Problem 67 - Algebra 2

Question

Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. \(h(x)=|x-2|\)

Step-by-Step Solution

Answer

The function is classified as an absolute value function.

Step 1: Analyzing the Function

The function given is \(h(x) = |x-2|\). This expression involves an absolute value operation.

Step 2: Identifying the Type

Absolute value functions are characterized by the use of absolute value symbols \(|x|\). They transform input values into their non-negative equivalents, causing a V-shaped graph. The given function \(h(x) = |x-2|\) fits this characteristic.

Key Concepts

Understanding the Absolute Value Function

The absolute value function is a type of mathematical function that is particularly useful for measuring distance and non-negative values. You will often see this function represented as

• The symbol \(|x|\), indicates absolute value, signifying the distance of a number from zero on the number line.
• All output values of an absolute value function are non-negative.

The function given in the exercise, \(h(x) = |x-2|\), is an example of this. In essence, it modifies all input values by transforming them into their non-negative counterparts. This behavior gives the absolute value function its characteristic V-shape when graphed. The vertex of this shape is a key feature, occurring where the expression inside the absolute value symbol equals zero—in the example \(x = 2\). This point marks a transition where the function's direction changes. Understanding this behavior is critical when graphing and working with absolute value functions.

The World of Function Types

In mathematics, functions can be categorized based on their characteristics and how they behave. Understanding function types makes it easier to identify and graph them. Here are some common types:

• **Constant Function**: It outputs the same value regardless of the input, representing a horizontal line on a graph.
• **Step Function**: This type of function jumps from one value to another, creating a series of flat, horizontal segments.
• **Piecewise Function**: Defined by multiple sub-functions, each applying to different parts of a function's domain, these graphs can have several distinct parts.

The absolute value function, such as \(h(x) = |x-2|\), is another important category. It is special due to its ability to reflect input values around a pivot point (or vertex), thus forming a V-shape on a graph. Identifying function types like this helps simplify your analysis and understanding of the equations' graphical behaviors.

Basic Principles of Graphing Functions

Graphing a function involves plotting its input values against output values on a coordinate plane. This visual representation helps us understand the function's behavior at a glance.

• First, determine key points: For an absolute value function like \(h(x) = |x-2|\), start by finding the vertex. The vertex here occurs at \(x = 2\), where the inside of the absolute value equals zero.
• Determine the shape: This function produces a V-shape. From the vertex, draw two symmetrical lines extending outward and upward.

Understanding how to graph these functions helps in identifying their key features and in predicting their behavior across different values. In graphing, always pay attention to these details as they reveal the function's unique characteristics and how they can impact real-life problems.