Evaluating F(x) = 2x² - 6x + 2 A Step-by-Step Guide

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This article delves into the process of evaluating a given quadratic function, f(x) = 2x² - 6x + 2, for specific input values. We will explore how to substitute these values into the function and perform the necessary calculations to determine the corresponding output values. This is a fundamental concept in algebra and is crucial for understanding the behavior and properties of quadratic functions. By carefully examining the function and its evaluation, we can identify the correct statement from the provided options. Understanding quadratic functions is essential, and this exploration will provide a solid foundation for more complex mathematical concepts. This detailed explanation will walk you through each step, ensuring clarity and a thorough understanding of the solution.

Evaluating f(3)

To begin, let's evaluate the function f(x) = 2x² - 6x + 2 at x = 3. This involves substituting 3 for every instance of x in the function's expression. This process is a cornerstone of function evaluation and helps us understand how the output of the function changes with different inputs. Let's break down the step-by-step evaluation:

  1. Substitute x with 3:

    • Replace every 'x' in the function with the value 3.
    • f(3) = 2(3)² - 6(3) + 2

  2. Calculate the exponent:

    • Evaluate 3 squared (3²), which is 3 * 3 = 9.
    • f(3) = 2(9) - 6(3) + 2

  3. Perform the multiplications:

    • Multiply 2 by 9, resulting in 18.
    • Multiply 6 by 3, resulting in 18.
    • f(3) = 18 - 18 + 2

  4. Perform the additions and subtractions from left to right:

    • Subtract 18 from 18, resulting in 0.
    • f(3) = 0 + 2
    • Add 0 and 2, resulting in 2.
    • f(3) = 2

Therefore, f(3) = 2. This means that when the input to the function is 3, the output is 2. This is a crucial data point for understanding the function's behavior. Mastering this evaluation process is fundamental for working with quadratic equations. By understanding how to substitute and simplify, you lay a solid foundation for more advanced mathematical concepts. This systematic approach ensures accuracy and clarity in your calculations.

Evaluating f(-4)

Next, we will evaluate the function f(x) = 2x² - 6x + 2 at x = -4. This process is similar to the previous evaluation but involves working with a negative input value. Handling negative numbers correctly is crucial in mathematical evaluations, particularly in the context of quadratic functions. Let's follow the same methodical steps:

  1. Substitute x with -4:

    • Replace every 'x' in the function with the value -4.
    • f(-4) = 2(-4)² - 6(-4) + 2

  2. Calculate the exponent:

    • Evaluate -4 squared ((-4)²), which is -4 * -4 = 16. Remember that a negative number squared is positive.
    • f(-4) = 2(16) - 6(-4) + 2

  3. Perform the multiplications:

    • Multiply 2 by 16, resulting in 32.
    • Multiply -6 by -4, resulting in 24. Remember that a negative number multiplied by a negative number is positive.
    • f(-4) = 32 + 24 + 2

  4. Perform the additions from left to right:

    • Add 32 and 24, resulting in 56.
    • f(-4) = 56 + 2
    • Add 56 and 2, resulting in 58.
    • f(-4) = 58

Therefore, f(-4) = 58. This result indicates that when the input to the function is -4, the output is 58. This, along with our previous calculation of f(3), provides valuable information about the function's behavior across different input values. Accurately evaluating functions with negative numbers is a key skill in algebra, and this example illustrates the process clearly. Understanding the rules of signs is critical for achieving the correct result. This thorough calculation ensures a strong grasp of function evaluation.

Determining the Correct Statement

Now that we have calculated f(3) = 2 and f(-4) = 58, we can identify the correct statement from the given options. This step involves comparing our calculated values with the provided statements and selecting the one that accurately reflects our findings. This exercise reinforces the importance of accurate evaluation and highlights how specific function values can help us understand the function's overall behavior. The ability to interpret results and relate them back to the original function is a crucial skill in mathematics.

Let's review the options:

  1. f(3) = 10 and f(-4) = -6: This is incorrect because we found f(3) = 2 and f(-4) = 58.
  2. f(3) = 10 and f(-4) = 18: This is also incorrect for the same reason.
  3. Answer is not provided: This would be incorrect if one of the other options was accurate.
  4. f(3) = 2 and f(-4) = 58: This matches our calculated values, so this is the correct statement.
  5. f(3) = 2 and f(-4) = 18: This is incorrect because f(-4) is not 18.
  6. f(3) = 2 and f(-4) = -6: This is also incorrect because f(-4) is not -6.

Therefore, the correct statement is option 4: f(3) = 2 and f(-4) = 58. This confirms our calculations and demonstrates the importance of precise function evaluation. By systematically working through each option, we were able to confidently identify the accurate answer. This process of verification and elimination is a valuable problem-solving strategy in mathematics.

Conclusion

In conclusion, we have successfully evaluated the quadratic function f(x) = 2x² - 6x + 2 for x = 3 and x = -4. Through step-by-step calculations, we determined that f(3) = 2 and f(-4) = 58. This allowed us to correctly identify option 4 as the true statement. This exercise highlights the significance of accurate function evaluation in understanding the behavior of mathematical functions. The ability to substitute values, perform calculations, and interpret the results is fundamental to success in algebra and beyond. Understanding quadratic equations and their evaluation is a crucial skill for solving various mathematical problems. This comprehensive exploration has provided a clear and thorough understanding of the process, solidifying the foundation for further mathematical studies.