Problem 46 - A Graphical Approach to College Algebra

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=|x+2|$$

Step-by-Step Solution

Answer

The graph of \( y = |x+2| \) is a 'V' shape with its vertex at (-2,0), shifted left from the origin.

Step 1: Identify the Base Function

The given function is \( y = |x+2| \). Recognize that the base function here is the absolute value function \( y = |x| \). This is one of the basic functions which we will transform in the following steps.

Step 2: Determine the Transformation

The expression \( x+2 \) inside the absolute value sign is a horizontal transformation. Specifically, \( |x+2| \) represents a horizontal shift to the left by 2 units. Therefore, we will shift the base graph of \( y = |x| \) two units to the left on the x-axis.

Step 3: Sketch the Base Graph

First, sketch the graph of the base function \( y = |x| \). This graph is a 'V' shape with its vertex at the origin (0,0), and it consists of two linear pieces: one increasing line for \( x \geq 0 \) and one decreasing line for \( x < 0 \).

Step 4: Apply the Horizontal Shift

Shift the 'V' shape graph of \( y = |x| \) to the left by 2 units. This means the new vertex of the graph will move from (0,0) to (-2,0). The lines of the 'V' will maintain their slopes with the left arm having a negative slope and the right arm having a positive slope, both originating from the new vertex (-2,0).

Step 5: Finalize the Graph

Finalize the sketch of the graph by ensuring you have the new vertex at (-2,0) and both lines protruding from it, creating a 'V' shape. The left arm of the V continues downward to infinity, and the right arm continues upward to infinity, reflecting the graph of the absolute base function shifted left.

Key Concepts

Absolute Value Function

The absolute value function, denoted by \( y = |x| \), is a fundamental component of mathematics. It transforms an input \( x \) into a non-negative output, effectively capturing the magnitude of a real number regardless of its sign. This function is visually recognizable by its distinct "V"-shaped graph. The arms of this "V" extend infinitely in both upward and downward directions.

• The graph has a vertex at the origin (0, 0), which means it starts from the center of the coordinate plane.
• For values of \( x \) less than zero, the graph decreases towards the left, reflecting the absolute value's effect of turning negative inputs into positive outputs.
• Conversely, for values of \( x \) greater than or equal to zero, it increases towards the right.

In essence, the absolute value function serves as a building block for more complex graph transformations.

Horizontal Shift

A horizontal shift refers to the movement of a graph along the x-axis. This type of transformation modifies the position but not the shape or orientation of a graph. In the equation \( y = |x + 2| \), the term \( x + 2 \) signifies a horizontal shift. Note this is a shift to the left rather than right.

• If the addition inside the absolute value were negative, for instance \( y = |x - 2| \), it would denote a shift to the right.
• Here, \( y = |x + 2| \) requires us to shift the graph of \( y = |x| \) leftward by 2 units.
• This means that every point on the original graph shifts two spaces left on the horizontal axis.

The concept of horizontal shifts is crucial for understanding how a graph can be relocated without altering its inherent structure.

Base Function

The term base function refers to the original, unmodified function that serves as the starting point for transformations. In this context, the base function is \( y = |x| \). This function is chosen because it is straightforward and foundational, making it easy to manipulate with transformations.

• The base function provides a reference point or a "blueprint" for transformations such as stretching, compressing, or shifting graphs.
• For the given function \( y = |x + 2| \), the base function \( y = |x| \) is transformed horizontally.
• By understanding the base function, you can predict and sketch more complex functions by applying different transformation rules.

In essence, the base function is your "starting line" in the race towards understanding graph transformations.

Vertex

The concept of a vertex is integral to functions, particularly the absolute value function. In its simplest form, the vertex of \( y = |x| \) is at the point \((0, 0)\). This is where the two linear segments of the "V" meet and change direction.

• The vertex is the point of minimum or maximum on the graph, depending on the nature of the transformation.
• In the scenario of \( y = |x + 2| \), the entire graph shifts left by 2 units, placing the new vertex at \((-2, 0)\).
• This new vertex is where the graph crosses the x-axis and establishes its foundational point moving forward and backward.

Understanding the vertex helps in quickly identifying the key features and any shifts that have occurred, making it central to mastering graph transformations.