Problem 16 - 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 \(f\). \(y=f(x)-2\)

Step-by-Step Solution

Answer

The graph of the function is a vertical shift of 2 units downward from the original graph of \( f(x) \).

Step 1: Identifying the Transformation

The given function is described as \( y = f(x) - 2 \). This indicates that we are working with a vertical transformation of the base function \( f(x) \). The transformation being described here is a vertical shift.

Step 2: Analyzing the Vertical Shift

In the function \( y = f(x) - 2 \), the '-2' represents a downward shift. Specifically, this indicates that the graph of \( f(x) \) will be shifted 2 units downward. This means each point on the graph of \( f(x) \) will have its y-coordinate decreased by 2.

Step 3: Describing the Transformed Graph

The transformation results in a graph that retains the original shape and orientation of \( f(x) \), only translated vertically. No changes occur to the x-coordinates of the points on the graph of \( f(x) \); only the y-coordinates are affected, being reduced by 2 units across all points.

Key Concepts

Vertical Shift

A vertical shift is a type of transformation that moves a graph up or down along the y-axis. To understand this, let's break down how it works step-by-step.Vertical shifts are determined by adding or subtracting a constant from a function. For example, in the function \( y = f(x) - 2 \), the term "-2" signifies a downward shift of the graph by 2 units.

• When a positive number is added, the entire graph rises. For instance, \( y = f(x) + c \) would move the graph up by \( c \) units.
• Conversely, subtracting a positive number, like \( y = f(x) - c \), shifts the graph down by \( c \) units.

These shifts change the y-coordinate of each point on the graph but keep the x-coordinates intact. It means the overall shape and "look" of the graph remain the same, just moved vertically up or down. This kind of transformation is straightforward because it only affects how high or low the graph is without altering its fundamental structure.

Function Transformation

Function transformation is a broader term that covers several types of changes to the graph of a function. These include shifting, reflecting, stretching, and compressing.

• Shifting: This involves moving the graph either vertically or horizontally without changing its shape.
• Reflecting: Reflecting a graph flips it over a line, such as the x-axis or y-axis.
• Stretching/Compressing: This alters the graph's width or height but maintains the original center point.

The purpose of transforming functions is to explore how these alterations impact the function's graph and to better understand different ways to visualize the function's behavior. By experimenting with these transformations, students can grasp how adjusting various parameters modifies the graph, thus enhancing their comprehension of functions and their properties.

Graph Translation

Graph translation specifically refers to shifting the graph of a function either horizontally or vertically.

• Vertical translation: This translates the graph up or down along the y-axis. It's exactly what occurs in the example \( y = f(x) - 2 \), where the graph is shifted 2 units downward.
• Horizontal translation: This moves the graph left or right along the x-axis. It is represented by \( y = f(x-h) \), where the graph shifts to the right by \( h \) units if \( h \) is positive and to the left if \( h \) is negative.

The concept of graph translation also retains the fundamental shape of the graph. Whenever a graph is translated, its orientation and general form remain unchanged. Only its position in the coordinate plane is altered. By mastering these translations, students gain valuable insight into manipulating and understanding the variety of graphs they may encounter.