Problem 2 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=|x|-2 $$

Step-by-Step Solution

Answer

The graph is a V-shape with vertex at (0, -2).

Step 1: Understand the Absolute Value Function

The function given is a transformation of the absolute value function, which is commonly represented as \(f(x) = |x|\). The absolute value function creates a V-shaped graph with its vertex at the origin (0,0).

Step 2: Identify the Transformation

The function given is \(f(x) = |x| - 2\). This represents a vertical shift of the absolute value function downwards by 2 units. Every point on the graph \(y = |x|\) will be lowered by 2 units.

Step 3: Find Critical Points

The vertex of the transformed graph will be at (0, -2) instead of (0, 0). To find other critical points, calculate \(f(x)\) for a few values of \(x\). For example, \(f(-2) = 0\), \(f(0) = -2\), and \(f(2) = 0\). These points help us sketch the graph accurately.

Step 4: Sketch the Graph

Plot the vertex at (0, -2). Then, plot the points identified in the previous step: (-2, 0) and (2, 0). Draw lines from these points to the vertex, forming a V-shape. The lines should be straight, as the absolute value function is linear on either side of the vertex.

Key Concepts

Absolute Value Functions

Absolute value functions are unique due to their characteristic V-shape. They originate from the basic function form of \( f(x) = |x| \), defining how the graph behaves. The absolute value, \(|x|\), takes any real number \(x\), and outputs its distance from zero on a number line. This means it's always non-negative, producing values of zero or positive numbers only. In a graph, \( f(x) = |x| \), results in a V-shape with its lowest point, called the vertex, at the origin (0,0).
This vertex represents the minimum point of the function because the absolute value is zero when \( x = 0 \) and increases symmetrically as \( x \) moves away from zero either in a positive or negative direction.
The key properties of absolute value functions include:

• Symmetry about the y-axis
• Vertex as a minimum value
• Linear increase on both sides of the vertex

These basic characteristics form the building blocks for transformations, such as shifts and stretches, that alter the graph's position or orientation.

Vertical Transformations

Vertical transformations modify the position of the entire graph in relation to the y-axis. In the function \( f(x) = |x| - 2 \), the "-2" signifies a downward vertical shift of the graph. It effectively lowers the entire graph of \( |x| \) by 2 units.
No horizontal position changes occur, but every output value \(y\) of the function \( |x| \) decreases by 2. This action drops the vertex from (0,0) to (0,-2).
Understanding vertical transformations allows one to predict and adapt a graph's layout:

• Note whether the transformation is a horizontal or vertical shift
• Identify the new vertex according to the transformation
• Maintain the same general shape, but in a different position

A vertical shift does not alter the graph's symmetry or linear rise on either side of the vertex, just its location. This shift mainly affects how the function translates in real-world applications, such as adjusting for different starting points or bases in calculations.

Critical Points in Graphs

Critical points are essential for understanding the behavior and shape of a graph. For absolute value functions, these include the vertex and additional plotted points that help outline the graph's structure. In our given function, \( f(x) = |x| - 2 \), the vertex becomes (0,-2), down from the original (0,0) due to the vertical shift.
Other critical points provide additional anchors when sketching the graph, such as (-2,0) and (2,0). Calculating these involves substituting values of \(x\) into the function to determine corresponding values of \(y\). The points give the graph its V-shape and ensure accurate representation.
To identify these points:

• Start with the vertex point
• Compute a few other points by using various \(x\) values
• Mark these on a graph to form the graph's complete shape

Strong understanding of critical points helps in drawing precise graphs and can assist in predicting how changes affect graph shapes in practical applications.