Question

Write the function that is finally graphed if the following transformations are applied in order to the graph of \(y=|x|\) \(\begin{array}{ll}\text { 1. Shift left } 3 \text { units. } & \text { 2. Reflect about the } x \text { -axis. }\end{array}\) 3. Shift down 4 units.

Short answer

\[ y = -|x + 3| - 4 \]

Step-by-step solution

Identify the base function

The base function given is: \[ y = |x| \]

Shift left 3 units

To shift the function left by 3 units, replace \[ x \text{ with } (x + 3) \] in the equation. The function becomes: \[ y = |x + 3| \]

Reflect about the x-axis

To reflect the function about the x-axis, multiply the entire function by -1: \[ y = -|x + 3| \]

Shift down 4 units

To shift the function down by 4 units, subtract 4 from the entire function: \[ y = -|x + 3| - 4 \]

Key concepts

Absolute Value Function

The absolute value function is a fundamental mathematical concept. It's written as \( y = |x| \). This function produces a V-shaped graph with a vertex at the origin (0,0). This graph is symmetric about the y-axis because the absolute value of both positive and negative inputs is positive.
For example:

• For \( x = 2 \), \( y = |2| = 2 \)
• For \( x = -2 \, \ y = |-2| = 2 \)

This means the graph rises to the right and left from the vertex. Understanding this basic function will help in visualizing transformations.

Horizontal Shift

Horizontal shifts move the graph left or right along the x-axis. When we shift a function to the left or right, we adjust the input variable x.
If we want to shift the graph of \( y = |x| \) to the left by 3 units, we replace \( x \) with \( (x + 3) \). Thus, the equation becomes:
\[ y = |x + 3| \]
This transformation moves the entire graph three units to the left. Each point on the graph will shift without altering the shape of the graph.

Reflection

Reflection transmits a mirror image of the graph across an axis.
Reflecting a graph across the x-axis changes the sign of the output. For example, reflecting \( y = |x + 3| \) about the x-axis results in:
\[ y = -|x + 3| \]
The reflection of the absolute value function flips the V-shape upside down. Therefore, all y-values become negative, transforming the graph to open downwards.

Vertical Shift

A vertical shift moves the graph up or down along the y-axis.
To shift a function down by 4 units, we subtract 4 from the entire equation. Given our reflected function \( y = -|x + 3|\r\), shifting it down by 4 units produces:
\[ y = -|x + 3| - 4 \]
This shift decreases all output values by 4, moving the entire graph downward. Each point on the graph descends while maintaining the shape of the graph.

Combining these transformations sequentially yields the final equation:
\[ y = -|x + 3| - 4 \]