Finding F(-3) For F(a) = -2a² - 5a + 4 A Step By Step Guide

Understanding function evaluation is a cornerstone of algebra, and in this article, we'll delve into the process of finding the value of a function at a specific point. Specifically, we're tackling the function f(a) = -2a² - 5a + 4 and our mission is to determine the value of f(-3). This involves substituting -3 for the variable 'a' throughout the function and then simplifying the resulting expression using the order of operations. This exercise is not just about getting the right answer; it's about solidifying your understanding of function notation, algebraic manipulation, and the fundamental principles that govern mathematical expressions. By the end of this exploration, you'll not only know the value of f(-3) but also have a clearer grasp of how functions work and how to evaluate them effectively.

Unveiling the Function: f(a) = -2a² - 5a + 4

Before we plunge into calculating f(-3), let's take a moment to dissect the given function, f(a) = -2a² - 5a + 4. This is a quadratic function, which means it's a polynomial function where the highest power of the variable 'a' is 2. Quadratic functions are ubiquitous in mathematics and have a wide range of applications, from describing projectile motion to modeling optimization problems. The general form of a quadratic function is f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. In our case, a = -2, b = -5, and c = 4. Understanding these coefficients is crucial because they influence the shape and position of the parabola that represents the graph of this function.

The term -2a² is the quadratic term, and it dictates the overall concavity of the parabola. Since the coefficient is negative (-2), the parabola opens downwards, meaning it has a maximum point. The term -5a is the linear term, and it affects the slope and position of the parabola. Finally, the constant term 4 represents the y-intercept of the parabola, which is the point where the graph intersects the y-axis. Recognizing these components and their roles will help you visualize the function and anticipate its behavior. This foundational understanding is essential not only for evaluating functions at specific points but also for solving equations, graphing functions, and analyzing their properties. Remember, a function is like a machine: you input a value (in this case, -3), and it processes it according to its rule (-2a² - 5a + 4) to produce an output.

The Substitution Step: Replacing 'a' with -3

Now comes the critical step of substituting -3 for 'a' in the function f(a) = -2a² - 5a + 4. This is where precision is key. It's not just about replacing 'a' with -3; it's about doing it correctly, paying close attention to parentheses and signs. When we substitute, we get f(-3) = -2(-3)² - 5(-3) + 4. Notice the use of parentheses around -3. This is crucial because it ensures that we square the entire value of -3, not just 3. Without the parentheses, we might incorrectly calculate (-3)² as -3², which would lead to a wrong answer.

The substitution step is the bridge between the abstract function definition and the concrete numerical evaluation. It's where we translate the symbolic representation into a specific calculation. This step highlights the power of function notation: it allows us to express a relationship between an input and an output in a concise and unambiguous way. By substituting a value for the variable, we're essentially asking the function, "What output do you produce when I give you this input?" This process of substitution is fundamental to understanding how functions work and how they can be used to model real-world phenomena. So, double-check your substitution, ensure you've used parentheses correctly, and prepare for the next step: simplification.

Order of Operations: A Step-by-Step Simplification

With the substitution complete, we now have the expression f(-3) = -2(-3)² - 5(-3) + 4. To evaluate this expression, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that we perform operations in the correct sequence, leading to a consistent and accurate result. First, we tackle the exponent: (-3)² = (-3) * (-3) = 9. Remember that a negative number multiplied by a negative number results in a positive number. So our expression becomes f(-3) = -2(9) - 5(-3) + 4.

Next, we perform the multiplications: -2(9) = -18 and -5(-3) = 15. Again, note that the product of two negative numbers is positive. Our expression now simplifies to f(-3) = -18 + 15 + 4. Finally, we perform the additions from left to right: -18 + 15 = -3, and then -3 + 4 = 1. Thus, we arrive at our final answer: f(-3) = 1. This step-by-step simplification demonstrates the importance of following the order of operations. Skipping a step or performing operations out of order can lead to a completely different result. By meticulously applying PEMDAS, we ensure that we're correctly evaluating the expression and arriving at the true value of f(-3).

The Grand Finale: f(-3) = 1

After carefully substituting -3 for 'a' in the function f(a) = -2a² - 5a + 4 and diligently applying the order of operations, we've arrived at our final answer: f(-3) = 1. This means that when the input to the function is -3, the corresponding output is 1. This single value encapsulates the entire process we've undertaken, from understanding the function's structure to performing the arithmetic calculations.

But the significance of f(-3) = 1 goes beyond just a numerical result. It represents a specific point on the graph of the function, the point (-3, 1). This point lies on the parabola that represents the quadratic function f(a) = -2a² - 5a + 4. Visualizing this point on the graph helps to solidify our understanding of the relationship between the input and output of the function. Furthermore, knowing the value of f(-3) can be useful in various applications, such as finding the roots of the equation, determining the vertex of the parabola, or modeling real-world scenarios. So, the answer f(-3) = 1 is not just a number; it's a piece of information that contributes to our broader understanding of the function and its properties. It's a testament to the power of mathematical tools and techniques in unlocking the secrets of functional relationships.

Mastering Function Evaluation: Key Takeaways

Our journey to find f(-3) for the function f(a) = -2a² - 5a + 4 has been more than just a calculation; it's been a valuable exercise in understanding and applying fundamental mathematical concepts. Here are some key takeaways from this exploration that will help you master function evaluation and tackle similar problems with confidence:

1. Understand Function Notation: Function notation, such as f(a), is a powerful tool for expressing relationships between inputs and outputs. It provides a concise and unambiguous way to define a function and specify its behavior. Recognizing that 'f(a)' represents the output of the function when 'a' is the input is crucial for understanding how to evaluate functions.

2. Precise Substitution: The substitution step is where we replace the variable in the function's expression with a specific value. Accuracy is paramount here. Always use parentheses when substituting values, especially negative numbers, to avoid errors with signs and exponents. Double-check your substitution to ensure you've replaced every instance of the variable with the correct value.

3. Order of Operations is King: The order of operations (PEMDAS) is the golden rule for simplifying mathematical expressions. Exponents before multiplication, multiplication before addition, and so on. Following this order ensures that you perform operations in the correct sequence, leading to a consistent and accurate result. Make PEMDAS your mantra when simplifying expressions.

4. Contextualize the Result: The final answer, like f(-3) = 1, is not just a number; it's a piece of information about the function. It represents a specific point on the graph of the function, and it can be used in various applications. Think about what the result means in the context of the problem and how it relates to the function's overall behavior.

By internalizing these key takeaways, you'll be well-equipped to tackle a wide range of function evaluation problems. Remember, practice makes perfect, so keep working through examples and challenging yourself to deepen your understanding of functions and their properties.