Analyzing And Sketching F(x) = (x+4)^2(x-2) / 2(x-1)^2(x-3) A Comprehensive Guide

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This article delves into a comprehensive analysis of the rational function f(x) = (x+4)^2(x-2) / 2(x-1)^2(x-3). We will explore its key characteristics, including intercepts, asymptotes, and intervals of increase and decrease, to ultimately sketch its graph. Understanding the behavior of rational functions is crucial in various fields, from calculus to engineering, as they model numerous real-world phenomena. Let's embark on this journey to dissect this function and visualize its graphical representation.

1. Identifying Intercepts

The intercepts of a function are the points where the graph crosses the x and y axes. Finding these points is a fundamental step in understanding the function's behavior and accurately sketching its graph. Intercepts provide key anchor points that help define the overall shape and position of the function.

1.1. X-Intercepts: Where f(x) = 0

To find the x-intercepts, we need to determine the values of x for which f(x) = 0. For a rational function, this occurs when the numerator is equal to zero. Therefore, we set the numerator of our function to zero and solve for x:

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

This equation has two solutions:

  • (x+4)^2 = 0 implies x = -4. Since the factor is squared, this root has a multiplicity of 2. This means the graph will touch the x-axis at x = -4 but not cross it.
  • x-2 = 0 implies x = 2. This root has a multiplicity of 1, so the graph will cross the x-axis at x = 2.

Therefore, the x-intercepts are (-4, 0) and (2, 0). The multiplicity of the root at x = -4 indicates a turning point on the x-axis, which is crucial for accurate sketching.

1.2. Y-Intercept: Where x = 0

To find the y-intercept, we need to determine the value of f(0). This means substituting x = 0 into the function:

f(0) = (0+4)^2(0-2) / 2(0-1)^2(0-3)

f(0) = (16)(-2) / 2(1)(-3)

f(0) = -32 / -6

f(0) = 16/3

Therefore, the y-intercept is (0, 16/3). This point provides another key reference for positioning the graph on the coordinate plane.

2. Determining Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. They provide crucial information about the function's behavior as x approaches infinity or certain finite values. Identifying asymptotes is essential for understanding the function's long-term behavior and accurately sketching its graph. There are three types of asymptotes we need to consider: vertical, horizontal, and oblique.

2.1. Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, but the numerator does not. These points represent values of x where the function becomes undefined, causing the graph to approach infinity or negative infinity. To find the vertical asymptotes, we set the denominator of our function to zero and solve for x:

2(x-1)^2(x-3) = 0

This equation has two solutions:

  • (x-1)^2 = 0 implies x = 1. Since the factor is squared, this asymptote has a multiplicity of 2, which will affect the behavior of the graph near this asymptote.
  • x-3 = 0 implies x = 3. This asymptote has a multiplicity of 1.

Therefore, the vertical asymptotes are the lines x = 1 and x = 3. The even multiplicity at x = 1 suggests that the graph will approach the asymptote from the same direction on both sides, while the odd multiplicity at x = 3 suggests opposite directions.

2.2. Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To determine the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. In our function:

f(x) = (x+4)^2(x-2) / 2(x-1)^2(x-3)

The numerator expands to a polynomial of degree 3: (x^2 + 8x + 16)(x - 2) = x^3 + 6x^2 - 32 The denominator expands to a polynomial of degree 3: 2(x^2 - 2x + 1)(x - 3) = 2(x^3 - 5x^2 + 7x - 3) = 2x^3 - 10x^2 + 14x - 6

Since the degrees of the numerator and denominator are equal (both are degree 3), the horizontal asymptote is the ratio of the leading coefficients:

y = (leading coefficient of numerator) / (leading coefficient of denominator) = 1 / 2

Therefore, the horizontal asymptote is the line y = 1/2. This asymptote provides a boundary for the function's long-term behavior as x approaches infinity.

2.3. Oblique Asymptotes

Oblique asymptotes (also called slant asymptotes) occur when the degree of the numerator is exactly one greater than the degree of the denominator. In our case, the degrees of the numerator and denominator are equal, so there is no oblique asymptote. This simplifies our analysis as we only need to consider horizontal behavior.

3. Analyzing Intervals of Increase and Decrease

Understanding where a function is increasing or decreasing is vital for sketching its graph accurately. This involves finding the function's critical points (where the derivative is zero or undefined) and analyzing the sign of the derivative in the intervals defined by these points. However, finding the derivative of our function using the quotient rule and simplifying it can be quite complex. For the purpose of this article, we will focus on the information we have gathered so far (intercepts and asymptotes) and use test points within the intervals defined by the x-intercepts and vertical asymptotes to determine the general trend of the function.

3.1. Test Intervals

The x-intercepts (-4 and 2) and vertical asymptotes (1 and 3) divide the x-axis into the following intervals:

  • (-∞, -4)
  • (-4, 1)
  • (1, 2)
  • (2, 3)
  • (3, ∞)

We will select a test point within each interval and evaluate f(x) to determine whether the function is positive (above the x-axis) or negative (below the x-axis) in that interval. This will give us a general sense of whether the function is increasing or decreasing.

3.2. Test Points and Function Values

Let's choose the following test points:

  • x = -5 in the interval (-∞, -4)
  • x = 0 in the interval (-4, 1)
  • x = 1.5 in the interval (1, 2)
  • x = 2.5 in the interval (2, 3)
  • x = 4 in the interval (3, ∞)

Now, we evaluate f(x) at each test point:

  • f(-5) = ((-5)+4)^2((-5)-2) / 2((-5)-1)^2((-5)-3) = (1)(-7) / 2(36)(-8) = -7 / -576 > 0 (Positive)
  • f(0) = (0+4)^2(0-2) / 2(0-1)^2(0-3) = (16)(-2) / 2(1)(-3) = -32 / -6 > 0 (Positive)
  • f(1.5) = ((1.5)+4)^2((1.5)-2) / 2((1.5)-1)^2((1.5)-3) = (30.25)(-0.5) / 2(0.25)(-1.5) = -15.125 / -0.75 > 0 (Positive)
  • f(2.5) = ((2.5)+4)^2((2.5)-2) / 2((2.5)-1)^2((2.5)-3) = (42.25)(0.5) / 2(2.25)(-0.5) = 21.125 / -2.25 < 0 (Negative)
  • f(4) = (4+4)^2(4-2) / 2(4-1)^2(4-3) = (64)(2) / 2(9)(1) = 128 / 18 > 0 (Positive)

3.3. Intervals and Function Sign

Based on the test points, we can summarize the sign of f(x) in each interval:

  • (-∞, -4): f(x) > 0
  • (-4, 1): f(x) > 0
  • (1, 2): f(x) > 0
  • (2, 3): f(x) < 0
  • (3, ∞): f(x) > 0

This information, along with the intercepts and asymptotes, provides a solid foundation for sketching the graph. We know where the function is above or below the x-axis, and we know its behavior near the asymptotes.

4. Sketching the Graph

Now, we can use all the information we've gathered to sketch the graph of f(x):

  1. Plot the intercepts: (-4, 0), (2, 0), and (0, 16/3).
  2. Draw the asymptotes: x = 1, x = 3, and y = 1/2. Use dashed lines to represent asymptotes.
  3. Consider the behavior near the vertical asymptotes:
    • At x = 1 (multiplicity 2), the graph will approach the asymptote in the same direction on both sides. Since the function is positive in both (-4, 1) and (1, 2), the graph will approach positive infinity on both sides of x = 1.
    • At x = 3 (multiplicity 1), the graph will approach the asymptote in opposite directions. Since the function is negative in (2, 3) and positive in (3, ∞), the graph will approach negative infinity as x approaches 3 from the left and positive infinity as x approaches 3 from the right.
  4. Consider the behavior near the horizontal asymptote: As x approaches positive or negative infinity, the graph will approach the line y = 1/2. The test points and intervals give us the general trend of whether the function approaches the asymptote from above or below.
  5. Connect the points and asymptotes smoothly, considering the intervals where the function is positive or negative and the multiplicity of the x-intercept at x = -4. The graph should touch the x-axis at x = -4 and cross it at x = 2.

By carefully considering all these factors, we can create a detailed and accurate sketch of the rational function f(x) = (x+4)^2(x-2) / 2(x-1)^2(x-3).

5. Conclusion

Analyzing rational functions involves a systematic approach of identifying intercepts, asymptotes, and intervals of increase and decrease. By meticulously examining these characteristics, we can gain a deep understanding of the function's behavior and accurately sketch its graph. The function f(x) = (x+4)^2(x-2) / 2(x-1)^2(x-3) serves as an excellent example of how these techniques can be applied to visualize complex functions. This process is not only valuable in mathematics but also in various fields that utilize mathematical modeling to understand real-world phenomena.