Question

Consider the graph of \(f(x)=|x|\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is shifted three units to the right and two units upward.

Short answer

The equation of the function after shifting the graph of \(f(x)=|x|\) three units to the right and two units upward is \(f(x)=|x-3|+2\).

Step-by-step solution

Identify the original function

Starting with the given function \(f(x)=|x|\).

Apply the horizontal shift

A shift three units to the right can be represented by replacing \(x\) with \(x-3\) in the original function. So, the function becomes \(f(x)=|x-3|\).

Apply the vertical shift

A shift two units upward is done by simply adding 2 to the function that had been shifted horizontally. It gives the final function as \(f(x)=|x-3|+2\).

Verify with a graphing utility

The transformation of function graphs should now be verified using a graphing utility. The original graph \(y = |x|\) and the transformed graph \( y = |x-3|+2\) should be plotted on the same graph for visual comparison and verification.

Key concepts

Understanding Absolute Value Functions

The absolute value function is a fundamental concept in mathematics.It is usually represented as \( f(x) = |x| \).The absolute value of a number is its distance from zero on the number line, making it always non-negative.
For example:

• \( |3| = 3 \)
• \( |-3| = 3 \)

As a graph, the absolute value function produces a "V" shape whose vertex is at the origin, (0,0).
The graph opens upwards and is symmetric about the y-axis.
Understanding the basic form of this function helps us to apply transformations such as shifts.
These transformations allow us to move the graph horizontally or vertically without affecting its shape.

Horizontal Shift of Graphs

A horizontal shift involves moving the graph of a function to the left or right. For a function such as \( f(x) = |x| \), replacing \( x \) with \( x - c \) results in a shift to the right by \( c \) units, while \( x + c \) results in a left shift by \( c \) units.
In the exercise, shifting the graph of \( f(x) = |x| \) three units to the right means replacing \( x \) with \( x - 3 \).
Thus, the modified function is \( f(x) = |x - 3| \).
This new graph has its vertex at (3,0), instead of (0,0).
By understanding this shift, you can move any absolute value graph horizontally while maintaining its distinctive V-shape.

Vertical Shift of Graphs

A vertical shift modifies the position of a graph by moving it up or down. To shift a graph such as \( f(x) = |x| \), we add or subtract a constant from the function.For the given exercise, after horizontally shifting the function to \( f(x) = |x-3| \), we then shift it vertically.
By adding 2 to the function, \( f(x) = |x-3| + 2 \) is achieved, resulting in the graph moving upward by two units.
This causes the vertex of the graph, post-horizontal shift, to rise from (3,0) to (3,2).
Vertical shifts are simple yet powerful as they maintain the overall shape and horizontal position of the graph.

Graph Verification Using a Graphing Utility

Verification is a crucial step to ensure graph transformations are correct.Graphing utilities allow for visual confirmation by plotting functions and observing changes.In the exercise, you start with plotting the original function \( y = |x| \) which has its vertex at the origin.
Then, plot the transformed function \( y = |x-3| + 2 \), noting it should shift 3 units right and 2 units up.
This results in the new vertex being at (3,2).
By comparing both graphs, the transformation can be visually confirmed if the expected shifts align as described.
This step, often facilitated by graphing calculators or computer software, reinforces understanding by providing a concrete representation of the abstract transformation concepts.