Graphing & Solving Y=|x|-(x-2)/|x-2|: A Step-by-Step Guide

by Viktoria Ivanova 59 views

Hey guys! Let's dive into graphing and solving the function y = |x| - (x-2)/|x-2|. This is a fun one because it involves absolute values, which can sometimes feel a bit tricky. But don't worry, we'll break it down step by step so it's super easy to understand. We'll go through how to think about the absolute value parts, how to graph the function, and discuss some of the interesting features of the graph. So, let’s get started!

Understanding the Function

Before we jump into graphing, let’s really understand what this function, y = |x| - (x-2)/|x-2|, is all about. The key here is the absolute value. Remember, the absolute value of a number is its distance from zero, which means it's always non-negative. So, |x| is x if x is greater than or equal to zero, and -x if x is less than zero. The other part, (x-2)/|x-2|, is a bit more interesting. We need to be careful about what happens when x is 2 because the denominator, |x-2|, becomes zero, which makes the expression undefined. For values other than 2, |x-2| is x-2 when x is greater than 2 and -(x-2) when x is less than 2. This is crucial for figuring out the behavior of our function.

To really grasp this, let’s consider a few cases. When x is greater than 2, both x and (x-2) are positive. Thus, |x| is just x, and |x-2| is x-2. So, the term (x-2)/|x-2| simplifies to (x-2)/(x-2), which equals 1. On the other hand, when x is less than 2, but still non-negative (i.e., between 0 and 2), |x| is still x, but |x-2| is -(x-2), so the term (x-2)/|x-2| becomes (x-2)/-(x-2), which equals -1. And finally, when x is less than zero, |x| becomes -x, and |x-2| is still -(x-2), so the term (x-2)/|x-2| remains -1. This case-by-case breakdown helps us to see how the function behaves differently across the number line, which is super important when we start graphing.

By carefully dissecting the absolute value components, we've transformed a somewhat intimidating function into a set of manageable pieces. This kind of analytical approach is often your best friend when dealing with functions that involve absolute values or piecewise definitions. Breaking it down into cases based on where the expressions inside the absolute values change sign gives you a clearer picture of the function's overall behavior. So, before we even think about the graph, we've already gained a solid understanding of the function's personality, its quirks, and where we need to be particularly cautious (like at x = 2).

Breaking it Down into Cases

Alright, let's formalize our understanding by breaking the function y = |x| - (x-2)/|x-2| down into distinct cases. This is a super useful technique when dealing with absolute values because it allows us to rewrite the function in a piecewise form, which is much easier to handle. We'll look at different intervals of x and see how the function behaves in each.

  • Case 1: x > 2

    When x is greater than 2, both |x| and |x-2| behave predictably. The absolute value of x, |x|, is simply x since x is positive. Similarly, since x is greater than 2, x-2 is also positive, so |x-2| is just x-2. Therefore, our function becomes:

    y = x - (x-2)/(x-2)

    Since (x-2)/(x-2) simplifies to 1 (as long as x isn't 2, which we've already accounted for), we get:

    y = x - 1

    This is a simple linear equation, which makes graphing much easier! It’s a straight line with a slope of 1 and a y-intercept of -1.

  • Case 2: 0 ≤ x < 2

    Now, let's consider the interval where x is non-negative but less than 2. In this case, |x| is still just x, but |x-2| behaves differently. Since x is less than 2, x-2 is negative, so |x-2| becomes -(x-2). Our function now looks like this:

    y = x - (x-2)/[-(x-2)]

    The term (x-2)/[-(x-2)] simplifies to -1 (again, as long as x isn't 2), so the function becomes:

    y = x - (-1)

    y = x + 1

    Again, we have another linear equation, a line with a slope of 1 and a y-intercept of 1. Notice how the absolute value in the original function has led to a different linear behavior in this interval compared to the previous one.

  • Case 3: x < 0

    Finally, let's look at what happens when x is less than zero. In this case, |x| is -x because we need to make the value non-negative. |x-2| is still -(x-2) since x-2 will be negative when x is negative. So, the function becomes:

    y = -x - (x-2)/[-(x-2)]

    As before, (x-2)/[-(x-2)] simplifies to -1, so we have:

    y = -x - (-1)

    y = -x + 1

    This is another linear equation, but this time it has a negative slope of -1 and a y-intercept of 1. The negative slope means the line will be decreasing as x increases.

By breaking the function down into these three cases, we've essentially transformed a complex problem into three simpler ones. We now have a piecewise function defined by three linear equations, each valid over a specific interval. This makes graphing much more straightforward because we know exactly what the function looks like in each part of its domain. Plus, understanding these cases helps us anticipate the shape of the graph and identify any points of discontinuity or sharp turns.

Graphing the Piecewise Function

Okay, guys, now for the fun part – graphing! We've already broken down the function y = |x| - (x-2)/|x-2| into three manageable cases, each with its own linear equation. Now we just need to put these pieces together on a coordinate plane. Remember, each case is valid for a specific interval of x, so we'll be drawing different line segments for each interval.

  • For x > 2: y = x - 1

    This is a line with a slope of 1 and a y-intercept of -1. But remember, this part of the function is only valid for x greater than 2. So, we'll start by plotting a point at x = 2. If we plug x = 2 into y = x - 1, we get y = 1. However, since this case is for x greater than 2, we'll use an open circle at the point (2, 1) to indicate that this point is not actually included in this piece of the function. Then, we draw a line with a slope of 1 extending to the right (for x > 2).

  • For 0 ≤ x < 2: y = x + 1

    This is another line with a slope of 1, but this time the y-intercept is 1. This segment is valid for x between 0 and 2 (including 0 but not 2). At x = 0, y = 0 + 1 = 1, so we plot a closed circle at (0, 1) because this point is included. At x = 2, y = 2 + 1 = 3, but since this case is for x less than 2, we'll use an open circle at (2, 3). Then, we draw the line segment connecting these points.

  • For x < 0: y = -x + 1

    This is a line with a slope of -1 and a y-intercept of 1. It's valid for all x less than 0. At x = 0, y = -0 + 1 = 1, but since this case is for x less than 0, we'll use an open circle at (0, 1). The line extends to the left (for x < 0) with a slope of -1.

When you put it all together, you'll see a graph that consists of three line segments. There's a jump, or discontinuity, at x = 2, where the function is undefined because of the division by zero in the original expression. The graph also has a sharp corner at x = 0, which is typical for functions involving absolute values. This corner occurs because the slope of the function changes abruptly at that point.

Graphing piecewise functions can seem a little daunting at first, but it's really just about carefully applying the correct equation to the correct interval. By paying attention to the endpoints of each interval and whether they should be included (closed circles) or excluded (open circles), you can create an accurate and informative graph. Plus, it's pretty cool to see how a single function can exhibit different behaviors across its domain!

Key Features of the Graph

Now that we've graphed the function y = |x| - (x-2)/|x-2|, let's take a closer look at the key features of the graph. Understanding these features will give us a deeper insight into the behavior of the function and how the absolute value components influence its shape. We'll focus on the discontinuities, the slope changes, and the overall trends of the graph.

  • Discontinuity at x = 2

    The most prominent feature of the graph is the discontinuity at x = 2. Remember, the original function has a term (x-2)/|x-2|. When x is 2, the denominator |x-2| becomes zero, making the entire expression undefined. This means there's a break in the graph at x = 2. On our graph, you'll see an open circle at the end of the line segment approaching x = 2 from the left (y = x + 1 approaches the point (2,3)) and another open circle where the line segment starts to the right of x = 2 (y = x - 1 starts at the point (2,1)). This jump in the graph visually represents the discontinuity.

  • Sharp Corner at x = 0

    Another interesting feature is the sharp corner at x = 0. This is a common characteristic of functions involving absolute values. The corner occurs because the slope of the graph changes abruptly at this point. To the left of x = 0, the function is defined by y = -x + 1, which has a slope of -1. To the right of x = 0, the function is defined by y = x + 1, which has a slope of +1. This sudden change in slope creates the sharp corner.

  • Piecewise Linear Segments

    The graph consists of three distinct linear segments, each corresponding to one of the cases we defined earlier. This piecewise linear nature is a direct result of the absolute value function. Each segment has a constant slope, but the slopes differ between the segments due to how the absolute values affect the sign of the terms.

  • Asymptotic Behavior

    Although this function doesn't have any traditional asymptotes (like vertical or horizontal asymptotes where the function approaches infinity), it does exhibit different behaviors as x approaches positive and negative infinity. As x becomes very large (approaches positive infinity), the function follows the line y = x - 1, so it increases without bound. As x becomes very negative (approaches negative infinity), the function follows the line y = -x + 1, which also increases without bound (since -x becomes a large positive number).

By examining these key features, we gain a deeper understanding of how the function behaves. We see how the absolute values create discontinuities and sharp corners, and how the piecewise nature of the function leads to different linear segments. Analyzing these features helps us connect the algebraic representation of the function with its graphical representation, giving us a more complete picture of its behavior.

Conclusion

So, guys, we've successfully graphed and analyzed the function y = |x| - (x-2)/|x-2|! We started by understanding the absolute value components, then broke the function down into cases to simplify it. We graphed each case and finally discussed the key features of the graph, like the discontinuity at x = 2 and the sharp corner at x = 0. Hopefully, this has demystified how to approach functions with absolute values and piecewise definitions. Remember, the key is to break it down, take it step by step, and don't be afraid to explore each piece individually. You got this!