Problem 30 - Brief Applied Calculus

Question

$$ \text { Graph each function } $$ $$ f(x)=|x+2|-2 $$

Step-by-Step Solution

Answer

The graph is a V-shape with vertex at (-2, -2), shifted 2 units left and 2 units down from \(|x|\).

Step 1: Identify the Parent Function

The parent function of the given function is the absolute value function, \( f(x) = |x| \). This is a V-shaped graph with its vertex at the origin (0,0).

Step 2: Determine Horizontal Shift

The function has \( x+2 \) inside the absolute value, indicating a horizontal shift. A +2 inside the function means the graph is shifted 2 units to the left. Thus, the new vertex will be at (-2,0).

Step 3: Determine Vertical Shift

There is a -2 outside the absolute value function. This results in a vertical shift. A -2 indicates that the graph is moved 2 units down, moving the vertex from (-2,0) to (-2,-2).

Step 4: Plot the Vertex

Plot the new vertex of the graph at (-2,-2) on the coordinate plane. This is where the V-shape of the graph will start.

Step 5: Sketch the Graph

Draw the V-shaped graph: Starting from the vertex (-2,-2), create lines extending upward and outward. The left arm extends from the vertex towards negative infinity on the y-axis and the right arm extends from the vertex towards positive infinity on the y-axis, symmetrical with respect to the vertex.

Key Concepts

Graph Transformation

Graph transformation is a process used to modify the graph of a function to achieve a desired outcome. In the context of absolute value functions, which are represented as V-shaped graphs, transformations can alter the graph's position or appearance. These transformations include:

• Translations (horizontal and vertical shifts)
• Reflections
• Stretching or compressing

In the exercise, the absolute value parent function is transformed into a new function, \[ f(x) = |x+2| - 2 \] Simple transformations such as shifts can be performed by applying simple arithmetic operations to the function definition. Each type of shift is described with specific changes to either input values (for horizontal shifts) or output values (for vertical shifts). Understanding these transformations helps to accurately graph complex functions based on simpler, parent functions.

Horizontal Shift

A horizontal shift occurs when a graph moves left or right from its original position. This happens due to additions or subtractions inside the function's argument. When dealing with the absolute value function, any change in the expression inside the absolute value brackets directly affects the horizontal position.
In the given exercise, the expression \( x+2 \) indicates a horizontal shift. To determine the direction and magnitude:

• Recognize that \( x+2 \) means shifting left, as opposed to the intuitive right movement.
• The graph shifts 2 units left because \( +2 \) is inside the function, effectively moving the vertex from \( (0, 0) \) to \( (-2, 0) \).

This might seem counterintuitive at first due to working inside the function's argument. However, once you understand this mechanism, predicting the movement becomes straightforward for any similar expression.

Vertical Shift

Vertical shifts involve moving the entire graph up or down. These are straightforward to identify and apply as they occur due to additions or subtractions outside of the function itself.
When analyzing the function \( f(x) = |x+2| - 2 \), the term \( -2 \) outside the absolute value indicates a vertical shift:

• Negative outside shifts the graph down, while positive moves it up.
• In this case, \( -2 \) means the V-shaped graph moves down by 2 units.
• This results in the vertex moving from \( (-2,0) \) to \( (-2,-2) \).

Vertical shifts are easier to visualize since they directly modify the output of the function, thus altering the function's vertical position across the entire graph. Understanding both horizontal and vertical shifts allows for full control over the positioning of a graph on the coordinate plane.