Evaluating H(-3) For H(x) = 4x^2 - 3x + 5 A Step By Step Guide
In this article, we will delve into the process of evaluating a quadratic function at a specific point. Specifically, we will focus on the function h(x) = 4x² - 3x + 5 and determine the value of h(-3). This involves substituting -3 for x in the function and simplifying the resulting expression. Understanding how to evaluate functions is a fundamental skill in algebra and calculus, and this example will provide a clear and concise demonstration of the process. Let's embark on this mathematical journey together and unravel the solution step by step.
Understanding Function Evaluation
Function evaluation is a core concept in mathematics, especially within algebra and calculus. Essentially, it's the process of finding the value of a function at a specific input. Think of a function as a machine: you feed it an input, and it spits out an output. The input is usually represented by a variable, like x, and the output is the value of the function at that x, denoted as f(x) or, in our case, h(x). To evaluate a function, you substitute the given input value for the variable in the function's expression and then simplify. This might involve arithmetic operations like addition, subtraction, multiplication, and division, as well as exponents and other mathematical operations, depending on the complexity of the function. The key is to follow the order of operations (PEMDAS/BODMAS) to ensure you arrive at the correct result. Function evaluation is not just an abstract mathematical concept; it has real-world applications in various fields, including physics, engineering, economics, and computer science. It allows us to model and analyze relationships between different quantities and make predictions based on these models. In this article, we are focusing on a quadratic function, but the principles of function evaluation apply to all types of functions, from simple linear functions to more complex trigonometric, exponential, and logarithmic functions. The ability to confidently evaluate functions is crucial for anyone pursuing further studies in mathematics or related fields. Moreover, it hones critical thinking and problem-solving skills that are valuable in any discipline. So, let's dive into our specific example and see how function evaluation works in practice.
The Quadratic Function h(x) = 4x² - 3x + 5
Our focus is on the quadratic function h(x) = 4x² - 3x + 5. Understanding the characteristics of quadratic functions is crucial for effectively evaluating them. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable x is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient a. In our case, a = 4, b = -3, and c = 5. Since a is positive, the parabola opens upwards. The term 4x² is the quadratic term, -3x is the linear term, and 5 is the constant term. Each term plays a role in shaping the parabola and determining its position on the coordinate plane. The coefficient a affects the steepness of the parabola; a larger absolute value of a makes the parabola narrower. The coefficient b influences the position of the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The constant term c represents the y-intercept, the point where the parabola intersects the y-axis. When we evaluate h(x) at a specific value of x, we are essentially finding the y-coordinate of the point on the parabola that corresponds to that x-coordinate. This gives us a specific point on the graph of the function. Understanding the structure of a quadratic function and the role of each coefficient helps us visualize the function's behavior and interpret the results of our evaluation. So, with this understanding in mind, let's proceed to evaluate h(-3).
Step-by-Step Evaluation of h(-3)
Now, let's embark on the step-by-step evaluation of h(-3) for the quadratic function h(x) = 4x² - 3x + 5. This process involves substituting -3 for x in the function's expression and then simplifying the resulting expression according to the order of operations. The first step is the substitution. We replace every instance of x in the function's expression with -3. This gives us: h(-3) = 4(-3)² - 3(-3) + 5. Notice how we've carefully placed the -3 in parentheses to avoid any confusion with the signs. This is crucial, especially when dealing with negative numbers and exponents. The next step is to simplify the expression. We begin by evaluating the exponent. Remember that (-3)² means -3 multiplied by itself, which equals 9. So, we have: h(-3) = 4(9) - 3(-3) + 5. Now, we perform the multiplications. 4 multiplied by 9 is 36, and -3 multiplied by -3 is 9. So, we have: h(-3) = 36 + 9 + 5. Finally, we perform the additions. 36 plus 9 is 45, and 45 plus 5 is 50. Therefore, h(-3) = 50. This is the value of the function h(x) when x is -3. It represents the y-coordinate of the point on the parabola h(x) = 4x² - 3x + 5 where the x-coordinate is -3. By carefully following these steps, we have successfully evaluated the function at the given point. This process can be applied to any function and any input value. The key is to substitute correctly, pay attention to signs, and follow the order of operations.
Detailed Calculation Breakdown
To solidify our understanding, let's break down the calculation of h(-3) in even greater detail. We start with the function: h(x) = 4x² - 3x + 5. Our goal is to find the value of h(-3), which means we need to substitute -3 for x in the function. Substitution is a fundamental operation in algebra, and it's crucial to perform it accurately. When we substitute, we get: h(-3) = 4(-3)² - 3(-3) + 5. Notice the use of parentheses around -3. This is important because it clarifies that we are squaring the entire quantity -3, not just 3. It also helps us keep track of the signs when we perform the multiplications. Now, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). The first operation we encounter is the exponent: (-3)². A negative number squared is a positive number, so (-3)² = (-3) * (-3) = 9. Substituting this back into our expression, we get: h(-3) = 4(9) - 3(-3) + 5. Next, we perform the multiplications. We have two multiplications to do: 4(9) and -3(-3). 4 multiplied by 9 is 36. A negative number multiplied by a negative number is a positive number, so -3 multiplied by -3 is 9. Our expression now becomes: h(-3) = 36 + 9 + 5. Finally, we perform the additions. 36 plus 9 is 45, and 45 plus 5 is 50. Therefore, we arrive at our final answer: h(-3) = 50. Each step in this calculation is crucial, and a small error in any step can lead to an incorrect result. By carefully breaking down the calculation and paying attention to detail, we can ensure accuracy and build our confidence in evaluating functions. This detailed breakdown not only shows the mechanics of the calculation but also reinforces the importance of understanding the underlying mathematical principles.
Result and Interpretation
After meticulously following the steps of function evaluation, we have arrived at the result: h(-3) = 50. But what does this result actually mean? In the context of the quadratic function h(x) = 4x² - 3x + 5, this value represents the y-coordinate of the point on the parabola that corresponds to the x-coordinate of -3. In other words, the point (-3, 50) lies on the graph of the function. To visualize this, imagine plotting the graph of the function h(x) on a coordinate plane. The parabola would open upwards (since the coefficient of x² is positive), and at the x-value of -3, the y-value would be 50. This point is located in the second quadrant of the coordinate plane, relatively high up due to the y-value of 50. The result also tells us something about the function's behavior. Since h(-3) is a positive value, it indicates that the function's output is positive when the input is -3. This is valuable information for understanding the function's overall trend and its relationship to the x-axis. In practical applications, this kind of evaluation can be used to model various real-world scenarios. For example, if h(x) represents the height of a projectile at time x, then h(-3) would represent the height of the projectile 3 units of time before our starting point (if such a negative time value made sense in the context of the problem). More generally, function evaluation allows us to make predictions and draw conclusions based on the mathematical model represented by the function. Understanding the meaning and interpretation of the result is just as important as the calculation itself. It connects the abstract mathematical concept to the real world and allows us to use functions as powerful tools for analysis and problem-solving. So, with a clear understanding of both the calculation and the interpretation, we have successfully evaluated h(-3) for the given quadratic function.
Conclusion
In conclusion, we have successfully navigated the process of evaluating the quadratic function h(x) = 4x² - 3x + 5 at x = -3. Through a step-by-step approach, we first substituted -3 for x in the function, then carefully simplified the resulting expression using the order of operations. This meticulous calculation led us to the result h(-3) = 50. This result signifies that the point (-3, 50) lies on the graph of the quadratic function, representing the function's y-value when the x-value is -3. Beyond the numerical answer, we emphasized the importance of interpreting the result within the context of the function. Understanding the relationship between the input and output values allows us to use functions as powerful tools for modeling and analyzing real-world phenomena. The principles and techniques demonstrated in this article are applicable to a wide range of functions, not just quadratic ones. Mastering function evaluation is a cornerstone of mathematical proficiency, and this example serves as a solid foundation for further exploration in algebra and calculus. By practicing these skills and developing a deep understanding of function behavior, you'll be well-equipped to tackle more complex mathematical challenges. Remember, mathematics is not just about memorizing formulas and procedures; it's about developing logical thinking, problem-solving skills, and the ability to connect abstract concepts to the world around us. So, continue to explore, question, and practice, and you'll unlock the power and beauty of mathematics.
