Identifying The Graph Of F(x) = (x-1)/(x^2-x-6)

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Identifying the Graph of f(x) = (x-1)/(x^2-x-6)

Hey guys! Let's dive into how to identify the graph of the function f(x) = (x-1)/(x^2 - x - 6). This is a classic problem in mathematics that combines algebra and graphical analysis. To tackle this, we'll break down the function, analyze its key features, and then match those features to a potential graph. By the end of this guide, you'll be able to confidently approach similar problems. So, buckle up and let’s get started!

1. Simplify and Factor

First, let's simplify the function. The given function is f(x) = (x-1)/(x^2 - x - 6). To understand it better, we should factor the denominator. The quadratic expression x^2 - x - 6 can be factored into (x - 3)(x + 2). Therefore, our function becomes:

f(x) = (x - 1) / ((x - 3)(x + 2))

This factorization helps us identify the function’s key characteristics, such as vertical asymptotes and zeros. Understanding these characteristics is crucial for accurately identifying the graph. Factoring simplifies the expression and makes it easier to analyze the behavior of the function as x approaches certain values. By factoring, we transform a complex rational function into a more manageable form, revealing critical information about its structure. It's like taking apart a machine to see how each piece contributes to the overall function. This step is fundamental in graphing rational functions and sets the stage for further analysis.

2. Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function equals zero, and the numerator does not. From the factored form f(x) = (x - 1) / ((x - 3)(x + 2)), we can see that the denominator is zero when x = 3 and x = -2. Thus, we have vertical asymptotes at x = 3 and x = -2. These asymptotes are vertical lines that the graph approaches but never touches. They define the boundaries of the function's domain and significantly influence its shape. When you look at potential graphs, immediately check for vertical lines at x = 3 and x = -2. The presence and correct placement of these asymptotes are key indicators of the correct graph. Recognizing vertical asymptotes is a fundamental skill in analyzing rational functions and provides essential clues for graph identification.

3. Find the Zeros (x-intercepts)

The zeros of the function, also known as x-intercepts, occur when the numerator equals zero. In our function, f(x) = (x - 1) / ((x - 3)(x + 2)), the numerator is x - 1. Setting this equal to zero gives us x = 1. So, there is a zero at x = 1. This means the graph crosses the x-axis at the point (1, 0). Zeros are crucial points for sketching or identifying the graph of a function. They tell us where the function's value is zero, and these points help anchor the graph on the coordinate plane. When examining potential graphs, verify that the graph indeed passes through the point (1, 0). The correct placement of this zero is another essential criterion for determining the correct graph. Identifying zeros helps complete the picture of the function's behavior and assists in accurate graph matching.

4. Determine the y-intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function: f(0) = (0 - 1) / ((0 - 3)(0 + 2)) = (-1) / (-6) = 1/6. Therefore, the y-intercept is at the point (0, 1/6). The y-intercept provides another fixed point that the graph must pass through. This is particularly useful in distinguishing between similar-looking graphs. When assessing potential graphs, confirm that the graph intersects the y-axis at y = 1/6. The accurate positioning of the y-intercept is a key detail that helps refine the graph identification process. It acts as a checkpoint, ensuring the graph aligns with the function's behavior near the y-axis. Finding the y-intercept is an essential step in completing the overall analysis and ensuring an accurate match.

5. Analyze the Behavior Near Asymptotes

To understand the function's behavior near the vertical asymptotes, we need to analyze the limits as x approaches these values from both the left and the right.

  • As x approaches -2 from the left (x → -2-): The term (x + 2) becomes a small negative number. The term (x - 1) is negative, and (x - 3) is also negative. So, the function becomes (-)/((-)(-)) = (-) / (+) = -. Thus, f(x) → -∞.
  • As x approaches -2 from the right (x → -2+): The term (x + 2) becomes a small positive number. The term (x - 1) is negative, and (x - 3) is negative. So, the function becomes (-)/((-)(+)) = (-) / (-) = +. Thus, f(x) → +∞.
  • As x approaches 3 from the left (x → 3-): The term (x - 3) becomes a small negative number. The term (x - 1) is positive, and (x + 2) is also positive. So, the function becomes (+)/((+)(-)) = (+) / (-) = -. Thus, f(x) → -∞.
  • As x approaches 3 from the right (x → 3+): The term (x - 3) becomes a small positive number. The term (x - 1) is positive, and (x + 2) is also positive. So, the function becomes (+)/((+)(+)) = (+) / (+) = +. Thus, f(x) → +∞.

This analysis tells us how the graph behaves as it gets very close to the vertical asymptotes. It's essential for understanding the direction the graph takes near these boundaries. When examining graphs, look for these behaviors: as x approaches -2 from the left, the graph should go down towards -∞, and as x approaches -2 from the right, it should go up towards +∞. Similarly, as x approaches 3 from the left, the graph should go down towards -∞, and as x approaches 3 from the right, it should go up towards +∞. This analysis adds a critical layer of detail to the graph identification process.

6. Check for Horizontal or Oblique Asymptotes

To determine if there are any horizontal or oblique asymptotes, we examine the degrees of the numerator and the denominator. In our function, f(x) = (x - 1) / (x^2 - x - 6), the degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0. This means as x approaches +∞ or -∞, the function approaches zero. When looking at the graph, confirm that the curve gets closer and closer to the x-axis (y = 0) as x moves far to the left or right. This behavior helps verify the overall trend of the function at extreme values of x. Horizontal asymptotes provide valuable information about the long-term behavior of the function and are essential for accurate graph identification.

7. Test Additional Points

To further refine our understanding, we can test additional points. For example, let's evaluate f(x) at x = -3 and x = 4:

  • f(-3) = (-3 - 1) / ((-3)^2 - (-3) - 6) = -4 / (9 + 3 - 6) = -4 / 6 = -2/3
  • f(4) = (4 - 1) / ((4)^2 - 4 - 6) = 3 / (16 - 4 - 6) = 3 / 6 = 1/2

So, we have the points (-3, -2/3) and (4, 1/2). These points provide additional anchors for the graph and help verify its shape in specific regions. When assessing potential graphs, check if the graph passes through or is close to these points. Additional points can help distinguish between graphs that may look similar based on asymptotes and intercepts alone. They provide concrete evidence of the function's behavior in certain intervals, reinforcing the accuracy of the identified graph. Testing additional points is a valuable technique for ensuring a comprehensive understanding and precise identification.

8. Match the Features to the Graph

Okay, guys, gathering all the information, we look for a graph that has:

  • Vertical asymptotes at x = -2 and x = 3.
  • A zero at x = 1.
  • A y-intercept at y = 1/6.
  • A horizontal asymptote at y = 0.
  • The correct behavior near the vertical asymptotes (approaching ±∞ from the left and right).
  • Points like (-3, -2/3) and (4, 1/2) fall on the graph.

By systematically analyzing the function and matching these features, you can confidently identify the correct graph. This step-by-step approach ensures a thorough understanding and accurate identification. Remember, it’s all about breaking down the problem and using each piece of information to narrow down the possibilities. Keep practicing, and you'll become a pro at identifying graphs of rational functions!