Problem 14 - Beginning and Intermediate Algebra

Question

Given the following pairs of functions, explain how the graph of \(g(x)\) can be obtained from the graph of \(f(x)\) using the transformation techniques. $$f(x)=|x|, g(x)=|x|+1$$

Step-by-Step Solution

Answer

The function \(g(x) = |x| + 1\) can be obtained from the function \(f(x) = |x|\) through a vertical translation. Since we are adding 1 to the output of \(f(x)\), the graph of \(g(x)\) will be shifted upwards by 1 unit compared to the graph of \(f(x)\). This upward shift represents the vertical translation, and visually shows how the graph of \(g(x)\) can be obtained from the graph of \(f(x)\).

Step 1: Identify the transformation applied

The transformation we can observe is the addition of a constant, 1, to the absolute value function, \(f(x)=|x|\). This transformation is represented by the function \(g(x)=|x|+1\). Since we are adding a constant to the output of the function without changing the variable itself, it is a vertical transformation.

Step 2: Determine the type of vertical transformation

A vertical transformation can be in the form of a stretch, compression, or translation. Since we're adding a constant (1) and not multiplying the function by a constant, it is a vertical translation, not a stretch or compression. The graph of \(g(x)\) will be a shift of the graph of \(f(x)\) along the vertical or y-axis.

Step 3: Determine the direction and magnitude of the vertical translation

In the function \(g(x) = |x|+1\), we are adding 1 to all the outputs of \(f(x)\). This means that the graph will be shifted upwards by 1 unit, as all the y-values will increase by 1. So, the direction of the vertical translation is upward, and the magnitude is 1 unit.

Step 4: Apply the vertical translation to the graph of \(f(x)\) to obtain the graph of \(g(x)\)

To obtain the graph of \(g(x)\) from the graph of \(f(x)\), we will move every point on the graph of \(f(x)\) upwards by 1 unit along the y-axis. This upward shift will create the graph of the function \(g(x) = |x|+1\) and visually show how the graph of \(g(x)\) can be obtained from the graph of \(f(x)\) using the vertical translation technique.

Key Concepts

Vertical Translation

Vertical translation is a type of graph transformation that involves shifting the entire graph of a function up or down along the y-axis. In other words, the shape of the graph remains unchanged but its vertical position is adjusted.

• When a positive constant is added to a function, the graph is shifted upward.
• If a negative constant is added, the graph is shifted downward.

Considering the 'determination' from the example, in the function \(g(x) = |x|+1\), we add 1 to each y-value of the absolute value function, \(f(x) = |x|\). This represents a vertical shift of the graph upwards by 1 unit. The form \(g(x)=f(x)+c\) directly illustrates this translation where \(c\) signifies the number of units shifted. The process involves no changes in the x-values, thus preserving the horizontal location, and only transforms the output values, resulting in a pure vertical move.

Absolute Value Function

The absolute value function, represented by \(f(x)=|x|\), is a fundamental mathematical concept with a distinctive V-shaped graph. This function measures the distance of a number from zero on the number line, always yielding a non-negative value. Characteristics include:

• The graph is symmetrical about the y-axis, making it an example of an even function.
• The vertex or the turning point is located at the origin \( (0,0) \).
• It is increasing on the interval \( [0, \infty) \) and decreasing on \( (-\infty, 0] \).

These properties make the absolute value function a versatile component in graph transformations. When modifying this function with transformation techniques, such as vertical translations, the geometric form of the function stays intact but its placement relative to the axes alters.

Function Transformation Techniques

Function transformation techniques are essential tools in modifying algebraic graphs, allowing for adjustments in shape, orientation, and position. These techniques expand our understanding of graphs and functions, making them vital in precalculus and calculus.Common transformation techniques include:

• **Vertical and Horizontal Translations**: Shift the graph up, down, left, or right.
• **Reflections**: Flip the graph across the x-axis or y-axis, inverting its orientation.
• **Stretching and Compression**: Alter the graph's appearance by lengthening or shortening along the x or y axis.

In the exercise provided, we focus on vertical translations, as seen with moving \(f(x)=|x|\) to \(g(x)=|x|+1\). By adding a constant, the entire graph's vertical placement shifts, a simple yet powerful transformation technique that maintains the original graph’s structure but adjusts its vertical alignment.