Question

a. Describe how you would construct the graph of \(f(|x|)\) from the graph of \(y=f(x)\). b. Use the result of part (a) to sketch the graph of \(y=\sin |x|\).

Short answer

To construct the graph of \(f(|x|)\) from the graph of \(y = f(x)\): 1. Keep any portion of the graph of \(y = f(x)\) where \(x \geq 0\) unchanged. 2. Reflect the portion of the graph of \(y = f(x)\) with negative x-values about the y-axis. To sketch \(y = \sin |x|\), draw the positive half of the sine function (\(y = \sin(x)\) for \(x \geq 0\)) and reflect it about the y-axis, resulting in a symmetric S-shape graph oscillating between -1 and 1.

Step-by-step solution

Understand the effect of absolute value

The absolute value function, denoted by \(|x|\), returns the non-negative value of the input variable. Specifically, \(|x|\) equals \(x\) if \(x \geq 0\) and \(-x\) if \(x < 0\). Thus, taking the absolute value of the input variable in a function has the effect of reflecting the negative portion of the input domain about the y-axis. This reflection creates a symmetric graph about the y-axis.

Identify the steps to transform the graph

To construct the graph of \(f(|x|)\) from the graph of \(y = f(x)\), follow these steps: 1. Keep any portion of the graph of \(y = f(x)\) where \(x \geq 0\) unchanged. 2. Reflect the portion of the graph of \(y = f(x)\) with negative x-values about the y-axis. These steps will result in the graph of \(f(|x|)\), which is symmetric about the y-axis. b. Sketching the graph of \(y = \sin |x|\)

Draw the graph of \(y = \sin (x)\) for \(x \geq 0\)

For \(x \geq 0\), the graph of \(y = \sin (x)\) starts at the origin and resembles an S-shape. It oscillates between -1 and 1, crossing the x-axis at \(x = n\pi\), where \(n\) is an integer. For this step, simply draw the "positive" half of the sine function.

Reflect the graph about the y-axis

Apply the steps from part (a) to create the graph of \(y = \sin |x|\). Reflect the part of the graph drawn in Step 1 about the y-axis, creating a "mirror image." The resulting graph will also resemble an S-shape and will continue oscillating between -1 and 1 in the negative x-domain. The resulting symmetric graph represents the function \(y = \sin |x|\).

Key concepts

Transformation of Functions

When graphing functions, it's important to grasp the concept of transformation. A transformation adjusts a function's graph with respect to its position, size, and orientation. There are a few fundamental types of transformations:

• Translation: Moves the graph horizontally or vertically.
• Reflection: Flips the graph over a particular line, usually an axis.
• Stretch or Compression: Alters the graph's steepness or width.

The transformation relevant to our exercise is reflection. To transform the graph of a function involving absolute value, such as from \( y = f(x) \) to \( y = f(|x|) \), we reflect every point on the negative x-axis over the y-axis. This results in a new graph that exhibits symmetry, with the positive side serving as a mirror image to the negative side.

Absolute Value

The absolute value of a real number x, denoted as \( |x| \), represents its distance from zero on the number line, regardless of direction. This means that positive values remain as they are, while negative values are converted to positive. Mathematically put, if \( x \geq 0 \), then \( |x| = x \), but if \( x < 0 \), then \( |x| = -x \).

In terms of graphing \( y = f(|x|) \), incorporating absolute value creates a function that is insensitive to whether x is positive or negative; it always yields the output of \( f \) for the non-negative value of x. As a result, we observe that the graph of \( y = f(|x|) \) will be symmetric with respect to the y-axis since both \( x \) and \( -x \) result in the same output.

Symmetry in Graphs

Symmetry in graphs makes for simpler and more intuitive understanding of functions. If a graph is symmetric about the y-axis, then for every point \( (x, y) \) on the graph, the point \( (-x, y) \) is also on the graph. This type of symmetry indicates that a function’s value at a positive input is equal to its value at the corresponding negative input.

For an absolute value transformation, symmetry becomes a cornerstone of graph construction. A symmetric graph around the y-axis will always reflect this rule: if you have a point on the right-side of the graph, you'll find its reflected twin on the left. This drastically simplifies drawing the function \( f(|x|) \), as you only need the 'right half' of the graph to visualize the whole function.

Sine Function

The sine function is an important and ubiquitous concept in trigonometry, defined as \( y = \text{{sin}}(x) \) for \( x \) in radians. Its graph is characterized by periodic waves oscillating above and below the x-axis, with amplitude between -1 and 1 and a period of \( 2\text{π} \).

When the sine function is subjected to the absolute value transformation—resulting in \( y = \text{{sin}}(|x|) \)—the graph becomes a wave that folds onto itself in a complete symmetry around the y-axis. The usual 'up and down' waves in the negative domain are now mirrored to look like those in the positive domain. This creates a graph that is not only symmetrical but also continuous and repeatable, which means it can be easily replicated onward in both the positive and negative direction.