Analyzing The Function F(x) = 5x² - 8x + 6: A Step-by-Step Solution

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In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. A particularly interesting function to explore is f(x) = 5x² - 8x + 6, a quadratic expression that unveils a parabolic curve when graphed. To gain a deeper understanding of this function, we will embark on a comprehensive analysis, delving into various aspects that illuminate its behavior and characteristics.

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

The first step in our exploration involves evaluating f(x + h), which signifies the function's value when the input is shifted by a constant h. This transformation provides insights into how the function's output changes as the input is perturbed. To determine f(x + h), we substitute (x + h) for x in the original function:

f(x + h) = 5(x + h)² - 8(x + h) + 6

Expanding the squared term and distributing the constants, we arrive at:

f(x + h) = 5(x² + 2xh + h²) - 8x - 8h + 6

Further simplification yields:

f(x + h) = 5x² + 10xh + 5h² - 8x - 8h + 6

This expression represents the function's value when the input is incremented by h. The presence of terms involving h indicates how the output changes in response to variations in the input. Understanding this transformation is crucial for analyzing the function's sensitivity to input changes and its overall behavior.

To elaborate further, let's break down each term in the expanded form of f(x + h):

  • 5x²: This term represents the original quadratic component of the function, scaled by a factor of 5.
  • 10xh: This term captures the interaction between the original input x and the increment h. It signifies how the change in input affects the quadratic component of the function.
  • 5h²: This term represents the squared increment h, scaled by a factor of 5. It reflects the contribution of the input change to the quadratic component of the function.
  • -8x: This term corresponds to the linear component of the original function.
  • -8h: This term represents the change in the linear component due to the input increment h.
  • 6: This constant term remains unaffected by the input change.

By carefully examining each term, we can gain a deeper appreciation for how the function's output is influenced by both the original input x and the increment h. This understanding is essential for further analysis, such as calculating the difference quotient, which will be explored in the subsequent sections.

(b) Unveiling the Difference: f(x + h) - f(x)

Now, let's delve into the difference between f(x + h) and f(x), which provides valuable insights into the function's rate of change. This difference, denoted as f(x + h) - f(x), quantifies the change in the function's output when the input is perturbed by h. To compute this difference, we subtract the original function f(x) from the expression we derived for f(x + h):

f(x + h) - f(x) = (5x² + 10xh + 5h² - 8x - 8h + 6) - (5x² - 8x + 6)

Simplifying the expression by canceling out common terms, we obtain:

f(x + h) - f(x) = 10xh + 5h² - 8h

This expression represents the change in the function's output as a result of the input increment h. The terms involving h reveal how the function's value varies with respect to changes in the input. This difference is a crucial component in calculating the average rate of change, which we will explore in the next section.

To further dissect this difference, let's analyze each term:

  • 10xh: This term, as before, captures the interaction between the original input x and the increment h. It represents the change in the function's output due to the combined effect of x and h.
  • 5h²: This term signifies the change in the function's output solely due to the squared increment h. It reflects the non-linear component of the change.
  • -8h: This term represents the change in the function's output due to the linear component of the original function and the increment h.

By carefully examining these terms, we can gain a deeper understanding of how different aspects of the function contribute to the overall change in output. This knowledge is essential for analyzing the function's behavior and predicting its response to variations in the input.

(c) The Difference Quotient: Unveiling the Average Rate of Change

The difference quotient, represented as [f(x + h) - f(x)] / h, is a fundamental concept in calculus that provides the average rate of change of a function over an interval of length h. It quantifies how much the function's output changes, on average, for each unit change in the input. To calculate the difference quotient for our function, we divide the difference we computed in the previous section by h:

[f(x + h) - f(x)] / h = (10xh + 5h² - 8h) / h

Factoring out h from the numerator, we get:

[f(x + h) - f(x)] / h = h(10x + 5h - 8) / h

Canceling out the common factor h, we arrive at:

[f(x + h) - f(x)] / h = 10x + 5h - 8

This expression represents the average rate of change of the function f(x) = 5x² - 8x + 6 over an interval of length h. As h approaches zero, this expression converges to the instantaneous rate of change, which is the derivative of the function. The derivative provides valuable information about the function's slope and its behavior at a specific point.

To further understand the difference quotient, let's analyze its components:

  • 10x: This term represents the contribution of the quadratic component of the original function to the average rate of change. It indicates how the rate of change varies linearly with the input x.
  • 5h: This term captures the effect of the interval length h on the average rate of change. As h becomes smaller, this term diminishes, and the average rate of change approaches the instantaneous rate of change.
  • -8: This constant term represents the contribution of the linear component of the original function to the average rate of change.

By carefully examining these components, we can gain a deeper appreciation for how the average rate of change is influenced by both the function's structure and the interval length h. This understanding is essential for applying the difference quotient in various contexts, such as approximating the derivative and analyzing the function's behavior.

In conclusion, by meticulously exploring f(x + h), f(x + h) - f(x), and the difference quotient, we have gained valuable insights into the behavior and characteristics of the function f(x) = 5x² - 8x + 6. These concepts form the foundation for further analysis in calculus and other areas of mathematics.