Graph the rational function [tex]y=\frac{x^2-3}{x-1}[/tex]. Both branches of the rational function pass through which quadrant?
A. Quadrant 2
B. Quadrant 4
C. Quadrant 3
D. Quadrant 1
Answer (1)
Find the vertical asymptote: x = 1 .
Find the slant asymptote: y = x + 1 .
Analyze the behavior near the vertical asymptote: right branch approaches − ∞ , left branch approaches + ∞ .
Determine the quadrants each branch passes through: both branches pass through quadrant 1. Q u a d r an t 1
Explanation
Understanding the Problem The problem asks us to identify which quadrant both branches of the rational function y = x − 1 x 2 − 3 pass through. To do this, we need to understand the behavior of the function and its graph.
Finding the Vertical Asymptote First, let's find the vertical asymptote. This occurs when the denominator is zero: x − 1 = 0 , so x = 1 . This divides the graph into two branches.
Finding the Slant Asymptote Next, let's analyze the end behavior. Since the degree of the numerator (2) is one greater than the degree of the denominator (1), there is a slant asymptote. We can find it by performing polynomial long division: x − 1 x 2 − 3 = x + 1 − x − 1 2 So the slant asymptote is y = x + 1 .
Analyzing Behavior Near the Vertical Asymptote Now, let's analyze the behavior of the function near the vertical asymptote x = 1 . As x approaches 1 from the right ( 1"> x > 1 ), x − 1 is positive, so x − 1 − 2 is negative and approaches − ∞ . Thus, y approaches − ∞ . This means the right branch enters quadrant 4. As x approaches 1 from the left ( x < 1 ), x − 1 is negative, so x − 1 − 2 is positive and approaches + ∞ . Thus, y approaches + ∞ . This means the left branch enters quadrant 2.
Finding Intercepts Let's find the x -intercepts by setting y = 0 : x 2 − 3 = 0 , so x = ± 3 . The x -intercepts are approximately x = ± 1.73 . Let's find the y -intercept by setting x = 0 : y = 0 − 1 0 2 − 3 = 3 .
Analyzing the Right Branch Now, let's consider the right branch. Since 1"> x > 1 , and we know there's an x -intercept at x = 3 ≈ 1.73 , the right branch will eventually cross the x -axis and enter quadrant 1. For large positive x , y will be positive since the function approaches the slant asymptote y = x + 1 . Thus, the right branch is in quadrant 1 and quadrant 4.
Analyzing the Left Branch Now, let's consider the left branch. Since x < 1 , and we know there's an x -intercept at x = − 3 ≈ − 1.73 , the left branch will pass through quadrant 2 and quadrant 3. For large negative x , y will be negative since the function approaches the slant asymptote y = x + 1 . Thus, the left branch is in quadrant 2 and quadrant 3.
Determining Common Quadrant From the python calculation tool, we found that the right branch passes through quadrants 1 and 4, and the left branch passes through quadrants 1, 2, and 3. The common quadrant is quadrant 1.
Final Answer Therefore, both branches of the rational function pass through quadrant 1.
Examples
Rational functions are used in various fields, such as physics, engineering, and economics, to model relationships between variables. For example, in physics, they can describe the behavior of lenses and mirrors. In economics, they can model cost-benefit ratios. Understanding the behavior of rational functions, including their asymptotes and intercepts, helps in analyzing and predicting the outcomes in these real-world scenarios. By analyzing the quadrants through which the function passes, we gain insights into the sign and magnitude of the relationship between the variables involved.
