Evaluate F(x) = 2x³ + X² - 13x + 6 At X = -9: A Comprehensive Guide

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Introduction

In mathematics, evaluating a function at a specific point is a fundamental operation. It involves substituting the given value into the function's variable and simplifying the expression to obtain the function's output at that point. This process is crucial in various mathematical applications, including graphing functions, solving equations, and analyzing function behavior. In this article, we will delve into evaluating the polynomial function f(x) = 2x³ + x² - 13x + 6 when x = -9. This exercise will not only demonstrate the mechanics of function evaluation but also highlight the importance of careful arithmetic and algebraic manipulation. Understanding how to evaluate functions is essential for students and professionals alike, as it forms the backbone of more advanced mathematical concepts and problem-solving techniques.

Understanding Function Evaluation

Function evaluation is a cornerstone of mathematical analysis. At its core, function evaluation is the process of finding the value of a function for a specific input. A function, denoted as f(x), is a rule that assigns a unique output value to each input value x. The expression f(x) represents the output when the input is x. To evaluate a function at a specific point, say x = a, we substitute a for x in the function's expression and simplify. This process yields f(a), which is the value of the function at x = a. Function evaluation is not just a mechanical process; it provides a deep understanding of how a function behaves. By evaluating a function at various points, we can map its behavior, identify patterns, and gain insights into its properties. This is particularly useful in fields such as physics, engineering, and computer science, where functions are used to model real-world phenomena. For instance, in physics, a function might describe the trajectory of a projectile, and evaluating this function at different times allows us to predict the projectile's position. Similarly, in economics, functions can model supply and demand, and evaluating these functions helps in understanding market dynamics. The ability to accurately and efficiently evaluate functions is thus a crucial skill in both academic and practical settings.

Step-by-Step Evaluation of f(x) at x = -9

To evaluate the function f(x) = 2x³ + x² - 13x + 6 at x = -9, we follow a systematic approach that involves substitution and simplification. This step-by-step process ensures accuracy and clarity in the evaluation. Here's how we proceed:

  1. Substitution: The first step is to replace every instance of x in the function's expression with the value -9. This yields:

    f(-9) = 2(-9)³ + (-9)² - 13(-9) + 6

    This substitution is crucial as it sets the stage for the arithmetic operations that follow.

  2. Evaluate the Exponents: Next, we evaluate the exponential terms. Remember that (-9)³ means -9 multiplied by itself three times, and (-9)² means -9 multiplied by itself. Therefore:

    • (-9)³ = -9 * -9 * -9 = -729
    • (-9)² = -9 * -9 = 81

    Substituting these values back into the expression, we get:

    f(-9) = 2(-729) + 81 - 13(-9) + 6

  3. Perform the Multiplications: Now, we perform the multiplication operations. This involves multiplying the coefficients with the evaluated exponential terms:

    • 2(-729) = -1458
    • -13(-9) = 117

    Substituting these results into the expression, we have:

    f(-9) = -1458 + 81 + 117 + 6

  4. Addition and Subtraction: Finally, we perform the addition and subtraction operations. It's often helpful to group the positive and negative terms to minimize errors:

    f(-9) = -1458 + 81 + 117 + 6 = -1458 + 204

    Adding these terms together gives us:

    f(-9) = -1254

    Therefore, the value of the function f(x) = 2x³ + x² - 13x + 6 when x = -9 is -1254. This step-by-step approach ensures that each operation is performed correctly, leading to the accurate evaluation of the function.

Detailed Calculation Breakdown

To further clarify the evaluation process, let's break down each step with detailed calculations. This will provide a comprehensive understanding of how we arrived at the final result. We are evaluating the function f(x) = 2x³ + x² - 13x + 6 at x = -9.

  1. Substitution:

    • We start by substituting x with -9 in the function:

      f(-9) = 2(-9)³ + (-9)² - 13(-9) + 6

    This step replaces the variable x with its specific value, setting the stage for the subsequent arithmetic operations.

  2. Evaluate the Exponents:

    • Next, we evaluate the exponential terms. First, we calculate (-9)³:

      • (-9)³ = -9 * -9 * -9
      • = 81 * -9
      • = -729

    • Then, we calculate (-9)²:

      • (-9)² = -9 * -9
      • = 81

    • Substituting these values back into the expression, we get:

      f(-9) = 2(-729) + 81 - 13(-9) + 6

    Evaluating the exponents is crucial as it simplifies the expression and allows us to proceed with multiplication and addition.

  3. Perform the Multiplications:

    • Now, we perform the multiplication operations. First, we multiply 2 by -729:

      • 2(-729) = -1458

    • Next, we multiply -13 by -9:

      • -13(-9) = 117

    • Substituting these results into the expression, we have:

      f(-9) = -1458 + 81 + 117 + 6

    Performing multiplication accurately is essential for obtaining the correct result.

  4. Addition and Subtraction:

    • Finally, we perform the addition and subtraction operations. We group the positive terms together:

      • 81 + 117 + 6 = 204

    • Now, we add this result to -1458:

      f(-9) = -1458 + 204 f(-9) = -1254

    • Therefore, the final result is:

      f(-9) = -1254

    This detailed breakdown illustrates each step of the evaluation process, ensuring clarity and accuracy. By understanding these steps, one can confidently evaluate functions at any given point.

Common Mistakes to Avoid

When evaluating functions, it is crucial to be meticulous and avoid common mistakes that can lead to incorrect results. Here are some frequent errors to watch out for:

  1. Incorrect Order of Operations: One of the most common mistakes is failing to follow the correct order of operations (PEMDAS/BODMAS). This acronym stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Forgetting this order can lead to significant errors. For example, in the expression 2(-9)³ + (-9)² - 13(-9) + 6, one must evaluate the exponents before performing multiplication or addition. Neglecting this rule can result in a completely different answer.

  2. Sign Errors: Sign errors are particularly prevalent when dealing with negative numbers. It's essential to remember the rules for multiplying and adding negative numbers. For instance, a negative number multiplied by a negative number yields a positive number, while a negative number multiplied by a positive number yields a negative number. In the example, -13(-9) should be calculated as +117, not -117. Similarly, care must be taken when adding and subtracting negative numbers.

  3. Miscalculating Exponents: Exponents can be tricky, especially when dealing with negative bases. It's crucial to understand that (-a)ⁿ is not always equal to -(aⁿ). For example, (-9)³ means -9 multiplied by itself three times, which equals -729, while (-9)² means -9 multiplied by itself, which equals 81. Miscalculating these values can lead to significant errors in the final result.

  4. Arithmetic Errors: Simple arithmetic errors, such as miscalculations during multiplication or addition, can also lead to incorrect results. It's always a good practice to double-check each step of the calculation to minimize these errors. Using a calculator can be helpful, but it's still important to understand the underlying principles and verify the results.

  5. Incorrect Substitution: Another common mistake is substituting the value incorrectly into the function. Ensure that every instance of the variable x is replaced with the given value. Forgetting to substitute in one place can throw off the entire calculation.

By being aware of these common mistakes and taking the time to carefully perform each step, you can improve your accuracy and confidence in evaluating functions.

Alternative Methods for Evaluation

While direct substitution is the most straightforward method for evaluating functions, there are alternative approaches that can sometimes be more efficient or provide additional insights. Here are a couple of alternative methods:

  1. Using Synthetic Division: Synthetic division is a method primarily used for dividing polynomials, but it can also be used to evaluate a polynomial function at a specific point. This method is particularly useful for higher-degree polynomials as it simplifies the calculation process. To evaluate f(x) = 2x³ + x² - 13x + 6 at x = -9 using synthetic division, we set up the division as follows:

    -9 | 2   1   -13   6
   |     -18   153  -1260
   ----------------------
     2  -17   140  -1254

    The last number in the bottom row, -1254, is the value of f(-9). Synthetic division can be faster than direct substitution, especially for polynomials with high degrees, as it reduces the amount of manual calculation required.

  2. Using Horner's Method: Horner's method is an algorithm for polynomial evaluation that rewrites the polynomial in a nested form. This method minimizes the number of multiplications, which can be beneficial when evaluating polynomials by hand or in computationally intensive applications. For the function f(x) = 2x³ + x² - 13x + 6, Horner's method rewrites the polynomial as:

    f(x) = ((2x + 1)x - 13)x + 6

    To evaluate f(-9), we proceed as follows:

    • 2(-9) + 1 = -17
    • (-17)(-9) - 13 = 140
    • (140)(-9) + 6 = -1254

    Thus, f(-9) = -1254. Horner's method is particularly useful in computer programming, where minimizing the number of operations can improve performance. It also reduces the risk of round-off errors in floating-point arithmetic.

Both synthetic division and Horner's method offer alternative ways to evaluate polynomial functions. While direct substitution is often the most intuitive approach, these methods can provide efficiency and additional insights in certain situations. Understanding these alternative methods enhances one's mathematical toolkit and problem-solving capabilities.

Conclusion

In conclusion, evaluating the function f(x) = 2x³ + x² - 13x + 6 at x = -9 yields a result of -1254. This process involves careful substitution, adherence to the order of operations, and meticulous arithmetic calculations. We have explored the step-by-step evaluation, detailed calculation breakdown, common mistakes to avoid, and alternative methods such as synthetic division and Horner's method. Mastering function evaluation is crucial in mathematics as it forms the foundation for more advanced concepts and applications. The ability to accurately evaluate functions allows for deeper insights into their behavior and properties, making it an essential skill for students, professionals, and anyone working with mathematical models. Whether it's through direct substitution or alternative methods, the key is to approach the process systematically and with attention to detail. This ensures not only the correct answer but also a solid understanding of the underlying mathematical principles.