Problem 48 - 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. $$y=|x|-3$$

Step-by-Step Solution

Answer

The graph of \( y = |x| - 3 \) is a downward-shifted V-shaped graph with vertex at (0, -3).

Step 1: Identify the Base Function

The given function is \( y = |x| - 3 \). Identify that the base function is \( y = |x| \), which is the absolute value function.

Step 2: Understand the Base Graph

Recall that the graph of \( y = |x| \) is a V-shaped graph with the vertex at the origin (0, 0). It opens upwards with two linear pieces.

Step 3: Analyze the Transformation

The transformation being applied to the base function is \( -3 \), which indicates a downward vertical shift by 3 units. This transformation moves the graph of \( y = |x| \) vertically downwards.

Step 4: Sketch the Transformed Graph

To sketch \( y = |x| - 3 \), start by drawing the graph of \( y = |x| \) with a vertex at (0, 0). Then, shift every point on this graph down by 3 units to get the new vertex at (0, -3). The 'V' shape remains the same.

Step 5: Verify the New Vertex

Check the new position of the vertex. Originally at (0, 0), it has moved to (0, -3) due to the transformation \( -3 \). Ensure each corresponding point from the base graph is translated downwards by 3 units.

Key Concepts

Absolute Value Function

The absolute value function is represented as \( y = |x| \). This function creates what is known as a V-shaped graph. The basic property of the absolute value function is that it outputs the magnitude of the input number, which is always non-negative. This idea can be related to thinking of how far a number is from zero on the number line, without considering the direction

For example, both \( |3| \) and \( |-3| \) equal 3, because 3 is three units away from zero, whether to the right or the left. In terms of graphing, you may imagine two rays starting from a common point (the vertex) and extending upward—one along the positive xx-axis and another along the negative xx-axis.

• The vertex of the graph is typically at the origin (0,0).
• The arms of the V-shaped graph open upwards.

Understanding the absolute value function is foundational for performing graph transformations like shifts and reflections.

Vertical Shifts

Vertical shifts occur when we add or subtract a constant value from a function. For our function \( y = |x| - 3 \), the constant -3 indicates a vertical shift.

Types of Vertical Shifts:

• Upward Shift: Adding a constant will shift the graph of a basic function upwards. For example, \( y = |x| + 2 \) means every point on the \( y = |x| \) graph moves up by 2 units.
• Downward Shift: Subtracting a constant moves the graph down. In this exercise, \( y = |x| - 3 \) shows that every point of the translatable feature is shifted down by 3 units.

Vertical shifts do not alter the shape of the graph. They adjust only its position on the y-axis. In the case of \( y = |x|-3 \), the entire V-shaped graph of \( y = |x| \) descends while maintaining its original structure.

Graph Sketching

Sketching a graph involves moving from understanding the function and transformation to visual representation. Begin with the base function—in this instance, \( y = |x| \)—and understand its typical shape and location. For an absolute value function, this is a V-shape with the vertex at the origin (0, 0).

To sketch transformations:

• Identify the basic shape of the function. For \( y = |x| \), it's a V-shape.
• Determine any transformations needed based on added constants. For \( y=|x|-3\), it's a downward shift of 3 units.
• Translate the entire shape according to these transformations. This means moving the vertex from (0, 0) to (0, -3).

While sketching, it's helpful to verify the new vertex position and plot a few key points to ensure accuracy. The symmetry and shape remain constant, helping to maintain consistency across such visual transformations.