Problem 26 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-|x+2|$$

Step-by-Step Solution

Answer

The graph is a V-shape opening downwards with vertex at \((-2, 0)\).

Step 1: Understand the Function

The function given is \(f(x) = -|x+2|\). This is an absolute value function with a horizontal shift of 2 units to the left and a reflection in the x-axis due to the negative sign.

Step 2: Set Up the Vertex

The vertex of the function \(f(x) = -|x + 2|\) is at \(x = -2\). This is because the absolute value function starts at the point where the expression inside becomes zero, which is \(-2\) in this case.

Step 3: Determine Key Points

To graph \(f(x)\), find some key points around the vertex. For example, if \(x = -2\), \(f(-2) = -|(-2) + 2| = 0\). Choose additional points such as \(x = -1\) and \(x = -3\), we get \(f(-1) = -|1| = -1\) and \(f(-3) = -|1| = -1\).

Step 4: Sketch the Graph

Plot the vertex \((-2, 0)\) and the additional points \((-1, -1)\) and \((-3, -1)\). Connect the points in a V-shape, opening downwards because of the negative sign outside the absolute value.

Key Concepts

Absolute Value Functions

Absolute value functions are unique mathematical expressions that appear in the form of \( f(x) = |x| \). These functions depict distances and are significant because they always yield non-negative results. An absolute value function graphed forms a characteristic "V" shape.
Each side of the "V" can either rise or fall depending on changes applied to the function. Key components of the function can alter this shape, such as a negative sign directly affecting the direction the "V" points. It’s important to understand these details as they are fundamental in manipulating and graphing these functions.

Horizontal Shift

A horizontal shift involves moving the graph left or right along the x-axis. In the function \( f(x) = -|x+2| \), the term \( (x+2) \) indicates a horizontal shift.

• If this term had been \( (x-2) \), the graph would move 2 units to the right.
• In the given function, \( (x+2) \) means the graph shifts 2 units to the left.

These shifts do not alter the "V" shape but reposition it along the x-axis. The shift is crucial to determining where the vertex and the overall graph starts.

Vertex in Absolute Value Graphs

The vertex in an absolute value graph is a crucial point, serving as the "tip" or point of the "V". For \(f(x) = -|x+2|\), the expression inside the absolute value, \(x+2=0\), provides the x-coordinate of the vertex, which is \(x = -2\).

• The y-coordinate at the vertex can be found by evaluating \( f(-2) \), which gives \( f(-2) = 0 \).
• Thus, the vertex is at the point \((-2, 0)\).

Understanding where the vertex lies helps in constructing the graph accurately. Changes in this location are also influenced by reflections and shifts applied to the function.

Reflection in x-axis

A reflection in the x-axis transforms the direction in which the "V" of the absolute value function opens. In the function \( f(x) = -|x+2| \), the negative sign before the absolute value causes this reflection.

• Normally, \( |x| \) opens upwards, creating an upright "V".
• The negative sign flips it to open downwards, resembling an upside-down "V".

This reflection is critical as it influences the overall orientation and the key points related to the function. Such transformations need careful consideration when plotting the function on a graph.