Analyzing The Quadratic Function F(x) = 5x² - 8x + 6

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In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding various relationships. Among the diverse array of functions, quadratic functions hold a prominent position due to their widespread applications in fields ranging from physics and engineering to economics and computer science. In this comprehensive article, we embark on an in-depth exploration of the quadratic function f(x) = 5x² - 8x + 6, delving into its properties, behavior, and transformations. Our analysis will encompass evaluating the function at specific points, determining the difference quotient, and unraveling the significance of these calculations in the broader context of mathematical analysis.

(a) Evaluating f(x + h): Unveiling the Function's Transformation

Evaluating f(x + h) is a crucial step in understanding how the function f(x) transforms when its input is shifted by a constant value, h. This process involves substituting (x + h) for x in the original function, effectively shifting the graph of the function horizontally by h units. The resulting expression, f(x + h), provides insights into the function's behavior as its input changes, laying the groundwork for further analysis.

To evaluate f(x + h) for the given function f(x) = 5x² - 8x + 6, we meticulously replace each instance of x with (x + h), ensuring that the substitution is performed accurately. This yields the expression f(x + h) = 5(x + h)² - 8(x + h) + 6. The next step involves expanding and simplifying this expression to obtain a more manageable form. Expanding the squared term, (x + h)², results in x² + 2xh + h². Distributing the 5 and the -8, we get 5(x² + 2xh + h²) - 8(x + h) + 6 = 5x² + 10xh + 5h² - 8x - 8h + 6. This expanded form of f(x + h) reveals the individual contributions of x, h, and their combinations to the function's output.

The expanded expression for f(x + h), 5x² + 10xh + 5h² - 8x - 8h + 6, provides a clear representation of how the function's value changes as both x and h vary. The presence of terms involving both x and h, such as 10xh, indicates an interaction between the input variable x and the shift parameter h. This interaction is crucial in understanding the function's sensitivity to changes in its input and the resulting impact on its output. The constant term, 6, remains unaffected by the shift, highlighting its role as a vertical intercept of the function's graph.

In essence, evaluating f(x + h) allows us to dissect the function's behavior under horizontal transformations, providing a foundation for understanding concepts such as the derivative and the rate of change. This process forms the cornerstone of calculus and is indispensable in analyzing the dynamics of functions and their applications in various fields.

(b) Determining f(x + h) - f(x): Unveiling the Difference in Function Values

Determining f(x + h) - f(x) is a pivotal step in calculus, as it forms the basis for defining the derivative, a fundamental concept that quantifies the instantaneous rate of change of a function. This expression represents the difference in the function's values at two distinct points, x + h and x, providing insights into how the function's output changes over a small interval. The difference f(x + h) - f(x) is a crucial component in understanding the function's local behavior and its sensitivity to small variations in the input variable.

To calculate f(x + h) - f(x) for the given function f(x) = 5x² - 8x + 6, we leverage the previously computed expression for f(x + h), which is 5x² + 10xh + 5h² - 8x - 8h + 6. Subtracting the original function, f(x) = 5x² - 8x + 6, from this expression, we obtain f(x + h) - f(x) = (5x² + 10xh + 5h² - 8x - 8h + 6) - (5x² - 8x + 6). This step involves carefully distributing the negative sign and combining like terms to simplify the expression.

Upon simplification, the expression f(x + h) - f(x) becomes 10xh + 5h² - 8h. This simplified form reveals the key factors that contribute to the difference in function values. The term 10xh represents the interaction between the input variable x and the shift parameter h, indicating the function's sensitivity to changes in input. The term 5h² reflects the quadratic nature of the function and its influence on the difference in values. The term -8h represents the linear component of the function and its contribution to the difference. The constant terms cancel out during the subtraction, highlighting that the difference in function values is solely determined by the terms involving x and h.

The expression 10xh + 5h² - 8h encapsulates the essence of the function's change over a small interval, providing a foundation for understanding the concept of the derivative. By analyzing this expression, we can gain insights into how the function's rate of change varies with respect to the input variable x and the shift parameter h. This understanding is crucial in various applications, such as optimization problems, where the goal is to find the maximum or minimum value of a function.

(c) Computing (f(x + h) - f(x)) / h: Unveiling the Difference Quotient

Computing (f(x + h) - f(x)) / h, the difference quotient, is a cornerstone of calculus and serves as a precursor to the derivative. The difference quotient represents the average rate of change of the function f(x) over the interval [x, x + h]. It provides a measure of how the function's output changes per unit change in the input variable, offering valuable insights into the function's behavior and its sensitivity to variations in its input.

To compute the difference quotient for the given function f(x) = 5x² - 8x + 6, we utilize the previously calculated expression for f(x + h) - f(x), which is 10xh + 5h² - 8h. Dividing this expression by h, we obtain (f(x + h) - f(x)) / h = (10xh + 5h² - 8h) / h. This step involves factoring out h from the numerator and canceling it with the denominator, simplifying the expression.

Upon simplification, the difference quotient, (f(x + h) - f(x)) / h, becomes 10x + 5h - 8. This simplified form reveals the key components that determine the average rate of change of the function. The term 10x represents the linear component of the rate of change, indicating how the function's output changes proportionally to the input variable x. The term 5h reflects the quadratic nature of the function and its influence on the rate of change as the interval size h varies. The constant term -8 represents a constant contribution to the rate of change, independent of x and h.

The difference quotient, 10x + 5h - 8, provides a crucial approximation of the function's instantaneous rate of change at a specific point. As the interval size h approaches zero, the difference quotient converges to the derivative of the function, which represents the exact rate of change at that point. The derivative is a fundamental concept in calculus and is used extensively in various applications, such as optimization, curve sketching, and modeling physical phenomena.

In conclusion, the difference quotient serves as a bridge between the average rate of change and the instantaneous rate of change, providing a powerful tool for analyzing the behavior of functions and their applications in diverse fields.

By meticulously evaluating f(x + h), determining f(x + h) - f(x), and computing the difference quotient (f(x + h) - f(x)) / h for the quadratic function f(x) = 5x² - 8x + 6, we have gained a comprehensive understanding of its transformations, behavior, and rate of change. These calculations lay the groundwork for further analysis of the function's properties, such as its vertex, axis of symmetry, and concavity. Moreover, the concepts explored in this article are fundamental to calculus and have wide-ranging applications in various fields of science, engineering, and mathematics.