Graphing F(x) = 2/(x-1) + 4 A Step By Step Guide

by ADMIN 49 views

Which graph correctly represents the function f(x) = 2/(x-1) + 4?

Understanding the graphical representation of functions is a cornerstone of mathematics. Among the myriad of functions, rational functions, characterized by a variable in the denominator, hold a special intrigue due to their unique asymptotic behavior. This article delves into the intricacies of graphing the rational function f(x) = 2/(x-1) + 4, providing a step-by-step guide to understanding its key features and ultimately identifying its correct graphical representation.

Dissecting the Function: Key Characteristics

Before we even attempt to sketch the graph, it's crucial to understand the fundamental properties of the function f(x) = 2/(x-1) + 4. This function is a transformation of the basic rational function 1/x, and recognizing these transformations is key to visualizing the graph. Let's break down the components:

  • The Parent Function: 1/x: The bedrock of our function is the simple hyperbola 1/x. This function has a vertical asymptote at x = 0 (since division by zero is undefined) and a horizontal asymptote at y = 0 (as x approaches infinity, 1/x approaches zero). Its graph occupies the first and third quadrants.
  • The Vertical Stretch: 2/(x-1): The numerator '2' introduces a vertical stretch by a factor of 2. This means that the graph of 2/x will be twice as far from the x-axis as the graph of 1/x for the same x-value. However, we have 2/(x-1), indicating a horizontal shift as well.
  • The Horizontal Shift: 2/(x-1): The term (x-1) in the denominator signifies a horizontal shift of the graph 1 unit to the right. This is because the function becomes undefined when x = 1, creating a vertical asymptote at x = 1 instead of x = 0. This horizontal shift is a crucial element in determining the correct graph.
  • The Vertical Shift: + 4: The '+ 4' at the end of the function represents a vertical shift of the entire graph upwards by 4 units. This means the horizontal asymptote, which was originally at y = 0, will now be at y = 4. Understanding this vertical shift is essential for accurately identifying the graph.

In essence, f(x) = 2/(x-1) + 4 is a hyperbola derived from 1/x, stretched vertically by a factor of 2, shifted 1 unit to the right, and 4 units upwards. These transformations dictate the asymptotes and the overall shape of the graph.

Identifying Asymptotes: The Guiding Lines

Asymptotes are imaginary lines that a graph approaches but never touches. They act as crucial guidelines for sketching the graph of a rational function. As we've already discussed, f(x) = 2/(x-1) + 4 has both vertical and horizontal asymptotes.

  • Vertical Asymptote: The vertical asymptote occurs where the denominator of the rational expression equals zero. In this case, x - 1 = 0 implies x = 1. Thus, the vertical asymptote is the vertical line x = 1. This asymptote dictates the function's behavior as x approaches 1 from either side.
  • Horizontal Asymptote: The horizontal asymptote describes the function's behavior as x approaches positive or negative infinity. To find it, we consider the limit of the function as x approaches infinity. In this case, as x becomes very large, the term 2/(x-1) approaches zero, and the function approaches 4. Therefore, the horizontal asymptote is the horizontal line y = 4. This asymptote shows the function's long-term trend as x moves away from the origin.

Knowing the asymptotes allows us to narrow down the possible graphs significantly. The correct graph must approach these lines without ever crossing them.

Plotting Key Points: Filling in the Gaps

While asymptotes provide the framework, plotting a few key points helps refine the graph and confirm its shape. We can strategically choose x-values to get a good sense of the function's behavior.

  • Consider values near the vertical asymptote: Let's evaluate f(x) for x values slightly less than and greater than 1. For example:
    • x = 0: f(0) = 2/(0-1) + 4 = -2 + 4 = 2. This gives us the point (0, 2).
    • x = 2: f(2) = 2/(2-1) + 4 = 2 + 4 = 6. This gives us the point (2, 6).
  • Consider the y-intercept: The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. We already calculated f(0) = 2, so the y-intercept is (0, 2).
  • Consider the x-intercept (if any): The x-intercept is the point where the graph crosses the x-axis, which occurs when f(x) = 0. Let's solve for x:
    • 0 = 2/(x-1) + 4
    • -4 = 2/(x-1)
    • -4(x-1) = 2
    • -4x + 4 = 2
    • -4x = -2
    • x = 1/2
    • So the x-intercept is (1/2, 0).

By plotting these points – (0, 2), (2, 6), and (1/2, 0) – along with the asymptotes, the shape of the hyperbola becomes clear. We can see how the graph approaches the asymptotes and curves through the plotted points.

The Hyperbola's Shape: Putting It All Together

With the asymptotes and key points in hand, we can now confidently sketch the graph of f(x) = 2/(x-1) + 4. The graph will consist of two branches, characteristic of a hyperbola:

  • One branch will be in the first quadrant relative to the intersection of the asymptotes: This branch will approach the asymptotes x = 1 and y = 4 as x increases and x approaches 1 from the right. We know the point (2, 6) lies on this branch.
  • The other branch will be in the third quadrant relative to the intersection of the asymptotes: This branch will approach the asymptotes x = 1 and y = 4 as x decreases and x approaches 1 from the left. The points (0, 2) and (1/2, 0) lie on this branch.

The vertical stretch factor of 2 will make the hyperbola slightly “narrower” compared to the basic 1/x hyperbola. The horizontal and vertical shifts place the center of the hyperbola at the point (1, 4), which is the intersection of the asymptotes.

Common Mistakes to Avoid: Pitfalls in Graphing

Graphing rational functions can be tricky, and several common mistakes can lead to incorrect representations. Being aware of these pitfalls can help avoid them.

  • Misinterpreting Shifts: The most common mistake is misinterpreting the direction of horizontal shifts. Remember that (x - a) indicates a shift to the right by a units, while (x + a) indicates a shift to the left by a units. Carefully analyze the term in the denominator to avoid this error.
  • Ignoring the Vertical Stretch: Forgetting the vertical stretch factor can lead to a graph with the correct asymptotes but an incorrect shape. The coefficient in the numerator affects how “steep” the hyperbola is. Always consider the vertical stretch when sketching the graph.
  • Incorrectly Identifying Asymptotes: Failing to correctly determine the vertical and horizontal asymptotes will result in a fundamentally flawed graph. Double-check your calculations and reasoning for finding these key lines. Asymptotes are the foundation of the graph; get them right!.
  • Assuming the Graph Crosses Asymptotes: A key property of asymptotes is that the graph approaches them but never crosses them (except for some more complex rational functions, which are beyond the scope of this discussion). If your graph crosses an asymptote, it's a clear sign of an error.

Conclusion: Mastering the Art of Graphing Rational Functions

Graphing the rational function f(x) = 2/(x-1) + 4 requires a systematic approach, combining an understanding of transformations, asymptotes, and key points. By dissecting the function, identifying its key characteristics, plotting strategic points, and avoiding common mistakes, you can confidently sketch its graph and understand its behavior. This process not only helps visualize the function but also deepens your understanding of rational functions in general. Mastering the art of graphing rational functions opens doors to a deeper understanding of calculus, engineering, and other fields where these functions play a crucial role. The journey from function to graph is a rewarding one, bridging the gap between abstract equations and visual representations.