Problem 25 - Calculus with Concepts in Calculus

Question

Sketch the graph. List the intercepts and describe the symmetry (if any) of the graph. $$ |y|=|3 x| $$

Step-by-Step Solution

Answer

Intercept is at (0, 0). The graph has symmetry about the x-axis and y-axis.

Step 1: Understand the Absolute Value Equation

The equation given is \(|y| = |3x|\). This means the absolute values of both \(y\) and \(3x\) are identical, implying they must be either both positive or both negative, or both zero.

Step 2: Break Down the Equation

The equation \(|y| = |3x|\) can be broken down into two separate cases based on the properties of absolute values: 1. \(y = 3x\) 2. \(y = -3x\).

Step 3: Sketch the Graph

Using the two equations \(y = 3x\) and \(y = -3x\), plot both lines. These represent two lines in the coordinate plane: one line rising with a slope of 3 and another one falling with a slope of -3. Both lines pass through the origin (0,0).

Step 4: Determine the Intercepts

Since both lines pass through the origin, the intercepts in this case are simply at the origin: - **x-intercept:** (0, 0) - **y-intercept:** (0, 0).

Step 5: Analyze the Symmetry

The given graph is symmetric about both the x-axis and the y-axis. The origins and equal slopes in opposite directions imply reflective symmetry over these axes.

Key Concepts

Graph Symmetry

The concept of graph symmetry is crucial when analyzing the equation \(|y| = |3x|\). Symmetry refers to how one part of a graph mirrors another. In this equation, the graph exhibits symmetry about both the x-axis and the y-axis. This is because the equation represents two lines: \(y = 3x\) and \(y = -3x\). Each line mirrors the other across the x-axis and the y-axis. Thus, the graph is perfectly symmetrical.

• X-axis Symmetry: Reflects the line \(y = 3x\) into \(y = -3x\).
• Y-axis Symmetry: Although less intuitive from the line equations, this symmetry arises from equal slopes going in opposite directions, combined with the absolute value property ensuring mirrored form on both side of axes.

Understanding symmetry can make it easier to sketch graphs and predict how they behave across different quadrants.

Intercepts

Intercepts are key feature points where the graph intersects the coordinate axes. In the equation \(|y| = |3x|\), both lines, \(y = 3x\) and \(y = -3x\), intersect exactly at the origin (0, 0). Here, the intercepts are simple:

• X-intercept: (0, 0)
• Y-intercept: (0, 0)

Intercepts are both crucial and basic parts of graph sketching. They help provide anchor points around which you can better understand the shape and position of the graph. With the origin being the only intercept for both lines here, the graph remains centered around this single pivotal point.

Slope Analysis

Analyzing the slope gives insight into the direction and steepness of a line. In the absolute value equation \(|y| = |3x|\), the slopes of the resulting lines are set by the forms \(y = 3x\) and \(y = -3x\).

• Line \(y = 3x\): This line has a positive slope of 3, indicating it rises three units upward for every one unit moved to the right.
• Line \(y = -3x\): This line has a negative slope of -3, meaning it falls three units downward for every one unit moved to the right.

Recognizing slopes can help in sketching and interpreting graphs, as they determine how quickly the lines ascend or descend. For this equation, the steepness of the lines and their symmetrical yet opposite rise and fall give rise to the absolute value's visual representation.