertical Shift

Subtracting a constant from the function causes a vertical shift. Specifically, \(y = f(x) - 7\) represents a downward vertical shift of the graph of the function \(f(x)\) by 7 units.

Step 3: Describe the Graph Transformation

To describe the graph transformation, the entire graph of \(f(x)\) is moved 7 units downward on the y-axis. This means each point on the graph of \(f(x)\) will have its y-coordinate reduced by 7.

Key Concepts

Vertical Shift

A vertical shift refers to moving the entire graph of a function up or down along the y-axis. This shift occurs when a constant is added to or subtracted from the function's formula. In our given problem, we have the function transformation expressed as \(y = f(x) - 7\). Here, the constant \(-7\) indicates that the graph is shifted vertically.

One crucial thing to understand about vertical shifts is:

• If the constant is positive, like \(y = f(x) + c\), the shift is upward.
• If the constant is negative, as in \(y = f(x) - c\), the shift is downward.

In the context of the exercise, subtracting 7 from the function means every point on the graph of the original function \(f(x)\) moves down. The y-coordinate of each point decreases by 7 units.

This simple manipulation helps in adjusting the position of a graph without altering its shape or slope. Vertical shifts are solely concerned with translating the graph in a vertical direction.

Graph Transformation

Graph transformation refers to any alteration applied to the graph of a function, changing its position or shape. It includes not only vertical shifts but also horizontal shifts, stretches, compressions, and reflections. In our exercise, the focus is on a vertical shift characterized by \(y = f(x) - 7\).

Understanding graph transformations is essential for analyzing and predicting the behavior of functions. The main types of graph transformations include:

• Vertical shifts, which we discussed earlier.
• Horizontal shifts, moving graphs left or right.
• Vertical and horizontal stretches or compressions, changing the size of a graph.
• Reflections, flipping graphs over an axis.

In practice, identifying and describing these transformations allows one to easily visualize how a function's graph has been altered compared to its original form. In our specific problem, the transformation is simply moving the graph 7 units down without any impact on its overall shape.

Downward Shift

A downward shift is a specific type of vertical shift where a function's graph moves down along the y-axis. For the equation \(y = f(x) - 7\), the subtraction of 7 is responsible for this downward movement.

Key characteristics of a downward shift include:

• Each point on the graph's y-coordinate reduces by the same amount, which here is 7 units.
• The overall shape of the function remains unchanged; only its position is different.

Understanding downward shifts is helpful for interpreting how real-world data and functional models change position relative to their original state.

By mastering this concept, you can visualize the movement of graphs without needing to plot them manually. This knowledge is essential not only for mathematical problem-solving but also for applications such as engineering, physics, and economics, where function modeling and interpretation are critical.