Function With Zeros At -3, -1, 0, And 6

by ADMIN 40 views

In the realm of mathematics, understanding functions and their properties is crucial. One such property is the zeros of a function, which are the values of x for which the function y equals zero. Identifying a function given its zeros is a fundamental skill in algebra and calculus. This article delves into the process of determining the function that has zeros at -3, -1, 0, and 6. We will explore the relationship between zeros and factors of a polynomial function, and demonstrate how to construct the correct function from the given zeros. This article aims to provide a comprehensive understanding of how to identify a function based on its zeros, ensuring that readers can confidently tackle similar problems in the future. Understanding functions with specific zeros is a fundamental concept in algebra. The zeros of a function are the values of x that make the function equal to zero. These zeros are directly related to the factors of the polynomial that defines the function. For example, if a function has a zero at x = a, then (x - a) is a factor of the function. Conversely, if (x - a) is a factor, then a is a zero of the function. This relationship allows us to construct a polynomial function if we know its zeros. Given the zeros -3, -1, 0, and 6, we can determine the factors of the function. For the zero -3, the factor is (x - (-3)), which simplifies to (x + 3). For the zero -1, the factor is (x - (-1)), which simplifies to (x + 1). For the zero 0, the factor is simply x. For the zero 6, the factor is (x - 6). Therefore, the function can be written as a product of these factors multiplied by a constant a, which represents vertical scaling of the graph. Thus, a general form of the function is y = a * x * (x + 3) * (x + 1) * (x - 6). By recognizing the relationship between zeros and factors, we can efficiently construct a function with specified zeros, further enhancing our understanding of polynomial functions and their graphical representations. In this article, we will explore the process of determining the function that has zeros at -3, -1, 0, and 6. This involves understanding how zeros correspond to factors in a polynomial function. Each zero of a function corresponds to a factor in the function's equation. Specifically, if a function has a zero at x = a, then (x - a) is a factor of the function. Conversely, if (x - a) is a factor, then a is a zero of the function. This direct relationship allows us to construct a polynomial function if we know its zeros. Given the zeros -3, -1, 0, and 6, we can construct the factors of the function. For the zero -3, the factor is (x - (-3)), which simplifies to (x + 3). For the zero -1, the factor is (x - (-1)), which simplifies to (x + 1). For the zero 0, the factor is simply x. For the zero 6, the factor is (x - 6). Therefore, the function can be written as a product of these factors, multiplied by a constant ‘a’, which represents vertical scaling of the graph. Thus, a general form of the function is y = a * x * (x + 3) * (x + 1) * (x - 6). By recognizing this relationship between zeros and factors, we can efficiently construct a function with specified zeros, furthering our understanding of polynomial functions and their graphical representations. We can determine the correct function by examining the zeros and their corresponding factors.

Understanding Zeros and Factors

Zeros of a function are the points where the function intersects the x-axis, meaning the function's value is zero at these points. These zeros are intimately connected with the factors of the polynomial function. Specifically, if a function f(x) has a zero at x = a, then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then a is a zero of the function. This fundamental relationship is the cornerstone of constructing a polynomial function from its zeros. Let's break down this relationship further. Suppose we have a polynomial function, and we know its zeros are x1, x2, x3, ..., xn. Then, we can express the function in factored form as: f(x) = k(x - x1)(x - x2)(x - x3)...(x - xn), where k is a constant. This constant k accounts for vertical stretching or compression of the graph and can be any real number. The factored form makes it clear why these values are called zeros. When x equals any of the zeros (x1, x2, x3, ..., xn), one of the factors becomes zero, making the entire function zero. For example, if x = x1, then the factor (x - x1) becomes zero, and thus f(x) = 0. This understanding allows us to reverse the process as well. If we are given the factored form of a polynomial, we can easily find its zeros by setting each factor equal to zero and solving for x. This is because the product of factors will be zero only if at least one of the factors is zero. This concept is crucial in many areas of mathematics, including solving polynomial equations, graphing functions, and understanding the behavior of polynomial functions. The connection between zeros and factors is not only a computational tool but also a conceptual aid. It helps visualize the relationship between the algebraic representation of a function and its graphical representation. The zeros of a function correspond to the x-intercepts of its graph, providing a direct visual interpretation of the solutions to the equation f(x) = 0. Furthermore, understanding this relationship simplifies many algebraic manipulations. For example, if we need to find a polynomial function that passes through certain points, we can use the zeros to write the function in factored form and then use other points to determine the constant k. This approach is often more efficient than trying to construct the polynomial directly from its standard form (e.g., ax^n + bx^(n-1) + ... + c). Therefore, mastering the relationship between zeros and factors is essential for anyone studying polynomials and functions. It provides a powerful method for both constructing and analyzing polynomial functions, making it a foundational concept in algebra and beyond. This concept is the key to solving the problem at hand. By identifying the zeros, we can construct the factors and, consequently, the function itself.

Analyzing the Given Zeros: -3, -1, 0, and 6

Given the zeros -3, -1, 0, and 6, we can construct the factors of the polynomial function. Each zero corresponds to a factor in the form of (x - zero). For the zero -3, the corresponding factor is (x - (-3)) which simplifies to (x + 3). Similarly, for the zero -1, the factor is (x - (-1)) which simplifies to (x + 1). The zero 0 directly corresponds to the factor x. Finally, for the zero 6, the factor is (x - 6). Now that we have the factors, we can write the function in its factored form. The general form of a polynomial function with these zeros is given by: y = a * x * (x + 1) * (x + 3) * (x - 6), where a is a constant. This constant a can be any real number and represents a vertical scaling factor. If a is positive, the graph opens upwards for large positive x, and if a is negative, the graph opens downwards. The value of a does not affect the zeros of the function, only its vertical stretch or compression. To understand why this factored form works, consider what happens when x takes on one of the zero values. If x = -3, then the factor (x + 3) becomes zero, making the entire product zero, so y = 0. Similarly, if x = -1, the factor (x + 1) becomes zero, and y = 0. If x = 0, the factor x makes the product zero, resulting in y = 0. Lastly, if x = 6, the factor (x - 6) becomes zero, and y = 0. Thus, the function y = a * x * (x + 1) * (x + 3) * (x - 6) indeed has the specified zeros. The polynomial function represented by these factors will be a quartic function (a polynomial of degree 4) because there are four factors involving x. When we multiply these factors together, the highest power of x will be x^4. The shape of the graph of this quartic function will have up to three turning points, which correspond to local maxima and minima. The zeros -3, -1, 0, and 6 mark the points where the graph crosses or touches the x-axis. This provides a clear picture of the function's behavior. The intervals between the zeros determine where the function is positive or negative. For example, between -3 and -1, the function might be either positive or negative, depending on the sign of a and the other factors. By analyzing the zeros and factors, we gain significant insight into the function's behavior and graph. We can sketch a rough graph of the function by marking the zeros on the x-axis and considering the sign of a to determine the overall direction of the graph. This step is essential in identifying the correct function from a set of options. Understanding how these zeros and factors create the function's graph is a key skill in polynomial functions. We can now match the given zeros with the correct factored form, which helps to identify the correct function among the choices.

Evaluating the Answer Choices

Now, let's evaluate the given answer choices to determine which function has zeros at -3, -1, 0, and 6. We will examine each option and identify its zeros by setting each factor equal to zero. This process will allow us to match the answer choice with the specified zeros.

  • Option A: y = (x - 6)(x + 1)(x + 3)

    To find the zeros of this function, we set each factor equal to zero:

    • x - 6 = 0 => x = 6
    • x + 1 = 0 => x = -1
    • x + 3 = 0 => x = -3

    The zeros of Option A are -3, -1, and 6. This option is missing the zero at 0. Thus, Option A is not the correct answer.

  • Option B: y = x(x - 3)(x - 1)(x + 6)

    Setting each factor equal to zero, we get:

    • x = 0
    • x - 3 = 0 => x = 3
    • x - 1 = 0 => x = 1
    • x + 6 = 0 => x = -6

    The zeros of Option B are -6, 0, 1, and 3. These zeros do not match the given zeros of -3, -1, 0, and 6. Therefore, Option B is incorrect.

  • Option C: y = x(x - 6)(x + 1)(x + 3)

    Setting each factor equal to zero:

    • x = 0
    • x - 6 = 0 => x = 6
    • x + 1 = 0 => x = -1
    • x + 3 = 0 => x = -3

    The zeros of Option C are -3, -1, 0, and 6. This matches the given zeros exactly. Therefore, Option C is the correct answer.

  • Option D: y = (x - 3)(x - 1)(x + 6)

    Setting each factor equal to zero:

    • x - 3 = 0 => x = 3
    • x - 1 = 0 => x = 1
    • x + 6 = 0 => x = -6

    The zeros of Option D are -6, 1, and 3. This option does not include the zeros -3, -1, and 0. Thus, Option D is incorrect.

By systematically analyzing each answer choice, we have determined that Option C is the function that has zeros at -3, -1, 0, and 6. This methodical approach ensures accuracy and reinforces the understanding of the relationship between zeros and factors of a polynomial function.

Conclusion: The Correct Function

In conclusion, after analyzing the given answer choices and identifying the zeros of each function, we have determined that the correct function representing zeros at -3, -1, 0, and 6 is:

C. y = x(x - 6)(x + 1)(x + 3)

This function accurately reflects the relationship between zeros and factors in polynomial functions. The zeros -3, -1, 0, and 6 directly correspond to the factors (x + 3), (x + 1), x, and (x - 6), respectively. The product of these factors constructs the polynomial function that we were seeking. Understanding this fundamental concept is crucial for solving a wide range of problems in algebra and calculus. By recognizing the connection between zeros and factors, we can efficiently construct and analyze polynomial functions. This skill is not only valuable for academic success but also for practical applications in various fields such as engineering, physics, and computer science. This exercise demonstrates the importance of a methodical approach to problem-solving. By breaking down the problem into smaller steps, such as identifying the zeros, constructing the factors, and evaluating the answer choices, we can arrive at the correct solution with confidence. This step-by-step process ensures accuracy and minimizes the chances of making errors. Furthermore, understanding the underlying principles allows us to generalize the method and apply it to similar problems. For example, if we were given a different set of zeros, we could use the same process to construct the corresponding function. The ability to generalize problem-solving strategies is a hallmark of mathematical proficiency. In summary, identifying a function with specified zeros involves understanding the relationship between zeros and factors, constructing the factored form of the function, and systematically evaluating the answer choices. This process not only provides the correct answer but also reinforces our understanding of polynomial functions and their properties. The correct option, C, accurately captures the essence of this relationship, making it the definitive solution to the problem. Thus, a strong understanding of zeros and factors significantly enhances one's mathematical toolkit and enables the efficient solution of a variety of mathematical problems. This conclusion underscores the importance of mastering fundamental mathematical concepts and applying them systematically to solve complex problems.