Analysis Of The Function F(x) = (x^2 - X + 6) / (x^2 - 6x + 8)

This article provides an in-depth analysis of the function $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$. We will explore its key features, including the y-intercept, zeros, asymptotes, and graphical representation. By understanding these aspects, we can gain a comprehensive understanding of the function's behavior and characteristics. So, let's embark on this mathematical journey and uncover the fascinating properties of this rational function.

a) Unveiling the Y-Intercept

To determine the y-intercept of the function $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$, we need to find the value of the function when x = 0. In simpler terms, we are looking for the point where the graph of the function intersects the y-axis. This can be achieved by substituting x = 0 into the function's equation and evaluating the expression. The y-intercept is a crucial point as it provides us with the function's value at the origin, giving us a starting point for understanding its behavior. By identifying the y-intercept, we gain valuable insights into the function's vertical position on the coordinate plane, which aids in visualizing its overall graph and characteristics. The y-intercept often serves as a reference point for analyzing the function's behavior and its relationship to the coordinate axes. Let's proceed by plugging in x = 0 into the equation:

f(0) = \frac{0^2 - 0 + 6}{0^2 - 6(0) + 8} = \frac{6}{8} = \frac{3}{4}$. Therefore, the y-intercept of the function is (0, 3/4). This point signifies where the graph of the function intersects the y-axis, offering us a visual anchor point. The y-intercept is a valuable characteristic of a function, providing insight into its behavior and position on the coordinate plane. It allows us to understand the function's value when x is zero, which can be crucial in various applications and analyses. Recognizing the y-intercept is often the first step in sketching or interpreting the graph of a function. This point serves as a reference, helping us visualize the function's overall shape and behavior in relation to the coordinate axes. In many real-world scenarios, the y-intercept can represent an initial value or a starting condition, making it a practical and meaningful aspect of the function. Understanding the y-intercept is essential for gaining a comprehensive understanding of the function and its applications. ## b) Finding the Zeros of the Function In this section, we will embark on the process of finding the **zeros** of the function $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$. The zeros of a function are the values of x for which the function equals zero, that is, f(x) = 0. In graphical terms, these are the points where the function's graph intersects the x-axis. Finding the zeros is a fundamental aspect of understanding a function's behavior, as they reveal where the function changes its sign or crosses the horizontal axis. To find the zeros of a rational function like this one, we need to determine when the numerator is equal to zero, as the fraction will be zero only if the numerator is zero. So, we need to solve the equation: $x^2 - x + 6 = 0$. This is a quadratic equation, and we can attempt to solve it using the quadratic formula or by factoring. However, let's first calculate the discriminant to determine the nature of the roots. The discriminant, denoted as Δ, is given by the formula: $Δ = b^2 - 4ac$, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our case, a = 1, b = -1, and c = 6. Plugging these values into the discriminant formula, we get: $Δ = (-1)^2 - 4(1)(6) = 1 - 24 = -23$. Since the discriminant is negative (Δ < 0), the quadratic equation has no real roots. This implies that the numerator, x^2 - x + 6, never equals zero for any real value of x. Consequently, the function f(x) has no real zeros. In the context of the graph, this means that the graph of the function does not intersect the x-axis at any point. The absence of real zeros provides valuable information about the function's behavior and its position relative to the x-axis. The fact that the function has no real zeros indicates that its graph remains either entirely above or entirely below the x-axis. This characteristic is crucial in understanding the function's overall shape and its range of values. The absence of zeros can also have practical implications in various applications, depending on the context of the function. In summary, by determining that the function has no real zeros, we gain a significant understanding of its behavior and graphical representation. This information helps us to visualize the function's trajectory and to interpret its properties in a meaningful way. ## c) Determining the Asymptotes The **asymptotes** of a function are lines that the graph of the function approaches as x approaches infinity or negative infinity, or as x approaches certain specific values. Identifying asymptotes is crucial for understanding the end behavior and overall shape of the function's graph. There are typically three types of asymptotes: vertical, horizontal, and oblique (or slant) asymptotes. For the function $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$, we will explore each type of asymptote. ### Vertical Asymptotes Vertical asymptotes occur where the denominator of the rational function is equal to zero, and the numerator is not zero at the same point. To find the vertical asymptotes, we need to solve the equation: $x^2 - 6x + 8 = 0$. This is a quadratic equation that can be factored as: $(x - 2)(x - 4) = 0$. Thus, the solutions are x = 2 and x = 4. Now, we need to check if the numerator is non-zero at these points. When x = 2, the numerator is: $2^2 - 2 + 6 = 4 - 2 + 6 = 8$, which is not zero. When x = 4, the numerator is: $4^2 - 4 + 6 = 16 - 4 + 6 = 18$, which is also not zero. Therefore, the function has vertical asymptotes at x = 2 and x = 4. These vertical lines indicate that the function approaches infinity (or negative infinity) as x gets closer to 2 and 4. Understanding vertical asymptotes is essential for sketching the graph of the function, as they define the boundaries where the function exhibits extreme behavior. Vertical asymptotes also help in determining the domain of the function, as the function is undefined at these points. The presence of vertical asymptotes can significantly impact the function's behavior and its applications in real-world scenarios. In summary, vertical asymptotes are critical features of rational functions, and their identification is a key step in analyzing the function's properties. ### Horizontal Asymptotes Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find horizontal asymptotes, we need to examine the degrees of the numerator and the denominator. In our case, the function is: $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$. The degree of both the numerator and the denominator is 2. When the degrees of the numerator and the denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. Therefore, the horizontal asymptote is: $y = \frac{1}{1} = 1$. This means that as x approaches infinity or negative infinity, the function approaches the line y = 1. The horizontal asymptote provides insight into the function's long-term behavior. It indicates the value that the function tends towards as x becomes very large or very small. Understanding the horizontal asymptote is crucial for sketching the graph of the function and for analyzing its behavior in various applications. The horizontal asymptote also helps in determining the range of the function, as it defines a boundary that the function approaches but may not necessarily cross. The concept of horizontal asymptotes is fundamental in the study of rational functions and their behavior over large intervals of x. In summary, the horizontal asymptote is an important characteristic of the function, providing valuable information about its end behavior and overall trend. ### Oblique (Slant) Asymptotes Oblique or slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In our function, $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$, the degrees of the numerator and the denominator are both 2, which are equal. Since the degree of the numerator is not one greater than the degree of the denominator, there is no oblique asymptote. Oblique asymptotes provide information about the function's behavior as x approaches infinity or negative infinity when the function does not have a horizontal asymptote. The absence of an oblique asymptote can also be a significant characteristic of the function, as it indicates that the function does not exhibit a linear trend in its end behavior beyond a horizontal asymptote. Understanding the conditions for oblique asymptotes is important in the comprehensive analysis of rational functions. In summary, the absence of an oblique asymptote in this case simplifies the analysis of the function's end behavior, as we only need to consider the horizontal asymptote. By analyzing the vertical and horizontal asymptotes, we have gained a thorough understanding of the function's behavior as x approaches specific values and as x approaches infinity. This knowledge is essential for accurately sketching the graph of the function and for interpreting its properties in various contexts. ## d) Graphing the Function To effectively **graph** the function $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$, we will utilize the information we've gathered in the previous sections. This includes the y-intercept, zeros, and asymptotes. The graph will visually represent the function's behavior and characteristics. ### Key Information for Graphing: * **Y-intercept**: (0, 3/4) This is where the graph intersects the y-axis. * **Zeros**: The function has no real zeros, meaning the graph does not intersect the x-axis. * **Vertical Asymptotes**: x = 2 and x = 4. These are vertical lines that the function approaches but never crosses. * **Horizontal Asymptote**: y = 1. This is a horizontal line that the function approaches as x goes to positive or negative infinity. * **Additional Points**: To get a more accurate graph, we can calculate the function's value at a few additional points. For example, we can find f(1), f(3), and f(5) to understand how the function behaves between the asymptotes and beyond them. ### Steps to Sketch the Graph: 1. **Draw the Asymptotes**: Start by drawing the vertical asymptotes (x = 2 and x = 4) and the horizontal asymptote (y = 1) as dashed lines. These lines will serve as guides for the graph. 2. **Plot the Y-intercept**: Plot the point (0, 3/4) on the graph. This is where the graph crosses the y-axis. 3. **Plot Additional Points**: Calculate and plot a few additional points to understand the shape of the graph in different regions. For example: * f(1) = (1 - 1 + 6) / (1 - 6 + 8) = 6 / 3 = 2. Plot the point (1, 2). * f(3) = (9 - 3 + 6) / (9 - 18 + 8) = 12 / -1 = -12. Plot the point (3, -12). * f(5) = (25 - 5 + 6) / (25 - 30 + 8) = 26 / 3 ≈ 8.67. Plot the point (5, 8.67). 4. **Sketch the Graph**: Now, sketch the graph by connecting the points, keeping in mind the asymptotes. The graph will approach the asymptotes but never cross them. Since there are no real zeros, the graph will not cross the x-axis. The graph will have three distinct sections, divided by the vertical asymptotes. * To the left of x = 2, the graph approaches the horizontal asymptote y = 1 from above and passes through the point (0, 3/4) and (1, 2). * Between x = 2 and x = 4, the graph approaches the vertical asymptotes, passing through the point (3, -12). * To the right of x = 4, the graph approaches the horizontal asymptote y = 1 from above and passes through the point (5, 8.67). 5. **Verify the Graph**: Use graphing software or a calculator to verify the graph. This will ensure that the sketch is accurate and captures all the essential features of the function. The graph of the function will show the three sections, each approaching the asymptotes as x approaches specific values or infinity. The absence of zeros indicates that the graph does not cross the x-axis, and the y-intercept gives a starting point for the graph on the y-axis. By plotting additional points, we can refine the shape of the graph and ensure that it accurately represents the function's behavior. Graphing the function provides a comprehensive visual understanding of its properties and behavior. ## Conclusion In conclusion, we have conducted a thorough analysis of the function $f(x) = \frac{x^2 - x + 6}{x^2 - 6x + 8}$. We determined the y-intercept to be (0, 3/4), indicating where the function's graph intersects the y-axis. We found that the function has no real zeros, meaning it does not cross the x-axis. The vertical asymptotes were identified as x = 2 and x = 4, showing where the function approaches infinity. The horizontal asymptote was found to be y = 1, indicating the function's behavior as x approaches infinity. By utilizing this information, we can accurately sketch the graph of the function, which provides a visual representation of its behavior and characteristics. This comprehensive analysis allows for a deeper understanding of the function and its properties, making it easier to apply in various mathematical and real-world contexts.