Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Use symmetry to complete the graph of the equation. \(y\) -axis symmetry $$y=|x|-2$$

Step-by-Step Solution

Answer

To graph \(y=|x|-2\), graph the function \(y=|x|\), then shift each point down by two units. Because the |x| function is symmetrical about the y-axis, mirror the points to the other side of the y-axis to complete the graph.

Step 1: Understanding the parent function

The function \(y=|x|\) is the parent function or the base of our given function. It is necessary to understand its graph. It forms a 'V' shaped graph which is symmetric about y-axis. It starts from the origin (0,0). When x is greater than or equal to 0, the value of y is equal to x. When x is less than 0, the value of y is equal to -x. Thus, for all values of x, y is always positive.

Step 2: Applying the transformation

The given function is \(y=|x|-2\). This is a vertical shift downwards by 2 from the function y=|x|. Let's take the graph of y=|x| and move each point down by 2 units. This is how we apply vertical transformations.

Step 3: Completing the graph using symmetry

Since the graph of \(y=|x|\) is symmetric about the y-axis, the graph of \(y=|x|-2\) should be symmetric too. This means that if we have a point (x, y) on the graph, the point (-x, y) will also be on the graph. We use this symmetry to complete the other half of the graph.

Key Concepts

Graph Transformations

Graph transformations refer to changes made to the graph of a function. These changes can include translations, reflections, stretches, and compressions. Understanding graph transformations helps in predicting how a graph changes when certain modifications are applied.

For example, consider the function \(y = |x|\). This is known as the parent function for our given problem. When we apply transformations, we are essentially modifying this base graph. Knowing how these modifications work is key to sketching the altered graph correctly.

• Translation: Shifts the graph horizontally or vertically.
• Reflection: Flips the graph across an axis.
• Stretch/Compression: Alters the shape of the graph by scaling it.

In the context of the given exercise, applying a vertical shift translates the graph up or down without altering its horizontal symmetry.

Vertical Shifts

Vertical shifts are one of the simplest types of graph transformations. They involve moving the entire graph up or down by a certain number of units. The key feature to remember is that vertical shifts only affect the \(y\)-coordinates, leaving the \(x\)-coordinates unchanged.

In the equation \(y = |x| - 2\), the "-2" indicates a vertical shift 2 units downward. This means every point on the original \(y = |x|\) graph is moved down by two units.

• If a positive number is added to \(y = |x|\), such as \(y = |x| + 2\), the graph shifts upward by 2 units.
• A subtraction, like \(y = |x| - 2\), shifts everything downward.

This transformation doesn't alter the symmetry of the graph, which remains symmetric around the y-axis in this case.

Absolute Value Functions

Absolute value functions create a distinctive 'V' shaped graph. This is due to the absolute value signs, which turn negative inputs into positive outputs. As a result, the graph is non-negative and typically symmetric about the y-axis.

The parent function \(y = |x|\) begins at the origin \((0,0)\) and has two linear pieces. The left side corresponds to \(y = -x\) and the right side corresponds to \(y = x\).

• The absolute value causes both halves of the graph to meet at the origin, forming a point.
• For any point \((x, y)\), there is always a corresponding point \((-x, y)\) due to symmetry.

In the modified function \(y = |x| - 2\), this 'V' shape is simply shifted downwards. The symmetry about the y-axis is not affected. Even after the vertical shift, the graph remains symmetric as its shape is altered uniformly.