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Magazine Chapter 2: Problem 56
Discuss how to graph the function \(y=|x|+\mid x-\) 3|. Carry out your ideas.
Short Answer
Expert verified
The graph of \( y = |x| + |x - 3| \) consists of three linear sections connected at \( x = 0 \) and \( x = 3 \), forming V-like structures.
Step by step solution
01
Understand Absolute Value Functions
The absolute value function \( y = |x| \) transforms negative input values into positive ones, resulting in a V-shaped graph with the vertex at the origin (0, 0). For a function of the form \( y = |x - a| \), the graph is shifted horizontally \( a \) units. In the function \( y = |x - 3| \), this indicates a shift to the right by 3 units.
02
Break Down the Function
The given function is \( y = |x| + |x - 3| \). This means we are adding the absolute value function \( |x| \) and the shifted absolute value function \( |x - 3| \). This results in a combined effect of two V-shaped graphs.
03
Analyze the Function in Each Interval
The function can be analyzed in three intervals based on the break points of the absolute value expressions:- For \( x < 0 \): both \( |x| \) and \( |x-3| \) can be replaced by \(-x\) and \(3-x\) respectively, so \( y = -x + (3-x) = 3 - 2x \).- For \( 0 \le x < 3 \): \( |x| = x \) and \( |x-3| = 3-x \), leading to \( y = x + (3 - x) = 3 \).- For \( x \, \ge \, 3 \): both values can be replaced by positive expressions \( x \) and \( x-3 \), resulting in \( y = x + (x - 3) = 2x - 3 \).
04
Graph Each Interval
Leverage the interval analysis:- For \( x < 0 \), plot the linear function \( y = 3 - 2x \), which is a decreasing line.- For \( 0 \le x < 3 \), plot the constant function \( y = 3 \).- For \( x \ge 3 \), plot the increasing line \( y = 2x - 3 \). Combine all sections to capture behavior across all \( x \) values.
05
Check for Continuity and Verify
Determine if the sections of the graph connect seamlessly. For instance:- At \( x = 0 \), function transitions from \( y = 3 \) (from the segment for \( 0 \leq x < 3 \)) and aligns with the vertical drop from \( x < 0 \).- At \( x = 3 \), both sections \( 0 \le x < 3 \) and \( x \ge 3 \) match value \( y = 3 \) ensuring continuity without gaps or jumps at these points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Function
The absolute value function, represented as \( y = |x| \), is a fundamental concept in graphing. This function transforms any negative input into a positive output. Essentially, it reflects all negative parts of the input across the x-axis, creating a V-shape in the graph. For the basic absolute value function \( y = |x| \), the vertex, or the point where the direction changes, is located at the origin (0, 0). To graph functions like \( y = |x - a| \), you simply shift the V-shaped graph horizontally. This means that if \( a = 3 \), as in \( y = |x - 3| \), the graph shifts right by 3 units.
Piecewise Functions
Piecewise functions are those where different expressions apply to different portions of the domain. For the function \( y = |x| + |x - 3| \), different expressions capture the function's behavior in varying intervals based on the absolute value inputs. Specifically, this function can be analyzed across three intervals:
- When \( x < 0 \), both segments \( |x| \) and \( |x - 3| \) yield negative expressions, and the function becomes \( y = 3 - 2x \).
- For \( 0 \leq x < 3 \), the first segment \( |x| \) is positive, and \( |x - 3| \) remains negative, leading to \( y = 3 \).
- When \( x \geq 3 \), both become positive, simplifying the function to \( y = 2x - 3 \).
Continuity in Functions
Continuity in functions refers to the smooth transition of graph parts without breaks or gaps. For the function \( y = |x| + |x - 3| \), checking for continuity is crucial at the intervals鈥� breaking points, particularly at \( x = 0 \) and \( x = 3 \). Evaluating continuity involves ensuring that the function value remains the same from different segments. At \( x = 0 \), the function moves from \( 3 - 2x \) to \( 3 \), and despite the change, the y-values align at 3. Similarly, at \( x = 3 \), both expressions from the segments \( 0 \leq x < 3 \) and \( x \geq 3 \) evaluate to \( y = 3 \). This seamless connection reflects a continuous function without jumps.
Function Analysis
Analyzing a function involves breaking it down within specified intervals and understanding each section's behavior. For \( y = |x| + |x - 3| \), the analysis looks at how each component contributes as \( x \) changes:
- For \( x < 0 \): The function transformations result in a linear component, \( y = 3 - 2x \), indicating a downward slope.
- For \( 0 \leq x < 3 \): The behavior of the function flattens out to a constant \( y = 3 \), showing no change in slope.
- For \( x \geq 3 \): It results in an upward linear expression \( y = 2x - 3 \), demonstrating an increasing slope.
Graph Interpretation
Graph interpretation is about understanding the visual representation of a function. For \( y = |x| + |x - 3| \), interpreting the graph involves piecing together each interval鈥檚 behavior into a coherent view. The graph is built piece by piece:
- For \( x < 0 \): The graph portrays a descending line.
- Between \( 0 \leq x < 3 \): It is a consistent flat line at \( y = 3 \).
- When \( x \geq 3 \): It rises, indicating increasing value.
