Analyzing The Rational Function F(x) = 3(x-2)(x+4) / (x^2 - 4x - 21)

In this article, we will thoroughly analyze the rational function f(x) = 3(x-2)(x+4) / (x^2 - 4x - 21). Our analysis will encompass determining key features such as the vertical intercept, horizontal intercepts, vertical asymptotes, and horizontal asymptotes. Understanding these characteristics is crucial for sketching the graph of the function and comprehending its behavior. Let's delve into each aspect step by step.

a. Determining the Vertical Intercept (x, y)

The vertical intercept, also known as the y-intercept, is the point where the graph of the function intersects the y-axis. This occurs when the x-coordinate is equal to zero. To find the vertical intercept, we substitute x = 0 into the function and evaluate f(0). This process allows us to determine the y-coordinate of the point where the graph crosses the vertical axis. The y-intercept is a fundamental characteristic of a function, providing a direct visual reference on the coordinate plane.

Let's substitute x = 0 into the given rational function:

f(0) = 3(0-2)(0+4) / (0^2 - 4(0) - 21)

f(0) = 3(-2)(4) / (-21)

f(0) = -24 / -21

f(0) = 8/7

Therefore, the vertical intercept is (0, 8/7). This point indicates where the graph of the function crosses the y-axis, providing a crucial reference point for sketching the graph and understanding the function's behavior near the y-axis. The vertical intercept, alongside other key features, contributes to a comprehensive understanding of the function's graphical representation and its mathematical properties. Identifying the vertical intercept is a standard procedure in function analysis, often serving as a starting point for further exploration of the function's characteristics.

b. Determining the Horizontal Intercepts (x, y)

The horizontal intercepts, also known as the x-intercepts or roots, are the points where the graph of the function intersects the x-axis. These points occur when the y-coordinate, or f(x), is equal to zero. To find the horizontal intercepts, we set the function f(x) equal to zero and solve for x. This involves finding the values of x that make the numerator of the rational function equal to zero, as the denominator cannot be zero for the function to be defined. The x-intercepts are crucial for understanding the function's behavior as it crosses the horizontal axis and are essential for accurately sketching the graph.

To find the horizontal intercepts, we set f(x) = 0:

0 = 3(x-2)(x+4) / (x^2 - 4x - 21)

For a fraction to be zero, the numerator must be zero. Thus,

3(x-2)(x+4) = 0

This equation is satisfied when either (x-2) = 0 or (x+4) = 0.

Solving for x, we get:

x - 2 = 0 => x = 2

x + 4 = 0 => x = -4

Therefore, the horizontal intercepts are (2, 0) and (-4, 0). These points represent where the graph of the function crosses the x-axis, providing valuable information about the function's roots and behavior near the horizontal axis. The x-intercepts, together with other key features like asymptotes and intercepts, help to paint a complete picture of the function's graph and its mathematical properties. Determining the x-intercepts is a standard step in analyzing rational functions and is critical for both graphing and understanding the function's solutions.

c. Finding the Vertical Asymptotes

Vertical asymptotes are vertical lines that the graph of the function approaches but never touches. They occur at values of x where the denominator of the rational function is equal to zero, and the numerator is not zero. These asymptotes indicate points where the function becomes undefined, leading to dramatic changes in its behavior. Identifying vertical asymptotes is crucial for understanding the function's domain and accurately sketching its graph. The vertical asymptotes effectively divide the graph into distinct sections, guiding the overall shape and direction of the curve.

To find the vertical asymptotes, we need to find the values of x for which the denominator is zero:

x^2 - 4x - 21 = 0

We can factor the quadratic expression:

(x - 7)(x + 3) = 0

This equation is satisfied when either (x - 7) = 0 or (x + 3) = 0.

Solving for x, we get:

x - 7 = 0 => x = 7

x + 3 = 0 => x = -3

We must check that the numerator is not also zero at these values. The numerator is 3(x-2)(x+4). At x=7, the numerator is 3(7-2)(7+4) = 3(5)(11) = 165, which is not zero. At x=-3, the numerator is 3(-3-2)(-3+4) = 3(-5)(1) = -15, which is not zero.

Therefore, the vertical asymptotes are the lines x = 7 and x = -3. These vertical lines indicate where the function approaches infinity or negative infinity, shaping the graph's overall structure and behavior. The presence and location of vertical asymptotes are fundamental characteristics of rational functions, essential for both analysis and graphical representation. They highlight the function's discontinuities and provide critical guidance for sketching the curve.

d. Finding the Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x approaches positive or negative infinity. They provide information about the function's long-term behavior and how it stabilizes as x moves towards extreme values. The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator of the rational function. Identifying horizontal asymptotes is essential for understanding the function's behavior at the ends of the graph and for accurately sketching the function's overall shape.

To find the horizontal asymptote, we compare the degrees of the numerator and the denominator.

The given function is:

f(x) = 3(x-2)(x+4) / (x^2 - 4x - 21)

First, we expand the numerator:

3(x-2)(x+4) = 3(x^2 + 4x - 2x - 8) = 3(x^2 + 2x - 8) = 3x^2 + 6x - 24

So, the function becomes:

f(x) = (3x^2 + 6x - 24) / (x^2 - 4x - 21)

The degree of the numerator is 2 (the highest power of x is x^2), and the degree of the denominator is also 2. When the degrees of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1.

Therefore, the horizontal asymptote is y = 3/1 = 3.

The horizontal asymptote is the line y = 3. This line indicates the value that the function approaches as x tends to positive or negative infinity, providing crucial information about the function's long-term behavior. The horizontal asymptote helps to define the overall shape of the graph and is essential for accurately sketching the function. Understanding the horizontal asymptote is a key component in the comprehensive analysis of rational functions, helping to predict the function's behavior over a wide range of x values.

In summary, by analyzing the vertical intercept, horizontal intercepts, vertical asymptotes, and horizontal asymptotes, we have gained a comprehensive understanding of the behavior of the rational function f(x) = 3(x-2)(x+4) / (x^2 - 4x - 21). This analysis provides a solid foundation for accurately sketching the graph of the function and predicting its behavior under various conditions.