Zeros, Multiplicity, And Graph Behavior Of F(x) = -2(x-1)(x+8)^2

by Alex Johnson 65 views

Understanding the behavior of polynomial functions is a fundamental concept in mathematics. Specifically, finding the zeros of a polynomial, determining their multiplicities, and understanding how the graph interacts with the x-axis at these zeros are crucial skills. In this article, we'll explore these concepts using the example function f(x) = -2(x-1)(x+8)². We will walk through the process step by step to ensure clarity and a comprehensive understanding.

Identifying Zeros of the Polynomial Function

To begin, let's delve into the process of identifying the zeros of the polynomial function. The zeros of a polynomial function are the values of x for which the function f(x) equals zero. In simpler terms, these are the points where the graph of the function intersects or touches the x-axis. For a polynomial in factored form, like our example f(x) = -2(x-1)(x+8)², finding the zeros is straightforward. We set each factor equal to zero and solve for x.

Our function, f(x) = -2(x-1)(x+8)², has two main factors involving x: (x-1) and (x+8)². The constant factor, -2, does not affect the zeros since it never equals zero. Let's examine each significant factor:

  1. (x - 1) = 0
    • Adding 1 to both sides of the equation, we get x = 1. This tells us that x = 1 is a zero of the function. At this point, the graph of the function will interact with the x-axis.
  2. (x + 8)² = 0
    • Taking the square root of both sides, we have x + 8 = 0. Subtracting 8 from both sides gives us x = -8. Thus, x = -8 is another zero of our function. This zero is particularly interesting because the factor (x + 8) is squared, which will influence how the graph behaves at this point.

So, we have found two zeros for the polynomial function f(x) = -2(x-1)(x+8)²: x = 1 and x = -8. These points are critical in understanding the graph's behavior. The next step is to determine the multiplicity of each zero, which will give us further insight into how the graph interacts with the x-axis at these points. By understanding the zeros, we lay the groundwork for a comprehensive analysis of the polynomial function's graph.

Determining the Multiplicity of Each Zero

Next, let's focus on determining the multiplicity of each zero. The multiplicity of a zero is the number of times its corresponding factor appears in the factored form of the polynomial. This concept is crucial because it dictates how the graph of the polynomial behaves at each zero, specifically whether it crosses the x-axis or merely touches it and turns around. For our example function, f(x) = -2(x-1)(x+8)², we've already identified the zeros as x = 1 and x = -8. Now, let's determine their multiplicities.

  1. Zero at x = 1
    • The zero x = 1 comes from the factor (x - 1). This factor appears only once in the polynomial. Therefore, the multiplicity of the zero x = 1 is 1. When a zero has a multiplicity of 1, the graph of the polynomial will cross the x-axis at that point. This is a straightforward intersection, meaning the function's value changes sign (from negative to positive or vice versa) as it passes through x = 1.
  2. Zero at x = -8
    • The zero x = -8 originates from the factor (x + 8)². Notice that this factor is squared, which means it appears twice. Hence, the multiplicity of the zero x = -8 is 2. A zero with a multiplicity of 2 indicates that the graph will touch the x-axis at this point but will not cross it. Instead, the graph will turn around, behaving like a quadratic function (parabola) near this zero. This is because the squared factor ensures that the function's sign does not change as it approaches and leaves x = -8.

In summary, we've established that the zero x = 1 has a multiplicity of 1, and the zero x = -8 has a multiplicity of 2. Understanding these multiplicities is essential for sketching or interpreting the graph of the polynomial function. The multiplicity tells us not only where the graph touches the x-axis but also how it behaves at those points – whether it crosses or simply touches and turns around. This information, combined with other characteristics such as the leading coefficient and the degree of the polynomial, provides a comprehensive picture of the function's graphical representation.

Graph Behavior at Each Zero and Interaction with the X-axis

Understanding the graph behavior at each zero and its interaction with the x-axis is the final piece in our analysis of the polynomial function f(x) = -2(x-1)(x+8)². We've identified the zeros and their multiplicities, which now allows us to describe how the graph behaves at these critical points. The way the graph interacts with the x-axis at each zero is directly determined by the multiplicity of that zero.

  1. Behavior at x = 1

    • As we determined earlier, the zero x = 1 has a multiplicity of 1. This means that the graph of the function will cross the x-axis at x = 1. When a graph crosses the x-axis, the function's value changes sign. In other words, the graph goes from being below the x-axis (where f(x) is negative) to above the x-axis (where f(x) is positive), or vice versa, as it passes through x = 1. The crossing is a clear indication that the function transitions from negative to positive or positive to negative values at this point. Graphically, this looks like a direct intersection of the curve with the x-axis.

  2. Behavior at x = -8

    • The zero x = -8 has a multiplicity of 2. This implies that the graph touches the x-axis at x = -8 but does not cross it. Instead, the graph turns around at this point. This behavior is often described as the graph being tangent to the x-axis at x = -8. The function's value does not change sign as it approaches and leaves x = -8. If the graph is above the x-axis just before x = -8, it will touch the x-axis and then turn back upward. Conversely, if it is below the x-axis, it will touch and turn back downward. This touching and turning behavior is characteristic of zeros with even multiplicities, such as 2, 4, 6, and so on.

To summarize, at x = 1, the graph of f(x) crosses the x-axis, indicating a change in the sign of the function. At x = -8, the graph touches the x-axis and turns around, meaning the function maintains its sign around this point. Understanding these behaviors allows us to sketch a more accurate representation of the polynomial function's graph. By combining the information about zeros, multiplicities, and graph behavior, we gain a comprehensive understanding of how the polynomial function behaves across its domain. This knowledge is invaluable for various applications in mathematics, science, and engineering, where polynomial functions are used to model a wide range of phenomena.

In conclusion, by finding the zeros of the polynomial function f(x) = -2(x-1)(x+8)², determining their multiplicities, and analyzing the graph's behavior at each zero, we have gained a thorough understanding of this function. The zero x = 1 with multiplicity 1 results in the graph crossing the x-axis, while the zero x = -8 with multiplicity 2 causes the graph to touch the x-axis and turn around. This process highlights the importance of these concepts in analyzing and graphing polynomial functions. For further reading on polynomial functions, you might find helpful resources on websites like Khan Academy's Algebra II section.