Problem 23 - Algebra 2

Question

Graph each function. Identify the domain and range. \(g(x)=|x|-4\)

Step-by-Step Solution

Answer

Domain: All real numbers; Range: \(y \geq -4\).

Step 1: Understand the Absolute Value Function

The function \(g(x) = |x| - 4\) is based on the absolute value function \(f(x) = |x|\), which is a V-shaped graph symmetric about the y-axis with its vertex at the origin (0,0). The graph of \(f(x) = |x|\) opens upwards.

Step 2: Transform the Graph

The function \(g(x) = |x| - 4\) shifts the original absolute value graph downwards by 4 units. The vertex, which was originally at (0,0), moves to (0,-4). This is because the '-4' in the function subtracts 4 from the y-values of the entire function.

Step 3: Graph the Function

Start by plotting the vertex of the transformed function at the point (0,-4). Then, draw the two linear rays emanating from the vertex. The left ray will have a slope of -1 and the right ray will have a slope of 1, just like the basic \(f(x) = |x|\) graph.

Step 4: Identify the Domain

The absolute value function \(g(x) = |x| - 4\) is defined for all real numbers. Therefore, the domain of \(g(x)\) is all real numbers, represented as \(-\infty < x < \infty\).

Step 5: Identify the Range

Since the lowest point on the graph (the vertex) is at (0,-4) and the V-shape opens upwards, the smallest value \(g(x)\) can take is -4. Thus, the range of \(g(x)\) is \(y \geq -4\).

Key Concepts

Domain and Range

When working with the absolute value function, understanding the domain and range is crucial. The domain of a function refers to all the possible input values, while the range refers to all possible output values. For the function \(g(x) = |x| - 4\), let's explore these aspects.

• Domain: Absolute value functions are defined for all real numbers, meaning there are no restrictions on the x-values you can use. Therefore, the domain of \(g(x)\) is all real numbers. You might see this written as \(-\infty < x < \infty\).
• Range: To find the range, consider where the lowest point or vertex of the graph lies. In this case, the vertex starts at (0, -4) and the graph opens upwards, indicating that the smallest value the function can produce is -4. Thus, the range of \(g(x)\) is \(y \geq -4\).

Understanding and identifying these can help you in sketching the graph and analyzing the function's behavior.

Function Transformation

The concept of function transformation involves shifting, stretching, compressing, or reflecting the basic graph of a function to create a new function. For \(g(x) = |x| - 4\), we focus on shifting.

• Vertical Shifting: The expression \(\lvert x \rvert - 4\) indicates a downward shift of the basic absolute value function \(f(x) = |x|\). This specific transformation means every point on the graph of \(f(x)\), which originally has its vertex at (0,0), moves 4 units down on the y-axis to settle at (0,-4).
• Horizontally Untouched: There are no changes to the x-values in this scenario, so the graph is only affected vertically, not stretched or compressed.

Learning these transformations helps visualize how a graph shifts from its parent function without needing to plot every point manually.

Absolute Value Graph Characteristics

Absolute value functions exhibit distinct characteristics that make their graphs easy to recognize. Understanding these traits allows you to sketch the graph quickly and accurately. Let's isolate some of these features for \(g(x) = |x| - 4\).

• Shape: The graph takes on a V-shape, with the vertex being the sharp point at the bottom of the curve. For \(g(x)\), since it is moved down 4 units, the vertex changes from (0,0) to (0,-4).
• Symmetry: One of the key characteristics is its symmetry about the y-axis. This means if you were to fold the graph along the y-axis, both halves would match up perfectly.
• Direction: Despite the transformation, the graph still opens upward, maintaining the original orientation of the basic absolute value graph.

By understanding these characteristics, you can not only plot these graphs more effectively but also predict how changes to the formula will affect the graph's appearance.