Problem 34 In Problems $9-34$, sketch the... [FREE SOLUTION]

Chapter 2: Problem 34

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. $$ y=\left\\{\begin{array}{ll} -x, & x<0 \\ 1-|x-1|, & 0 \leq x \leq 2 \\ x-2, & x>2 \end{array}\right. $$

Short Answer

Expert verified

The function has $x$- and $y$-intercept at $(0,0)$ and is continuous everywhere.

Step by step solution

Analyze Each Piece

The function is defined piecewise with three distinct segments. For $ x < 0 $, the function is $ y = -x $, which is a straight line with a negative slope passing through the origin. For $ 0 \leq x \leq 2 $, the function is $ y = 1 - |x-1| $. This represents a 'V' shaped graph, peaking at $ x = 1 $. For $ x > 2 $, the function is $ y = x - 2 $, a straight line with a slope of 1.

Determine Intercepts

To find the $ x $-intercepts, set $ y = 0 $ in each piece of the function. The $ x $-intercept for $ y = -x $ is $ x = 0 $. For $ y = 1 - |x-1| $, solve $ 1 - |x-1| = 0 $, giving $ x = 0 $ and $ x = 2 $. For $ y = x - 2 $, solve $ x - 2 = 0 $, giving $ x = 2 $. The $ y $-intercept is found by evaluating the function at $ x = 0 $, which gives $ y = -0 = 0 $. The intercept is $ (0,0) $.

Check for Discontinuities

A function is discontinuous at points that don't seamlessly connect between segments. Check transition points: at $ x = 0 $, both segments meet at $ y = 0 $; at $ x = 2 $, both segments meet at $ y = 0 $. Since the function values agree at these points, the function is continuous, with no discontinuities.

Sketch the Graph

Plot the piecewise function segment by segment. 1. For $ x < 0 $, sketch a line through the origin with a slope of $-1$.2. For $ 0 \leq x \leq 2 $, sketch the 'V' shape peaking at $ y = 1 $ (at $ x = 1 $).3. For $ x > 2 $, sketch a line starting from $ (2,0) $ with a slope of $ 1 $.Combine these into a continuous graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Sketching for Piecewise Functions

When sketching piecewise functions, each segment is drawn separately and then combined to form the entire graph. For the function given in this exercise, there are three distinct segments:

For the region where $ x < 0 $, the function is $ y = -x $. This part of the graph is a straight line with a downward slope passing through the origin. It's like drawing a line from the origin downwards to the left.

Next, for $ 0 \leq x \leq 2 $, the function changes to $ y = 1 - |x-1| $. This creates a 'V' shape in the graph, peaking at $ x = 1 $. Picture the graph climbing up to 1 at $ x = 1 $ and then descending back to 0 at $ x = 2 $.

Finally, for $ x > 2 $, the function becomes $ y = x - 2 $. This segment is another straight line, but this time it rises from the point $ (2,0) $ with a slope of 1, moving upwards to the right.

To sketch the graph effectively, plot each piece with consideration for their individual shapes and slopes. Connect them smoothly, ensuring continuity if specified by the function's conditions.

Understanding Discontinuity in Piecewise Functions

Discontinuity occurs in a function when you have a break or jump between the segments of a graph. While working with piecewise functions, checking for discontinuities is crucial to understand the overall behavior of the function.

Discontinuity can be found at the transition points where one segment of the piecewise function ends, and another begins.

In our function, these transition points are at $ x = 0 $ and $ x = 2 $. At both these points, we need to ensure that the function values match seamlessly.

At $ x = 0 $, both the segments $ y = -x $ and $ y = 1 - |x-1| $ meet at $ y = 0 $.

Similarly, at $ x = 2 $, the segments $ y = 1 - |x-1| $ and $ y = x - 2 $ also meet at $ y = 0 $.

Since both transition points in our function connect without any breaks or jumps, the function is continuous. Ensuring continuity can help correct misunderstandings in interpreting the graph of a piecewise function.

Finding Intercepts on the Graph

Intercepts are the points where the graph crosses the axes. They give critical information about the behavior of a function.

The $ x $-intercepts occur where the graph crosses the x-axis. For a piecewise function, you need to find this separately for each segment by setting $ y = 0 $.

For $ y = -x $, the $ x $-intercept is $ x = 0 $ because at the origin, both $ x $ and $ y $ are zero.

In $ y = 1 - |x-1| $, solving $ 1 - |x-1| = 0 $ results in $ x = 0 $ and $ x = 2 $, placing two $ x $-intercepts along the x-axis.

For $ y = x - 2 $, setting it to 0 gives the $ x $-intercept $ x = 2 $, confirming it reinforces the intercept already present.

The $ y $-intercept occurs where the graph crosses the y-axis, which is readily found by substituting $ x = 0 $. For this function, at $ x = 0 $, $ y = 0 $, hence the $ y $-intercept is $ (0,0) $.

Understanding where a graph crosses the axes is fundamental in graph sketching and analysis in piecewise functions, allowing better visualization and comparison of function behaviors across segments.

91影视

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. $$ y=\left\\{\begin{array}{ll} -x, & x<0 \\ 1-|x-1|, & 0 \leq x \leq 2 \\ x-2, & x>2 \end{array}\right. $$

Short Answer

Step by step solution

Analyze Each Piece

Determine Intercepts

Check for Discontinuities

Sketch the Graph

Key Concepts

Graph Sketching for Piecewise Functions

Understanding Discontinuity in Piecewise Functions

Finding Intercepts on the Graph

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