Evaluating F(x) = -2x^2 - 3x + 5 At X = -3

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Understanding Function Evaluation

In mathematics, function evaluation is a fundamental concept. It involves substituting a given value for the variable in a function and calculating the resulting value. In simpler terms, if you have a function, let's say f(x), and you want to evaluate it at a specific point, like x = a, you replace every instance of 'x' in the function's expression with 'a' and then simplify. This process allows us to determine the output of the function for a particular input. Function evaluation is a cornerstone of algebra and calculus, playing a crucial role in understanding the behavior and properties of mathematical functions. It's used extensively in various applications, from graphing functions to solving equations and modeling real-world phenomena.

The ability to accurately evaluate functions is critical for various mathematical tasks, including graphing, solving equations, and understanding the behavior of mathematical models. Whether you're dealing with linear, quadratic, trigonometric, or exponential functions, the core principle remains the same: substitute, simplify, and solve. It's a skill that builds the foundation for more advanced mathematical concepts and is essential for anyone pursuing studies in mathematics, science, or engineering. So, let's dive into a specific example to see how this process works in practice and demystify any complexities that might arise.

The Function: f(x) = -2x^2 - 3x + 5

The function we'll be working with is a quadratic function: f(x) = -2x^2 - 3x + 5. Quadratic functions are polynomial functions of degree two, meaning the highest power of the variable 'x' is 2. They have a characteristic U-shaped curve when graphed, known as a parabola. The general form of a quadratic function is ax^2 + bx + c, where 'a', 'b', and 'c' are constants. In our case, a = -2, b = -3, and c = 5. These coefficients determine the shape, position, and direction of the parabola. The 'a' coefficient, in particular, dictates whether the parabola opens upwards (if a > 0) or downwards (if a < 0), as well as how wide or narrow the curve is. The 'b' coefficient influences the parabola's horizontal position, and the 'c' coefficient represents the y-intercept, which is the point where the parabola crosses the y-axis.

This specific function, f(x) = -2x^2 - 3x + 5, is a downward-opening parabola because the coefficient of x^2 is negative (-2). This means the parabola has a maximum point, also known as the vertex. Understanding the properties of quadratic functions is crucial for various applications, such as modeling projectile motion, optimizing processes, and designing structures. In this article, we're focusing on evaluating this function at a specific point, but it's important to recognize that function evaluation is just one aspect of the broader study of quadratic functions and their diverse applications in mathematics and beyond.

The Input Value: x = -3

Now that we have our function, f(x) = -2x^2 - 3x + 5, we need to decide at which point we want to evaluate it. In this case, the input value is given as x = -3. This means we are interested in finding the value of the function when 'x' is replaced with '-3'. The input value is a crucial element in function evaluation because it determines the specific location on the function's graph that we are examining. In the context of our quadratic function, choosing different input values will give us different points on the parabola.

Selecting an input value is often driven by the specific problem or application we are dealing with. For example, if we were modeling the height of a projectile over time, 'x' might represent time, and we might choose a specific time value (like x = -3) to determine the projectile's height at that moment. The input value can also be chosen to explore the function's behavior, such as finding its maximum or minimum value, identifying its roots (where the function equals zero), or understanding its symmetry. In this particular example, we've been given x = -3, so our task is to substitute this value into the function and compute the corresponding output. This process will give us a single point on the parabola, representing the function's value at that specific x-coordinate.

Step-by-Step Evaluation

Let's dive into the step-by-step evaluation of the function f(x) = -2x^2 - 3x + 5 at x = -3. This process involves carefully substituting the input value into the function and then simplifying the resulting expression according to the order of operations.

  1. Substitution: The first and most crucial step is to replace every instance of 'x' in the function with the value '-3'. This gives us: f(-3) = -2(-3)^2 - 3(-3) + 5 Notice how we've replaced 'x' with '-3' while maintaining the rest of the function's structure. It's essential to use parentheses around the negative value to ensure that the squaring operation is performed correctly.

  2. Exponents: The next step is to evaluate any exponents. In this case, we have (-3)^2, which means -3 multiplied by itself: (-3)^2 = (-3) * (-3) = 9 So, our expression now becomes: f(-3) = -2(9) - 3(-3) + 5

  3. Multiplication: Now, we perform the multiplication operations: -2(9) = -18 -3(-3) = 9 Our expression is further simplified to: f(-3) = -18 + 9 + 5

  4. Addition and Subtraction: Finally, we perform the addition and subtraction from left to right: -18 + 9 = -9 -9 + 5 = -4

Therefore, the final result of evaluating the function f(x) = -2x^2 - 3x + 5 at x = -3 is f(-3) = -4.

The Result: f(-3) = -4

After performing the step-by-step evaluation, we've arrived at the result: f(-3) = -4. This means that when the input value is x = -3, the output value of the function f(x) = -2x^2 - 3x + 5 is -4. In graphical terms, this corresponds to the point (-3, -4) on the parabola represented by the function. This point is a specific location on the curve, and its coordinates tell us the function's height at that particular x-value. The result, f(-3) = -4, is not just a numerical answer; it's a piece of information about the function's behavior.

Understanding the result in context is crucial. If, for example, this function represented the profit of a business as a function of the number of units sold, then f(-3) = -4 might indicate a loss of $4 when 3 units are not sold (note that in this context, a negative input might not have a direct real-world interpretation, but the mathematical result is still valid). Similarly, if the function modeled the height of a ball thrown in the air, f(-3) = -4 would not be physically meaningful since time cannot be negative in this scenario, and the height cannot be negative as well unless measured relative to a reference point below the ground. The interpretation of the result always depends on the specific application and the real-world context of the problem. In any case, the ability to accurately evaluate functions is essential for making informed decisions and predictions based on mathematical models.

Conclusion

In conclusion, evaluating the function f(x) = -2x^2 - 3x + 5 at the input value x = -3 has demonstrated the fundamental process of function evaluation. By carefully substituting the value and following the order of operations, we determined that f(-3) = -4. This result provides a specific point on the parabola represented by the function, highlighting the relationship between input and output values.

Function evaluation is a cornerstone of mathematics, enabling us to understand the behavior of functions and their applications in various fields. Whether you're working with quadratic functions, or any other type of mathematical expression, the ability to substitute values and simplify expressions is crucial for solving problems and making informed decisions. Mastering this skill opens the door to more advanced mathematical concepts and empowers you to tackle real-world challenges with confidence.

For further exploration of functions and their properties, consider visiting Khan Academy's Functions and Equations for comprehensive resources and practice exercises.