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Magazine Chapter 2: Problem 19
In Problems \(9-34\), sketch the graph of the given piece wise-defined function. Find any \(x\) - and \(y\) intercepts of the graph. Give any numbers at which the function is discontinuous. $$ y=2-|x| $$
Short Answer
Expert verified
No discontinuities, x-intercepts at (-2,0) and (2,0), y-intercept at (0,2).
Step by step solution
01
Understanding the Problem
We are given a piecewise function: \( y = 2 - |x| \). This function involves an absolute value, meaning the graph has different behaviors for \( x \geq 0 \) and \( x < 0 \). We need to sketch the graph, find the intercepts, and determine discontinuity points.
02
Identifying the Function Pieces
The function \( y = 2 - |x| \) remains symmetric around the y-axis due to the absolute value. For \( x \geq 0 \), \( |x| = x \), so the function simplifies to \( y = 2 - x \). For \( x < 0 \), \(|x| = -x \), and the function is \( y = 2 + x \). Thus, we have:- \( y = 2 - x \) for \( x \geq 0 \) - \( y = 2 + x \) for \( x < 0 \)
03
Sketching the Graph
The graph consists of two linear functions that meet at the y-axis. For \( x \geq 0 \), the line decreases with a slope of -1 starting from the point \( (0,2) \). For \( x < 0 \), the line increases with a slope of 1 also starting from \( (0,2) \). These two lines form a 'V' shape where \( x = 0 \) is the vertex (also the maximum point of the graph).
04
Finding the x-Intercept
To find the x-intercept, set \( y = 0 \). For \( y = 2 - x \), solve \( 2 - x = 0 \), which gives \( x = 2 \). For \( y = 2 + x \), solve \( 2 + x = 0 \), which gives \( x = -2 \). Thus, the x-intercepts are \( (-2, 0) \) and \( (2, 0) \).
05
Finding the y-Intercept
To find the y-intercept, set \( x = 0 \). Substitute into the function to get \( y = 2 - |0| = 2 \). Therefore, the y-intercept is \( (0, 2) \).
06
Checking for Discontinuities
The function \( y = 2 - |x| \) is continuous everywhere because it is a linear function on both sides of the y-axis. The lines meet at \( (0, 2) \) without any jumps or breaks. Thus, there are no points of discontinuity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Function
An absolute value function involves the expression \(|x|\), which represents the distance of a number \(x\) from zero on the number line. This distance is always non-negative. When graphed, an absolute value function typically forms a "V" shape.
- The absolute value splits the function into two parts based on the value of \(x\): for non-negative \(x\), the absolute expression evaluates to just \(x\) itself, while for negative \(x\), it evaluates to \(-x\).
- For example, in the function \(y = 2 - |x|\), the absolute value influences the function by creating two linear segments:
- For \(x \geq 0\), the function simplifies to \(y = 2 - x\).
- For \(x < 0\), it becomes \(y = 2 + x\).
Graph Sketching
Sketching the graph of a piecewise function like \(y = 2 - |x|\) involves plotting each linear piece separately and then combining them on the graph.
This function is composed of two linear equations: \(y = 2 - x\) for \(x \geq 0\), and \(y = 2 + x\) for \(x < 0\).
This function is composed of two linear equations: \(y = 2 - x\) for \(x \geq 0\), and \(y = 2 + x\) for \(x < 0\).
- Begin by plotting the y-intercept, which is the point where the graph crosses the y-axis. Here, it's at \((0, 2)\).
- Since \(y = 2 - x\) has a slope of -1, it will slope downwards from the y-intercept for positive \(x\)-values. Plot a few points and draw this line extending to the right.
- The equation \(y = 2 + x\) has a slope of 1, sloping upwards for negative \(x\)-values. Plot a few points and extend this line to the left.
Intercepts
Intercepts are critical points where the graph intersects the axes.
- The x-intercepts are found by setting \(y = 0\). For the function \(y = 2 - |x|\), this means solving for \(x\) when \(y = 0\). The solutions are:
- For \(y = 2 - x\), at \(x = 2\), giving us the point \((2, 0)\).
- For \(y = 2 + x\), at \(x = -2\), giving us the point \((-2, 0)\).
- The y-intercept is the point where the graph crosses the y-axis, found by setting \(x = 0\). In this case, substituting \(x = 0\) gives \(y = 2\), resulting in the point \((0, 2)\).
Discontinuity
Discontinuities in a function occur where there are breaks, jumps, or holes in the graph. However, the function \(y = 2 - |x|\) is continuous across all real numbers.
- Since each "piece" of the graph defined by the different expressions of the absolute value meets smoothly, there are no breaks at the point \(x = 0\).
- The graph does not have any abrupt jumps or undefined points.
