Problem 38 - Algebra 2

Question

Sketch the graph of each rational function. $$ y=-\frac{x}{(x-1)^{2}} $$

Step-by-Step Solution

Answer

The graph starts from the second quadrant, crosses the X-axis at the origin, then falls to negative infinity as it reaches \(x = 1\). After \(x = 1\), it increases from negative infinity from within the first quadrant and goes to positive infinity.

Step 1: Identify Key Features

Given the function \(y=-\frac{x}{(x-1)^{2}}\), we can identify the key features as follows: \n\n1. The vertical asymptote is where the denominator is zero which gives \(x = 1\).\n2. There's no horizontal asymptote as degree of numerator (1) is less than degree of denominator (2).\n3. The root of the function is the x-value for which \(y = 0\), which gives \(x = 0\).

Step 2: Determine the Behavior Near the Asymptotes and Zeroes

The behavior of the function near the vertical asymptote \(x = 1\) is:\n\n1. As \(x\) approaches 1 from the left (\(x < 1\)), the function decreases without bound (\(y\) approaches \(-\infty\)).\n2. As \(x\) approaches 1 from the right (\(x > 1\)), the function increases without bound (\(y\) approaches \(+\infty\)). \n\nAt the root of the function \(x = 0\), the function passes through the origin.

Step 3: Connect the Dots to Sketch the Graph

The function starts from the second quadrant, crosses the X-axis at the origin, then falls to negative infinity as it reaches \(x = 1\). After \(x = 1\), it increases from negative infinity from under the X-axis and goes to positive infinity and continues in the first quadrant.

Key Concepts

Vertical Asymptote

A vertical asymptote in a rational function is a line that the graph of the function approaches but never actually touches, as you move towards infinity in either the positive or negative direction. In the function \(y = -\frac{x}{(x-1)^{2}}\), the denominator is crucial to finding the vertical asymptote.

The vertical asymptote occurs where the denominator equals zero because division by zero is undefined. Therefore, for this function, set the denominator \((x-1)^{2} = 0\). Solving this equation, we find that the vertical asymptote is at \(x=1\). This means as \(x\) approaches 1, the function's value will either go towards positive infinity or negative infinity, depending on the side \(x\) is approaching from.

Vertical asymptotes indicate a dramatic behavior change and are critical in graphing rational functions. Always pay close attention to these points on your graph.

Roots of Functions

Roots of a function are the values of \(x\) that make the function equal to zero. For rational functions, this occurs where the numerator is zero, provided the denominator is not also zero at the same point. In the function \(y = -\frac{x}{(x-1)^{2}}\), examining the numerator \(x\), we see that it equals zero when \(x = 0\).

This value indicates the root of the function, often referred to as a zero or x-intercept. The function will cross the x-axis at this point.

• Root at \(x = 0\): This is where the graph will intersect the x-axis, making it a key feature in the sketch of the function.

Identifying roots is vital because it tells us where the function changes from positive values to negative values, or vice versa.

Behavior Near Asymptotes

Understanding how a function behaves as it approaches its asymptotes is pivotal for accurate graphing. In this function \(y = -\frac{x}{(x-1)^{2}}\), we have determined a vertical asymptote at \(x=1\). The behavior near this asymptote varies depending on the direction from which \(x\) approaches 1.

As \(x\) approaches 1 from the left (%x < 1%): The function value \(y\) decreases towards \(-\infty\). This indicates a sharp decline in the graph as it nears \(x = 1\) coming from the left side.

As \(x\) approaches 1 from the right (%x > 1%): The function value \(y\) increases towards \(+\infty\). This shows a steep incline as you approach \(x = 1\) from the right side.

These behaviors are crucial for sketching the entire curve. They help us predict how the graph should move and change around asymptotes, ensuring that the sketch reflects the actual nature of the function.