Evaluating F(-3) For The Function F(a) = -2a² - 5a + 4

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Introduction

In the realm of mathematics, functions play a pivotal role in describing relationships between variables. Evaluating a function at a specific point is a fundamental operation, allowing us to understand the function's behavior and output for a given input. In this comprehensive guide, we will delve into the process of evaluating the function f(a) = -2a² - 5a + 4 at a = -3. This step-by-step exploration will equip you with the knowledge and skills to confidently evaluate various functions at specific points. To accurately determine what is f(-3) for the function f(a) = -2a² - 5a + 4, we will need to substitute -3 for every instance of 'a' in the function's equation. This process will involve careful attention to the order of operations (PEMDAS/BODMAS) to ensure the correct result. Let's embark on this mathematical journey together and unlock the secrets of function evaluation.

Understanding the Function Notation and Structure

Before we dive into the evaluation process, let's first understand the notation and structure of the function f(a) = -2a² - 5a + 4. The notation f(a) represents the function named "f" with the input variable "a." The expression on the right-hand side of the equation, -2a² - 5a + 4, defines the rule or formula that the function applies to the input variable. This particular function is a quadratic function, characterized by the presence of the a² term. Quadratic functions are widely used in various fields, including physics, engineering, and economics, to model phenomena such as projectile motion, optimization problems, and growth patterns. The coefficients of the terms in the quadratic expression play a crucial role in determining the function's shape and behavior. The coefficient of the a² term (-2) affects the direction and steepness of the parabola, while the coefficients of the a term (-5) and the constant term (4) influence the function's position and intercepts. By carefully examining the structure of the function, we can gain valuable insights into its properties and how it will respond to different input values. Recognizing the function as a quadratic function allows us to anticipate its parabolic shape and understand the significance of its coefficients. In the following sections, we will apply our understanding of the function's structure to accurately evaluate it at a = -3.

Step-by-Step Evaluation of f(-3)

Now that we have a solid understanding of the function f(a) = -2a² - 5a + 4, let's proceed with the step-by-step evaluation at a = -3. This process involves substituting -3 for every instance of "a" in the function's equation and then simplifying the expression according to the order of operations (PEMDAS/BODMAS). The order of operations is crucial in ensuring that we arrive at the correct result. PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following this order diligently will prevent errors and ensure the accuracy of our calculations. Let's break down the evaluation into manageable steps:

  1. Substitution: Replace every "a" in the function with (-3). This gives us f(-3) = -2(-3)² - 5(-3) + 4.
  2. Exponents: Evaluate the exponent term. (-3)² = 9. So, the expression becomes f(-3) = -2(9) - 5(-3) + 4.
  3. Multiplication: Perform the multiplication operations from left to right. -2(9) = -18 and -5(-3) = 15. The expression now looks like this: f(-3) = -18 + 15 + 4.
  4. Addition: Perform the addition operations from left to right. -18 + 15 = -3 and -3 + 4 = 1. Therefore, f(-3) = 1.

By meticulously following these steps, we have successfully evaluated the function f(a) = -2a² - 5a + 4 at a = -3 and found that f(-3) = 1. This result represents the output of the function when the input is -3. In the next section, we will further discuss the interpretation and significance of this result.

Interpretation and Significance of the Result

We have determined that f(-3) = 1 for the function f(a) = -2a² - 5a + 4. But what does this result actually mean? In mathematical terms, f(-3) = 1 signifies that when the input value to the function is -3, the corresponding output value is 1. This can be visualized as a point on the graph of the function, where the x-coordinate is -3 and the y-coordinate is 1. This point, (-3, 1), lies on the parabola that represents the quadratic function f(a). The value of f(-3) provides us with a specific data point on the function's graph, helping us understand the function's behavior in that region. For quadratic functions, evaluating at different points allows us to trace the parabolic curve and identify key features such as the vertex, intercepts, and axis of symmetry. The vertex represents the maximum or minimum point of the parabola, depending on the sign of the leading coefficient (in this case, -2, indicating a downward-opening parabola). The intercepts are the points where the parabola crosses the x-axis (roots) and the y-axis. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. By evaluating the function at various points, we can gain a comprehensive understanding of its shape, position, and overall behavior. In practical applications, evaluating functions at specific points can provide valuable insights into real-world phenomena modeled by those functions. For example, if f(a) represents the height of a projectile at time a, then f(-3) = 1 would indicate the projectile's height at 3 seconds before the launch (assuming the model is valid for negative time values). Similarly, in economics, a function representing cost or revenue can be evaluated at different production levels to determine optimal values. In conclusion, the result f(-3) = 1 is not just a numerical answer; it's a piece of information that contributes to our understanding of the function's behavior and its potential applications.

Common Mistakes to Avoid

When evaluating functions, especially quadratic functions, it's crucial to be mindful of potential errors that can lead to incorrect results. Here are some common mistakes to avoid:

  1. Order of Operations Errors: Failing to follow the correct order of operations (PEMDAS/BODMAS) is a frequent source of errors. Remember to evaluate exponents before multiplication, and multiplication before addition. For instance, in the expression -2(-3)², it's essential to square -3 first, then multiply by -2. Incorrectly multiplying -2 by -3 first would lead to a wrong answer.
  2. Sign Errors: Dealing with negative signs can be tricky. Pay close attention to the signs when substituting values and performing operations. For example, remember that a negative number squared is positive, while a negative number multiplied by a negative number is also positive. In the expression -5(-3), the result is +15, not -15.
  3. Incorrect Substitution: Ensure that you substitute the input value correctly for every instance of the variable in the function. Missing a substitution or substituting in the wrong place will result in an inaccurate evaluation. Double-check your substitutions before proceeding with the calculations.
  4. Arithmetic Errors: Simple arithmetic mistakes, such as adding or multiplying numbers incorrectly, can derail the entire evaluation process. Take your time and double-check your calculations to minimize these errors. Use a calculator if necessary, but be sure to enter the values and operations correctly.
  5. Misinterpreting the Function Notation: Understanding the function notation is essential. f(a) represents the function named "f" with input variable "a." Make sure you understand what the function is asking you to do before attempting to evaluate it. If the function is f(a) = -2a² - 5a + 4, you should replace every "a" with the given input value.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in evaluating functions.

Practice Problems

To solidify your understanding of function evaluation, let's tackle a few practice problems. These examples will provide you with an opportunity to apply the steps and techniques we've discussed. Remember to carefully follow the order of operations, pay attention to signs, and double-check your calculations.

  1. Evaluate g(x) = x² + 3x - 2 at x = 2.
  2. Find h(t) = -4.9t² + 10t + 2 when t = 1.5.
  3. Determine p(r) = (r - 3)² + 5 at r = 0.
  4. Calculate q(m) = 2m³ - m + 1 for m = -1.
  5. If f(x) = |2x - 1|, what is f(-2)?

(Note: The absolute value function |x| returns the non-negative value of x.)

Take your time to work through these problems, showing your steps clearly. Once you have your answers, you can compare them to the solutions provided below to check your understanding. Practicing with various examples is the key to mastering function evaluation.

Solutions to Practice Problems

Here are the solutions to the practice problems presented in the previous section. Reviewing these solutions will help you identify any areas where you may need further clarification or practice.

  1. g(2) = (2)² + 3(2) - 2 = 4 + 6 - 2 = 8
  2. h(1.5) = -4.9(1.5)² + 10(1.5) + 2 = -4.9(2.25) + 15 + 2 = -11.025 + 15 + 2 = 5.975
  3. p(0) = (0 - 3)² + 5 = (-3)² + 5 = 9 + 5 = 14
  4. q(-1) = 2(-1)³ - (-1) + 1 = 2(-1) + 1 + 1 = -2 + 1 + 1 = 0
  5. f(-2) = |2(-2) - 1| = |-4 - 1| = |-5| = 5

If you encountered any difficulties while solving these problems, revisit the step-by-step evaluation process and common mistakes to avoid. Pay close attention to the order of operations, sign conventions, and substitution techniques. With consistent practice, you'll become proficient in evaluating functions with confidence.

Conclusion

In this comprehensive guide, we have explored the process of evaluating the function f(a) = -2a² - 5a + 4 at a = -3. We began by understanding the function's notation and structure, recognizing it as a quadratic function with a parabolic graph. We then meticulously followed the step-by-step evaluation process, substituting -3 for "a" and simplifying the expression according to the order of operations. This led us to the result f(-3) = 1, which we interpreted as the y-coordinate of a point on the function's graph when the x-coordinate is -3. We also discussed the significance of this result in the context of quadratic functions and their applications. Furthermore, we highlighted common mistakes to avoid when evaluating functions, such as order of operations errors, sign errors, and incorrect substitutions. By being aware of these pitfalls, you can improve your accuracy and confidence in function evaluation. To reinforce your understanding, we presented a set of practice problems with detailed solutions. Working through these examples allowed you to apply the techniques and concepts learned in this guide. Evaluating functions is a fundamental skill in mathematics and various related fields. It provides a powerful tool for understanding relationships between variables and modeling real-world phenomena. By mastering the principles and techniques discussed in this guide, you will be well-equipped to tackle more complex mathematical problems and apply your knowledge in diverse contexts. Whether you're studying algebra, calculus, or any other quantitative discipline, the ability to evaluate functions accurately and efficiently is an invaluable asset. Continue to practice and explore different types of functions, and you'll find yourself becoming increasingly comfortable and confident in your mathematical abilities. Remember, mathematics is a journey of discovery, and each step you take builds upon the previous one. Keep learning, keep practicing, and keep exploring the fascinating world of functions!