Evaluating F(3) F(2) And F(3+5) For The Function F(x) = 4x² - 2x - 7

Jul 13, 2025 by ADMIN 69 views

Evaluating the Quadratic Function f(x) = 4x² - 2x - 7: A Comprehensive Guide

In mathematics, functions are essential tools for modeling relationships between variables. Among the many types of functions, quadratic functions hold a special place due to their widespread applications in various fields, including physics, engineering, and economics. A quadratic function is defined as a polynomial function of degree two, which means it can be written in the general form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. In this article, we will delve into the process of evaluating a specific quadratic function, f(x) = 4x² - 2x - 7, for different values of x. We will explore how to substitute these values into the function and perform the necessary arithmetic operations to obtain the corresponding function values. Understanding how to evaluate functions is a fundamental skill in mathematics, as it allows us to analyze the behavior of functions and solve related problems. The ability to accurately evaluate a function for given input values is crucial for understanding its properties and applications. This article serves as a comprehensive guide to mastering this essential skill, providing step-by-step instructions and detailed explanations to ensure a clear understanding of the process. This guide is helpful for students learning about functions and their evaluations, as well as anyone seeking to refresh their knowledge of this important mathematical concept. By the end of this article, you will be able to confidently evaluate the quadratic function f(x) = 4x² - 2x - 7 for any given value of x, and apply this skill to other quadratic functions as well. By understanding this concept, you'll lay a solid foundation for more advanced topics in algebra and calculus. We'll start with a clear explanation of the function itself, and then guide you through the process of substituting values and simplifying the expression. Let's begin our journey into the world of function evaluation!

Understanding the Function f(x) = 4x² - 2x - 7

Before we dive into the evaluation process, let's take a moment to understand the function we're working with: f(x) = 4x² - 2x - 7. This is a quadratic function, which means it has a highest power of 2 for the variable x. The coefficients of the terms are as follows: the coefficient of x² is 4, the coefficient of x is -2, and the constant term is -7. These coefficients play a crucial role in determining the shape and position of the parabola, which is the graphical representation of a quadratic function. The general form of a quadratic function is f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. In our case, a = 4, b = -2, and c = -7. The coefficient 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). Since a = 4 in our function, the parabola opens upwards, indicating that the function has a minimum value. The 'b' coefficient affects the position of the axis of symmetry of the parabola, which is a vertical line that divides the parabola into two symmetrical halves. The 'c' coefficient represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. In our function, the y-intercept is -7. Understanding these coefficients and their roles is essential for analyzing the behavior of the quadratic function. For example, we can determine the vertex of the parabola, which is the point where the function reaches its minimum value. The x-coordinate of the vertex can be found using the formula x = -b / 2a. In our case, x = -(-2) / (2 * 4) = 2 / 8 = 1/4. To find the y-coordinate of the vertex, we substitute x = 1/4 into the function: f(1/4) = 4(1/4)² - 2(1/4) - 7 = 4(1/16) - 1/2 - 7 = 1/4 - 1/2 - 7 = -7.25. Therefore, the vertex of the parabola is (1/4, -7.25). The vertex represents the minimum point of the function, as the parabola opens upwards. Understanding the properties of the function, such as the coefficients, the vertex, and the y-intercept, allows us to visualize its graph and make predictions about its behavior. Now that we have a solid understanding of the function f(x) = 4x² - 2x - 7, we can proceed to evaluate it for different values of x.

Evaluating f(3)

To evaluate f(3), we substitute x = 3 into the function f(x) = 4x² - 2x - 7. This means replacing every instance of 'x' in the function's expression with the number 3. The substitution gives us: f(3) = 4(3)² - 2(3) - 7. Now, we need to follow the order of operations (PEMDAS/BODMAS) to simplify the expression. This means we first perform the exponentiation, then multiplication, and finally addition and subtraction. First, we calculate 3² (3 squared), which is 3 * 3 = 9. So, our expression becomes: f(3) = 4(9) - 2(3) - 7. Next, we perform the multiplications. 4 multiplied by 9 is 36, and 2 multiplied by 3 is 6. Our expression now looks like this: f(3) = 36 - 6 - 7. Finally, we perform the subtractions from left to right. 36 minus 6 is 30, and then 30 minus 7 is 23. Therefore, f(3) = 23. This means that when x is 3, the value of the function f(x) is 23. Graphically, this corresponds to the point (3, 23) on the parabola representing the function. Evaluating a function at a specific value gives us a single output value that corresponds to the input value. In this case, we have found that the output value is 23 when the input value is 3. This process of substitution and simplification is fundamental to understanding how functions work and how they can be used to model real-world phenomena. By substituting different values for x, we can explore how the function behaves and how its output changes in response to changes in the input. Now that we have successfully evaluated f(3), we can move on to evaluating the function at other values.

Evaluating f(2)

Next, let's evaluate the function f(x) = 4x² - 2x - 7 at x = 2. Similar to the previous example, we substitute x = 2 into the function. This gives us: f(2) = 4(2)² - 2(2) - 7. Again, we follow the order of operations (PEMDAS/BODMAS) to simplify the expression. First, we calculate 2² (2 squared), which is 2 * 2 = 4. So, the expression becomes: f(2) = 4(4) - 2(2) - 7. Next, we perform the multiplications. 4 multiplied by 4 is 16, and 2 multiplied by 2 is 4. The expression now looks like this: f(2) = 16 - 4 - 7. Finally, we perform the subtractions from left to right. 16 minus 4 is 12, and then 12 minus 7 is 5. Therefore, f(2) = 5. This means that when x is 2, the value of the function f(x) is 5. On the graph of the parabola, this corresponds to the point (2, 5). By evaluating the function at x = 2, we have found another point on the parabola. This process of evaluating the function at different values helps us to build a better understanding of its shape and behavior. Each point we find gives us a piece of the puzzle, allowing us to visualize the function's graph more accurately. Just as we did with f(3), evaluating f(2) involves a straightforward process of substitution and simplification. The key is to follow the order of operations carefully to ensure that we arrive at the correct result. The ability to evaluate functions at different points is a fundamental skill in mathematics, and it is essential for solving a wide range of problems. Now that we have evaluated f(3) and f(2), we can move on to the next challenge: evaluating f(3 + 5).

Evaluating f(3 + 5)

Now, let's evaluate the function f(x) = 4x² - 2x - 7 at x = 3 + 5. This might seem a bit more complex, but the process is essentially the same. First, we need to simplify the expression inside the parentheses: 3 + 5 = 8. So, we are actually evaluating f(8). Substituting x = 8 into the function, we get: f(8) = 4(8)² - 2(8) - 7. Following the order of operations (PEMDAS/BODMAS), we first calculate 8² (8 squared), which is 8 * 8 = 64. So, our expression becomes: f(8) = 4(64) - 2(8) - 7. Next, we perform the multiplications. 4 multiplied by 64 is 256, and 2 multiplied by 8 is 16. The expression now looks like this: f(8) = 256 - 16 - 7. Finally, we perform the subtractions from left to right. 256 minus 16 is 240, and then 240 minus 7 is 233. Therefore, f(8) = 233. This means that when x is 8, the value of the function f(x) is 233. On the graph of the parabola, this corresponds to the point (8, 233). As we have seen in the previous examples, evaluating a function involves substituting a value for x and simplifying the expression. In this case, we first simplified the expression inside the parentheses (3 + 5) to obtain the value of x that we needed to substitute. This is a common technique in mathematics, and it is important to pay attention to the order of operations to ensure that we arrive at the correct result. Evaluating f(3 + 5) demonstrates that we can substitute not only single numbers but also expressions into a function. This flexibility is one of the key strengths of functions, as it allows us to model complex relationships between variables. Now that we have successfully evaluated f(3), f(2), and f(3 + 5), we have a better understanding of how the function f(x) = 4x² - 2x - 7 behaves.

Conclusion

In this comprehensive guide, we have explored the process of evaluating the quadratic function f(x) = 4x² - 2x - 7 for different values of x. We have demonstrated how to substitute these values into the function and perform the necessary arithmetic operations to obtain the corresponding function values. We started by understanding the function itself, identifying its coefficients and their roles in determining the shape and behavior of the parabola. We then evaluated the function at x = 3, x = 2, and x = 3 + 5, carefully following the order of operations (PEMDAS/BODMAS) to ensure accurate results. Through these examples, we have seen that evaluating a function is a fundamental skill in mathematics that allows us to analyze the behavior of functions and solve related problems. This skill is essential for understanding how functions work and how they can be used to model real-world phenomena. The process of substitution and simplification is at the heart of function evaluation, and mastering this process is crucial for success in algebra and calculus. By evaluating the function at different values, we can gain insights into its properties and how its output changes in response to changes in the input. We have also seen that we can substitute not only single numbers but also expressions into a function, which expands the possibilities for modeling complex relationships. The ability to evaluate functions accurately and efficiently is a valuable tool in mathematics and its applications. It allows us to make predictions, solve equations, and gain a deeper understanding of the world around us. We hope that this guide has provided you with a clear and comprehensive understanding of how to evaluate the quadratic function f(x) = 4x² - 2x - 7. By mastering this skill, you will be well-equipped to tackle more advanced mathematical concepts and problems. Keep practicing and exploring different functions, and you will continue to build your mathematical expertise. Remember, the key to success in mathematics is a solid foundation in the fundamentals, and function evaluation is a crucial part of that foundation.