Maximum Turning Points For F(x) = (x+7)(x-1)(7x+5)
Introduction to Polynomial Functions and Turning Points
When delving into the world of polynomial functions, understanding their graphical behavior is paramount. A key aspect of this behavior is the presence of turning points. These points, also known as local maxima and minima, signify where the function's direction changes, transitioning from increasing to decreasing or vice versa. Determining the maximum possible number of turning points for a given polynomial function is a fundamental concept in calculus and pre-calculus mathematics. In this article, we will explore how to find the maximum possible number of turning points for a polynomial function, using the function f(x) = (x+7)(x-1)(7x+5) as a prime example. We will dissect the relationship between the degree of a polynomial and its turning points, providing a step-by-step approach to solving this type of problem. Understanding turning points is crucial for sketching accurate graphs of polynomial functions and for solving optimization problems in various fields, including engineering and economics. By mastering this concept, you will gain a deeper insight into the nature of polynomial functions and their applications in the real world. The ability to identify and analyze turning points allows us to predict the behavior of complex systems, making it an indispensable tool in mathematical modeling and analysis. Furthermore, this understanding lays the foundation for more advanced topics in calculus, such as finding critical points and using derivatives to analyze function behavior. This article will not only provide a solution to the specific problem but also equip you with the knowledge and skills to tackle similar problems confidently.
Understanding the Degree of a Polynomial Function
The degree of a polynomial function is the highest power of the variable in the polynomial. It is a critical piece of information because it directly relates to the maximum number of turning points the function can have. The degree of a polynomial is determined by expanding the function and identifying the highest power of the variable. For the given function, f(x) = (x+7)(x-1)(7x+5), we first need to expand the expression to find the degree. By multiplying out the factors, we get:
f(x) = (x+7)(x-1)(7x+5)
f(x) = (x^2 + 6x - 7)(7x + 5)
f(x) = 7x^3 + 5x^2 + 42x^2 + 30x - 49x - 35
f(x) = 7x^3 + 47x^2 - 19x - 35
From the expanded form, it is evident that the highest power of x is 3. Therefore, the degree of the polynomial f(x) is 3. The degree of a polynomial is not just a number; it provides valuable insights into the function's behavior. Specifically, it tells us about the end behavior of the graph, the maximum number of roots, and, importantly, the maximum number of turning points. A polynomial of degree n can have at most n-1 turning points. This relationship is a fundamental theorem in calculus and is essential for understanding the shape and characteristics of polynomial graphs. Recognizing the degree is the first step in determining the potential turning points and provides a framework for further analysis. In our example, the degree being 3 immediately suggests that the maximum number of turning points will be 2. This initial understanding is crucial for narrowing down the possible answers and guiding our subsequent calculations and interpretations.
The Relationship Between Degree and Turning Points
The relationship between the degree of a polynomial and its turning points is a cornerstone concept in understanding polynomial functions. A fundamental theorem in calculus states that a polynomial of degree n can have at most n - 1 turning points. This is because the turning points correspond to the local maxima and minima of the function, which occur where the derivative of the function is equal to zero. The derivative of a polynomial of degree n is a polynomial of degree n - 1, and a polynomial of degree n - 1 can have at most n - 1 real roots. These roots correspond to the critical points of the original function, which are potential locations for turning points.
For our function, f(x) = (x+7)(x-1)(7x+5), we have already determined that the degree is 3. Applying the theorem, the maximum number of turning points is 3 - 1 = 2. This means that the graph of the function can have at most two points where it changes direction, either from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum). Understanding this relationship is crucial because it allows us to predict the general shape of the graph without actually plotting it. A cubic function (degree 3) can have either two turning points or none. It cannot have more than two, and it must have an even number (including zero) of turning points. This is because the function must change direction an even number of times to return to its original direction as x approaches positive or negative infinity. In practical terms, this relationship is invaluable for sketching graphs, solving optimization problems, and analyzing the behavior of systems modeled by polynomial functions. It provides a quick and efficient way to estimate the complexity of a polynomial function's graph and identify key features such as local maxima and minima. This understanding forms the basis for more advanced techniques in calculus, such as finding critical points and using the first and second derivative tests to determine the nature of these points.
Determining the Maximum Turning Points for f(x) = (x+7)(x-1)(7x+5)
Now that we have established the relationship between the degree of a polynomial and its maximum number of turning points, we can apply this knowledge to our specific function, f(x) = (x+7)(x-1)(7x+5). As we previously determined, the degree of this polynomial is 3. According to the theorem, a polynomial of degree n can have at most n - 1 turning points. Therefore, the maximum number of turning points for our function f(x) is 3 - 1 = 2. This means the graph of the function can have, at most, two points where it changes direction. These turning points represent local maxima and minima, where the function transitions from increasing to decreasing or vice versa. To further illustrate this, consider the graph of a general cubic function. It typically has a shape that resembles an 'S' or a backward 'S', with two distinct turning points. However, it is also possible for a cubic function to have no turning points if its derivative has no real roots. In the case of f(x) = (x+7)(x-1)(7x+5), the presence of three distinct linear factors suggests that the function will indeed have two turning points. This is because each factor corresponds to a root of the polynomial, and the function will change sign at each root. The changes in sign, combined with the smooth, continuous nature of polynomial functions, necessitate the presence of turning points between the roots. It's important to note that while the maximum number of turning points is 2, the actual number could be less. To determine the exact number and location of the turning points, we would need to find the derivative of the function and analyze its critical points. However, the question asks for the maximum possible number, which we have confidently determined to be 2 based on the degree of the polynomial. This approach highlights the power of understanding fundamental mathematical principles and applying them to solve specific problems. By knowing the relationship between degree and turning points, we can quickly and efficiently determine the maximum possible number of turning points without resorting to more complex calculations.
Analyzing the Options: A, B, C, and D
Having determined that the maximum possible number of turning points for the graph of the function f(x) = (x+7)(x-1)(7x+5) is 2, we can now analyze the given options to select the correct answer. The options provided are:
A) 7 B) 2 C) 0 D) 3
Based on our previous analysis, we know that a polynomial of degree 3 can have at most 2 turning points. Therefore, option A (7) and option D (3) are incorrect because they exceed this maximum. Option C (0) is also a possibility for a cubic function, as it could have no turning points if its derivative has no real roots. However, the question asks for the maximum possible number of turning points. Option B (2) directly corresponds to the maximum number of turning points we calculated based on the degree of the polynomial. Thus, option B is the correct answer. This process of elimination and comparison highlights the importance of understanding the underlying mathematical principles. By knowing the relationship between the degree of a polynomial and its turning points, we can efficiently narrow down the options and select the correct answer. In a test-taking scenario, this approach can save valuable time and ensure accuracy. Furthermore, it demonstrates a deeper understanding of the concept rather than simply guessing or relying on memorization. The ability to justify the answer based on mathematical reasoning is a crucial skill in problem-solving and critical thinking. By analyzing each option in the context of the known mathematical principles, we can confidently arrive at the correct solution and demonstrate a comprehensive understanding of the topic.
Conclusion: The Maximum Possible Number of Turning Points
In conclusion, by understanding the relationship between the degree of a polynomial and its turning points, we have successfully determined the maximum possible number of turning points for the graph of the function f(x) = (x+7)(x-1)(7x+5). We began by expanding the function to identify its degree, which we found to be 3. We then applied the theorem that a polynomial of degree n can have at most n - 1 turning points. This led us to the conclusion that the maximum number of turning points for f(x) is 3 - 1 = 2. Finally, we analyzed the given options and confidently selected option B (2) as the correct answer. This process demonstrates the importance of a systematic approach to problem-solving in mathematics. By breaking down the problem into smaller, manageable steps, we were able to apply relevant mathematical principles and arrive at the solution. This approach not only provides the correct answer but also deepens our understanding of the underlying concepts. The concept of turning points is fundamental to understanding the behavior of polynomial functions and has wide-ranging applications in various fields. From sketching graphs to solving optimization problems, the ability to determine the maximum number of turning points is a valuable skill. Moreover, this problem highlights the connection between algebraic expressions and their graphical representations. By understanding the relationship between the degree of a polynomial and its turning points, we can gain insights into the shape and characteristics of the graph without actually plotting it. This connection between algebra and geometry is a central theme in mathematics and is essential for developing a comprehensive understanding of mathematical concepts. The ability to analyze and interpret mathematical functions is a key skill for success in higher-level mathematics and related fields.
Final Answer: B) 2
