Finding H(-1) For The Function H(x) = 3x³ - Ax

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In this article, we will delve into the intricacies of the given function, h(x) = 3x³ - ax, and embark on a journey to determine the elusive value of h(-1). The key to unlocking this mathematical puzzle lies in the information provided: h(2) = 20. We will meticulously dissect the function, unravel the significance of the given value, and employ a strategic approach to calculate the desired outcome. Let's embark on this mathematical exploration together.

Decoding the Function h(x) = 3x³ - ax

At its core, h(x) = 3x³ - ax is a polynomial function, specifically a cubic function due to the presence of the x³ term. This means that the graph of this function will exhibit a characteristic S-shaped curve, potentially with local maxima and minima. The function consists of two primary components:

  1. 3x³: This term signifies a cubic relationship, indicating that as x increases, this component will grow rapidly, either positively or negatively depending on the sign of x. The coefficient 3 scales the cubic term, influencing the steepness of the curve.

  2. -ax: This term represents a linear component, where a is a constant coefficient. The presence of -ax introduces a linear influence on the function's behavior, potentially shifting or tilting the curve. The value of a dictates the slope of this linear component.

The interplay between these two components shapes the overall behavior of the function. The cubic term governs the long-term trend, while the linear term introduces local variations. Understanding the impact of the coefficient a is crucial, as it will affect the specific shape and position of the cubic curve.

To fully grasp the function, let's consider how it behaves for different values of x. For large positive values of x, the 3x³ term will dominate, causing the function to increase rapidly. Conversely, for large negative values of x, the 3x³ term will become significantly negative, pulling the function downwards. The linear term, -ax, will either add or subtract from this cubic trend, depending on the sign of a and x.

Utilizing h(2) = 20 to Find the Value of 'a'

The information h(2) = 20 is a critical piece of the puzzle. It provides us with a specific point on the function's graph: when x = 2, the function's output is 20. This seemingly simple piece of data allows us to determine the value of the unknown coefficient, a. By substituting x = 2 and h(2) = 20 into the function's equation, we can establish an equation solely in terms of a:

20 = 3(2)³ - a(2)

Simplifying this equation, we get:

20 = 3(8) - 2a

20 = 24 - 2a

Now, we can isolate a by performing algebraic manipulations:

2a = 24 - 20

2a = 4

a = 2

Therefore, we have successfully determined the value of a to be 2. This revelation transforms our function from a general form into a specific equation:

h(x) = 3x³ - 2x

With the value of a in hand, we now have a complete and well-defined function. This is a significant step forward, as it allows us to calculate the value of h(x) for any given x. Our next mission is to leverage this newfound knowledge to find the elusive value of h(-1).

Calculating h(-1) with the Determined Value of 'a'

Now that we have established the specific form of the function, h(x) = 3x³ - 2x, calculating h(-1) is a straightforward process. We simply substitute x = -1 into the equation:

h(-1) = 3(-1)³ - 2(-1)

Let's break down this calculation step-by-step:

  1. (-1)³ = -1: The cube of -1 is -1, as -1 * -1 * -1 = -1.

  2. 3(-1) = -3: Multiplying the result of the previous step by 3 gives us -3.

  3. 2(-1) = -2: Multiplying 2 by -1 yields -2.

  4. -2(-1) = +2: Since there is a negative sign in front of 2x term, we need to substract negative 2, which is mathematically equal to adding positive 2.

Now, substituting these values back into the equation, we get:

h(-1) = -3 + 2

h(-1) = -1

Therefore, the value of h(-1) is -1. We have successfully navigated the mathematical landscape, deciphered the function, and arrived at our final destination. This result represents a specific point on the graph of the function, indicating the function's output when x is -1.

Conclusion: The Value of h(-1) is -1

In conclusion, by meticulously analyzing the given function, h(x) = 3x³ - ax, and leveraging the information h(2) = 20, we have successfully determined the value of h(-1). Our journey involved decoding the function's components, solving for the unknown coefficient a, and ultimately substituting x = -1 into the refined equation. The final result, h(-1) = -1, underscores the power of mathematical reasoning and problem-solving techniques. This exploration not only provides a concrete answer but also illuminates the intricate workings of polynomial functions and the significance of specific data points in unraveling their mysteries.