Evaluate F(x) = -2x^2 + 4x - 7 At X = -4

by Alex Johnson 41 views

In this article, we'll walk through how to evaluate the function f(x)=−2x2+4x−7{f(x) = -2x^2 + 4x - 7} at x=−4{x = -4}. This involves substituting −4{-4} for x{x} in the function and then simplifying the expression to find the value of f(−4){f(-4)}. Let's dive in!

Understanding the Function

Before we jump into the calculation, let's briefly understand the given function. The function is a quadratic function in the form f(x)=ax2+bx+c{f(x) = ax^2 + bx + c}, where a=−2{a = -2}, b=4{b = 4}, and c=−7{c = -7}. Quadratic functions create a parabolic curve when graphed. Our goal is to find the y{y}-value (or the function value) when x{x} is −4{-4}.

Step-by-Step Evaluation

To find f(−4){f(-4)}, we substitute −4{-4} for every instance of x{x} in the function:

f(−4)=−2(−4)2+4(−4)−7{f(-4) = -2(-4)^2 + 4(-4) - 7}

Now, let's break this down step-by-step:

  1. Calculate (-4)^2:

    Remember that squaring a negative number results in a positive number.

    (−4)2=(−4)×(−4)=16{(-4)^2 = (-4) \times (-4) = 16}

  2. Multiply -2 by 16:

    −2×16=−32{-2 \times 16 = -32}

  3. Multiply 4 by -4:

    4×(−4)=−16{4 \times (-4) = -16}

  4. Substitute these values back into the equation:

    f(−4)=−32−16−7{f(-4) = -32 - 16 - 7}

  5. Combine the terms:

    f(−4)=−32−16−7=−48−7=−55{f(-4) = -32 - 16 - 7 = -48 - 7 = -55}

Therefore, f(−4)=−55{f(-4) = -55}.

Detailed Calculation

Let's go through the calculation again, ensuring each step is clear:

  1. Original function: f(x)=−2x2+4x−7{f(x) = -2x^2 + 4x - 7}

  2. Substitute x with -4: f(−4)=−2(−4)2+4(−4)−7{f(-4) = -2(-4)^2 + 4(-4) - 7}

  3. Evaluate (-4)^2: (−4)2=16{(-4)^2 = 16}

  4. Multiply -2 by 16: −2×16=−32{-2 \times 16 = -32}

  5. Multiply 4 by -4: 4×−4=−16{4 \times -4 = -16}

  6. Substitute the results: f(−4)=−32+(−16)−7{f(-4) = -32 + (-16) - 7}

  7. Simplify: f(−4)=−32−16−7{f(-4) = -32 - 16 - 7} f(−4)=−48−7{f(-4) = -48 - 7} f(−4)=−55{f(-4) = -55}

Thus, when x=−4{x = -4}, the value of the function f(x)=−2x2+4x−7{f(x) = -2x^2 + 4x - 7} is −55{-55}.

Practical Implications and Uses

Evaluating functions at specific points is a fundamental skill in algebra and calculus. Understanding how to substitute values into functions is crucial for various applications, such as:

  • Graphing functions: By evaluating a function at multiple points, you can plot those points on a graph and sketch the curve of the function.
  • Modeling real-world scenarios: Functions are often used to model real-world phenomena. Evaluating the function at a specific point can provide insights into the state of the system at that particular input value.
  • Optimization problems: In calculus, finding the maximum or minimum value of a function often involves evaluating the function at critical points.
  • Computer programming: In programming, functions are used extensively, and evaluating them with different inputs is a core part of writing and testing code.

Common Mistakes to Avoid

When evaluating functions, there are several common mistakes to watch out for:

  • Incorrectly squaring negative numbers: Remember that (−x)2=x2{(-x)^2 = x^2}, so (−4)2=16{(-4)^2 = 16}, not −16{-16}.
  • Sign errors: Pay close attention to the signs of the numbers, especially when adding and subtracting negative numbers.
  • Order of operations: Follow the correct order of operations (PEMDAS/BODMAS) to ensure accurate calculations.
  • Forgetting to substitute in all instances of x: Ensure you replace every x{x} in the function with the given value.

Elaborating on Quadratic Functions

Our function f(x)=−2x2+4x−7{f(x) = -2x^2 + 4x - 7} is a quadratic function. Quadratic functions are polynomials of degree 2 and have a general form of f(x)=ax2+bx+c{f(x) = ax^2 + bx + c}, where a{a}, b{b}, and c{c} are constants. The graph of a quadratic function is a parabola. Understanding the properties of quadratic functions, such as finding the vertex, axis of symmetry, and roots, is essential in algebra.

In this specific function:

  • The coefficient a=−2{a = -2} indicates that the parabola opens downwards (because a<0{a < 0}).
  • The vertex of the parabola can be found using the formula x=−b/(2a){x = -b / (2a)}. In this case, x=−4/(2×−2)=1{x = -4 / (2 \times -2) = 1}. The y{y}-coordinate of the vertex can be found by evaluating f(1){f(1)}.
  • The constant term c=−7{c = -7} represents the y{y}-intercept of the parabola.

Alternative Methods

While direct substitution is the most straightforward method, there are no significant alternative methods for evaluating this specific function at a given point. However, understanding the properties of the function can provide additional insights.

Conclusion

In summary, evaluating the function f(x)=−2x2+4x−7{f(x) = -2x^2 + 4x - 7} at x=−4{x = -4} involves substituting −4{-4} for x{x} in the function and simplifying the expression. By following the steps outlined above, we found that f(−4)=−55{f(-4) = -55}. This process is fundamental in mathematics and has wide-ranging applications in various fields. Remember to pay attention to the order of operations and avoid common mistakes to ensure accurate calculations.

For further learning and to deepen your understanding of functions and their evaluations, consider exploring resources like Khan Academy's Algebra I course, which offers comprehensive lessons and practice exercises.