Question

Describe the sequence of transformations from \(f(x)=|x|\) to \(g .\) Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=|x-2|+2\)

Short answer

The sequence of transformations from \(f(x)=|x|\) to \(g(x)=|x-2|+2\) involves a 2 units shift to the right and a 2 units shift upwards. The graph of \(g(x)=|x-2|+2\) follows a similar V-shape pattern as \(f(x)=|x|\) but is located at (2,2).

Step-by-step solution

Identify the Transformations

Transformations of the given functions can be written in the general form \(g(x)=a| x-h | +v\), where 'h' represents horizontal shift, 'v' represents vertical shift, 'a' represents stretching or shrinking factor, and 'a<0' represents reflection in x-axis. In \(g(x)=|x-2|+2\), the value of 'h' is 2, which indicates that the graph is shifted to the right by 2 units. The value of 'v' is 2, indicating the graph is shifted upwards by 2 units. The values of 'a' and 'r' are not given, indicating there is no stretching/shrinking or reflection.

Sketch the Parent Function \(f(x)=|x|\)

The absolute function \(f(x)=|x|\) is a V-shaped graph with the vertex at the origin (0,0). The graph increases linearly from the vertex along the y-axis towards the positive x and y directions.

Apply the Transformations to Sketch \(g(x)=|x-2|+2\)

First apply the horizontal shift. Move every point on \(f(x)\) to the right by 2 units to get \(|x-2|\). Then apply the vertical shift. Move every point on \(|x-2|\) upwards by 2 units to get \(|x-2|+2\). The vertex of \(g(x)\) is now at (2,2). The graph of \(g(x)\) maintains the V-shape of \(f(x)\) but is located at the new vertex (2,2).

Verify with a Graphing Utility

You can verify the sketch by using a graphing utility. The plot of \(g(x)=|x-2|+2\) should match the sketch made with the above transformations. The vertex of the graph should be at the point (2,2).

Key concepts

Absolute Value Function

The absolute value function, represented as \(f(x) = |x|\), is a fundamental mathematical concept. It is unique due to its distinctive "V"-shaped graph. This graph has its vertex point at the origin, (0,0), where the graph changes direction.
The graph of \(f(x) = |x|\) consists of two linear pieces. For \(x\) values greater than or equal to zero, the function is simply \(x\), forming a line with a positive slope. For \(x\) values less than zero, the function is \(-x\), creating a line with a negative slope.

The absolute value function's shape and vertex make it a popular starting point for practicing transformations. Understanding this function and its graphing properties will provide a foundation for transforming and sketching more complex functions, such as \(g(x) = |x-2| + 2\).

Horizontal Shift

A horizontal shift in a function involves moving its graph left or right on the coordinate plane. For the function \(g(x) = |x-h|\), this transformation is characterized by the parameter \(h\).
If \(h\) is positive, as in \(g(x) = |x-2|\), the graph shifts to the right by \(h\) units, meaning every point on the original graph of \(f(x) = |x|\) is moved 2 units right. Conversely, if \(h\) were negative, the graph would shift left by the opposite amount.

In the transformation from the parent function \(f(x) = |x|\) to \(g(x) = |x-2|\), the horizontal shift results in the vertex moving from (0,0) to (2,0). Linearly increasing and decreasing slopes remain unchanged, preserving the shape of the graph while altering its position on the x-axis.

Vertical Shift

A vertical shift involves moving the graph of a function up or down the y-axis. This shift is controlled by an added or subtracted constant at the end of the function, noted as \(v\) in the expression \(g(x) = |x-h| + v\).
In our example, \(g(x) = |x-2| + 2\), the value of \(v\) is 2. This results in the entire graph moving upwards by 2 units.

By applying this shift to \(g(x) = |x-2|\), the new vertex is relocated from (2,0) to (2,2) on the coordinate plane. While the direction and slope of the graph remain the same, the vertical shift effectively raises all points on the graph, keeping its "V" shape intact.

Graph Sketching

Graph sketching is a critical skill in understanding the transformation of functions. It involves visually representing the function on a graph, which aids in grasping how transformations change the parent function's behavior and position.
To sketch \(g(x) = |x-2| + 2\), start by sketching the basic "V"-shaped absolute value graph \(f(x) = |x|\). Then apply the horizontal shift by moving it 2 units to the right, followed by a vertical shift by moving it 2 units up.

Steps for Sketching

• Draw the parent function \(f(x) = |x|\).
• Apply the horizontal shift to move to \(g(x) = |x-2|\).
• Apply the vertical shift to achieve \(g(x) = |x-2| + 2\).
• Ensure the vertex is at (2,2).

Using these steps, one can clearly visualize and confirm the resulting graph is accurate. Always compare with graphing utilities for precision and correctness.