Exploring F(x) = -1 + 6x^2 Calculating F(a), F(a+h) And The Difference Quotient
#Introduction In this article, we will delve into the function f(x) = -1 + 6x². This quadratic function presents interesting properties and behaviors that we can explore through various calculations and analyses. Our primary goal is to calculate specific values related to this function, namely f(a), f(a+h), and the difference quotient (f(a+h) - f(a))/h. Understanding these values will provide insights into the function's behavior, particularly its rate of change and how it varies with different inputs. We'll break down each calculation step-by-step, ensuring a clear and comprehensive understanding for readers of all backgrounds. This exploration will not only enhance our understanding of this specific function but also strengthen our grasp of general function evaluation and analysis techniques.
Our first task is to calculate the value of the function f(x) when the input is a. This means we need to substitute x with a in the function's expression. The function is given by f(x) = -1 + 6x². Substituting x with a, we get f(a) = -1 + 6a². This expression represents the value of the function at the point x = a. It's a straightforward substitution, but it's a crucial first step in understanding the function's behavior. The value f(a) essentially tells us the y-coordinate on the graph of the function corresponding to the x-coordinate a. This single value, while simple to calculate, serves as a building block for more complex analyses, such as calculating the average rate of change of the function over an interval. Understanding how to evaluate a function at a specific point is fundamental in calculus and other areas of mathematics. Furthermore, the expression f(a) = -1 + 6a² highlights the quadratic nature of the function, where the output is dependent on the square of the input. This quadratic relationship leads to the characteristic parabolic shape of the function's graph. By understanding the algebraic representation of f(a), we can begin to visualize and interpret the function's behavior graphically.
Next, we need to calculate f(a+h). This involves substituting (a+h) for x in the function f(x) = -1 + 6x². So, we have f(a+h) = -1 + 6(a+h)². This step is slightly more complex than calculating f(a) because we need to expand the squared term. Let's break it down: (a+h)² = (a+h)(a+h) = a² + 2ah + h². Now, we substitute this back into our expression for f(a+h): f(a+h) = -1 + 6(a² + 2ah + h²). Finally, we distribute the 6: f(a+h) = -1 + 6a² + 12ah + 6h². This expression for f(a+h) is crucial for understanding how the function's value changes when we slightly perturb the input a by a small amount h. The h term represents a small change or increment in the x-value. By calculating f(a+h), we are essentially finding the y-value of the function at a point slightly shifted from a. This concept is fundamental in calculus, particularly in the definition of the derivative, which measures the instantaneous rate of change of a function. The expansion and simplification of f(a+h) reveal the various components that contribute to the function's value at this slightly shifted point. Specifically, the terms involving h (i.e., 12ah and 6h²) indicate how the function's value changes as we move away from x = a. This calculation sets the stage for the next step, where we will use f(a) and f(a+h) to calculate the difference quotient.
The difference quotient, (f(a+h) - f(a))/h, is a fundamental concept in calculus. It represents the average rate of change of the function f(x) over the interval from a to a+h. To calculate this, we'll use the expressions we found earlier: f(a) = -1 + 6a² and f(a+h) = -1 + 6a² + 12ah + 6h². First, let's find the difference f(a+h) - f(a): f(a+h) - f(a) = (-1 + 6a² + 12ah + 6h²) - (-1 + 6a²). Notice that the -1 and 6a² terms cancel out: f(a+h) - f(a) = 12ah + 6h². Now, we divide this difference by h: (f(a+h) - f(a))/h = (12ah + 6h²)/h. We can factor out an h from the numerator: (f(a+h) - f(a))/h = h(12a + 6h)/h. Finally, we cancel the h terms, assuming h is not zero: (f(a+h) - f(a))/h = 12a + 6h. This simplified expression, 12a + 6h, is the difference quotient for the function f(x) = -1 + 6x². It tells us how much the function's value changes, on average, for a small change h in the input. The difference quotient is a precursor to the derivative, which is found by taking the limit of the difference quotient as h approaches zero. In this case, as h approaches 0, the difference quotient 12a + 6h approaches 12a, which is the derivative of f(x) at x = a. The difference quotient provides a crucial link between the average rate of change and the instantaneous rate of change, which is a cornerstone of differential calculus. Furthermore, understanding the difference quotient helps in approximating the function's behavior near a specific point. By choosing a small value for h, we can use the difference quotient to estimate the change in f(x) for a small change in x. This approximation technique has wide applications in various fields, including physics, engineering, and economics.
In this exploration, we successfully calculated the following values for the function f(x) = -1 + 6x²:
- f(a) = -1 + 6a²
- f(a+h) = -1 + 6a² + 12ah + 6h²
- (f(a+h) - f(a))/h = 12a + 6h
These results provide a comprehensive understanding of the function's behavior at a specific point a and in its vicinity. The value f(a) gives us the function's output at x = a, while f(a+h) describes the output at a slightly shifted point. The difference quotient, (f(a+h) - f(a))/h, quantifies the average rate of change of the function over the interval from a to a+h. This quantity is particularly significant as it forms the basis for the concept of the derivative in calculus. The process of calculating these values demonstrates the fundamental techniques of function evaluation and algebraic manipulation. By substituting variables, expanding expressions, and simplifying results, we gain a deeper appreciation for the function's underlying structure. The difference quotient, in particular, highlights the connection between average rates of change and instantaneous rates of change, which is a central theme in calculus. Furthermore, the results we obtained can be used for various applications. For instance, we can use the difference quotient to approximate the function's change over a small interval or to estimate the function's value at a point near a. These approximation techniques are widely used in numerical analysis and other fields. Overall, this exploration has not only provided us with specific values related to the function f(x) = -1 + 6x² but has also reinforced our understanding of fundamental mathematical concepts and techniques.
By meticulously calculating f(a), f(a+h), and the difference quotient (f(a+h) - f(a))/h for the function f(x) = -1 + 6x², we have gained valuable insights into its behavior and properties. These calculations exemplify the core principles of function evaluation and algebraic manipulation. The difference quotient, in particular, serves as a crucial stepping stone towards understanding the concept of the derivative, a cornerstone of calculus. This exploration not only provides concrete results for this specific function but also reinforces the general methodology for analyzing functions and their rates of change. The techniques we employed, such as substitution, expansion, and simplification, are applicable to a wide range of mathematical problems. Furthermore, the understanding gained from this analysis can be extended to other areas of mathematics and science. For instance, the concept of the rate of change is fundamental in physics, where it is used to describe velocity and acceleration. In economics, it is used to analyze marginal cost and marginal revenue. The ability to calculate and interpret these quantities is essential for problem-solving and decision-making in various fields. In conclusion, this exploration of the function f(x) = -1 + 6x² has provided a valuable learning experience. By systematically calculating key values and understanding their significance, we have strengthened our grasp of fundamental mathematical concepts and techniques. This knowledge will serve as a solid foundation for further studies in calculus and related fields.
