Problem 46 - A Graphical Approach to Precalculus with Limits

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 is a V-shape opening upwards with its vertex at (-2,0).

Step 1: Identify the Basic Function

The function given is \( y = |x+2| \). The basic function here is the absolute value function \( y=|x| \), which is a V-shaped graph that opens upwards and has a vertex at the origin (0,0).

Step 2: Identify the Translation Operation

The function \( y = |x+2| \) involves a horizontal translation of the basic function \( y = |x| \). The expression \( x+2 \) indicates a horizontal shift to the left by 2 units. This means that every point on the graph of \( y = |x| \) will move 2 units left.

Step 3: Sketch the Translated Graph

Start by sketching the V-shaped graph of the basic function \( y = |x| \) with its vertex at (0,0). Then, move the entire graph 2 units to the left. This places the new vertex at (-2,0).

Step 4: Verify Key Points

For further accuracy, verify a few key points. For example, the point (1,1) on the graph of \( y=|x| \) will move to (-1,1) on the graph of \( y=|x+2| \). Similarly, the point (-1,1) will move to (-3,1). These shifts ensure the translation is correctly applied.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in graphical transformations. It is denoted as \( y = |x| \) and is recognized by its distinct V-shaped graph that opens upwards. The absolute value function essentially measures the distance of any point from zero on a number line, always yielding non-negative results. This V-shape originates from the unique property where both negative and positive inputs result in positive outputs.

• Vertex: The point where the two lines of the "V" meet is called the vertex, positioned at the point (0,0) for the basic function.
• Symmetry: This graph is symmetrical concerning the y-axis, reflecting the inherent properties of absolute value in having identical outputs for positive and negative inputs of equal magnitude.
• Piecewise Definition: Algebraically, the absolute value function can be expressed as a piecewise function: \( |x| = x \) if \( x \geq 0 \), and \( |x| = -x \) if \( x < 0 \).

This function serves as a foundational tool for students learning about transformations due to its simplicity and visually intuitive nature.

Horizontal Translation

A horizontal translation involves shifting a graph left or right along the x-axis without altering its shape or orientation, which is crucial in understanding graphical function transformations. The equation \( y = |x+2| \) illustrates a horizontal translation.

• Formula Impact: In this function, replacing \( x \) with \( x+2 \) shifts the graph of \( y=|x| \) two units to the left. The general rule is that \( y = |x+a| \) moves the graph left by "a" units if "a" is positive and right if "a" is negative.
• Vertex Change: For \( y=|x| \), the vertex is at (0,0). After a horizontal shift of 2 units left, it becomes (-2,0).

Horizontal translations provide a straightforward way to alter the position of graphs, making them essential for creating functions that fit specific scenarios or meet certain constraints.

Sketching Graphs

Sketching graphs by hand is a skill that involves transforming a basic graph step-by-step to visualize how changes in the equation affect its shape and position. To sketch \( y = |x+2| \), begin with the basic graph of \( y = |x| \).

• Step-by-step Approach: First, draw the V-shaped graph of the basic \( y=|x| \) function. Identify its vertex at (0,0).
• Applying Translation: Shift the entire graph 2 units to the left—this means moving each point on the graph, including the vertex, which becomes (-2,0).
• Verify Points: Check a few key points to ensure accuracy. For example, the point (1,1) on \( y=|x| \) moves to (-1,1) on \( y=|x+2| \).

This methodical approach ensures that you understand the freedom translations provide. It shows visually how the function has changed through its graph.