Problem 19 - College Algebra

Question

For the following exercises, graph the given functions by hand. $$ y=|x|-2 $$

Step-by-Step Solution

Answer

Graph the V-shaped function \(y=|x|-2\) with vertex at (0,-2).

Step 1: Understand the Basic Graph

The basic graph we will draw here is for the function \(y = |x|\). This is a V-shaped graph that has a vertex or a point at the origin (0,0). It opens upwards with two parts: the right part is the line \(y = x\) and the left part is the line \(y = -x\).

Step 2: Apply the Transformation

The function given is \(y = |x| - 2\). This is the absolute value function \(y = |x|\) shifted downward by 2 units. So, take the entire graph of \(y = |x|\) and move every point on the graph 2 units down.

Step 3: Plot Key Points

Begin by plotting the vertex of the transformed graph at (0, -2), since it was moved down 2 units. Then plot additional points. For example, for x = 1, y = 1-2 = -1; for x = -1, y = 1-2 = -1. This will keep the same V shape but start at (0, -2).

Step 4: Draw the Graph

With these points plotted, draw lines from the vertex (0, -2) rising rightwards through (1, -1) and leftwards through (-1, -1). These two lines will continue as y = x - 2 for x ≥ 0, and y = -x - 2 for x < 0.

Step 5: Verify

Finally, make sure the graph maintains its V shape and labels the vertex at (0, -2). Check the key points to ensure accuracy, from both the left and right arms of the V intersecting properly through these points.

Key Concepts

Transformations of Functions

Transformations of functions are crucial for adapting a basic graph into a different shape or position on the coordinate plane. Types of transformations include:

• Vertical shifts: Moving the graph up or down parallel to the y-axis.
• Horizontal shifts: Moving the graph left or right parallel to the x-axis.
• Reflections: Flipping the graph over a particular axis.
• Stretching or compressing: Changing the graph's shape by stretching or compressing it along the x or y-axis.

For instance, the function transformation from \( y = |x| \) to \( y = |x| - 2 \) involves a vertical shift downward by 2 units. This means every point on the original graph moves two units down, but maintains its distinct V-shape.

V-shaped Graphs

V-shaped graphs are characteristic of absolute value functions like \( y = |x| \). The "V" is formed because absolute value functions reflect both positive and negative entries into positive outputs.

• Vertex: The point at the tip of the V, serving as a pivot point. For \( y = |x| \), this vertex is at the origin (0, 0).
• Arms of the V: These are linear components that extend from the vertex. One arm represents \( y = x \) and the other \( y = -x \), both extending indefinitely.

In transformations, the vertex may shift along the coordinate plane, resulting in our transformed example at the vertex (0, -2), while the V-shape persists.

Graphing Techniques

Graphing techniques help visualize functions reliably and accurately. Begin with plotting key points to provide a framework for your graph:

• Identify the vertex: Start with the vertex; for the transformed graph \( y = |x| - 2 \), this is at (0, -2).
• Plot additional points: Use simple values, like \( x = 1 \) or \( x = -1 \), to determine additional coordinates such as (1, -1) and (-1, -1).
• Connect the points: Draw lines that connect these coordinates, extending the arms of the V.

Check the graph for symmetry, particularly important for absolute value functions, ensuring both sides are mirror images relative to the vertex. This method provides consistency in maintaining the proper shape of the function.

Algebraic Functions

Algebraic functions, like absolute value functions, combine constants, variables, and operations. The function \( y = |x| - 2 \) uses the absolute value operation, which alters how regular functions behave.

• Absolute value operation: Converts all negative inputs to positive, resulting in the distinctive V-shape.
• Linear components: Each arm of the V graphed corresponds to linear equations like \( y = x - 2 \) for \( x \geq 0 \) and \( y = -x - 2 \) for \( x < 0 \).
• Variable manipulation: By shifting the function down, we alter the constant term, highlighting these graphing transformations.

Understanding the underlying algebra helps in manipulating functions to achieve the desired graph form, as demonstrated in the transformed absolute value function.