Analyzing The Quadratic Expression -1.5(40-60)^2+15,700 A Comprehensive Guide

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This article delves into the mathematical expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700, providing a comprehensive analysis and interpretation. We will explore the expression's structure, its potential applications, and the mathematical principles it embodies. This exploration aims to not only understand the numerical result but also to appreciate the underlying mathematical concepts. Our primary focus will be on dissecting the components of the expression, understanding their individual roles, and then synthesizing this knowledge to comprehend the expression as a whole. We will consider the expression within the broader context of quadratic functions and their properties.

Understanding the Expression's Components

To effectively analyze the expression, we need to break it down into its constituent parts. The expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 comprises several key elements, each contributing significantly to the final value. Let's examine each component individually:

  1. The Constant Term: The expression begins with the constant -1.5. This value acts as a coefficient that scales the squared term that follows. The negative sign indicates that the parabola represented by the quadratic expression will open downwards, and the magnitude (1.5) dictates the steepness of the curve. A larger magnitude would result in a steeper parabola, while a smaller magnitude would produce a flatter curve. This constant plays a crucial role in determining the overall shape and orientation of the quadratic function represented by the expression. The negative sign is particularly important as it indicates that the expression will have a maximum value rather than a minimum.

  2. The Parenthetical Term: The term (40-60) represents a difference. Here, we are subtracting 60 from 40, resulting in -20. This difference is a critical part of the expression as it forms the base of the squared term. The result of this subtraction, -20, will be squared in the subsequent step, which significantly impacts the overall value of the expression. The difference (40-60) highlights the concept of displacement or shift in the context of quadratic functions. This shift is essential for understanding the location of the vertex of the parabola represented by the expression.

  3. The Squared Term: The expression (-20)^2 signifies the squaring operation. Squaring -20 yields 400. This squaring operation is fundamental to the quadratic nature of the expression. Squaring a negative number results in a positive number, which in this case is 400. The squared term introduces the non-linear behavior characteristic of quadratic functions. It is the squared term that gives the expression its parabolic shape when graphed. Understanding the impact of the squared term is crucial for predicting the behavior of the expression as the values within the parentheses change.

  4. The Multiplication: Next, we multiply -1.5 by the result of the squared term, 400. This multiplication yields -600. The multiplication by -1.5 scales the squared term, and the negative sign flips the parabola vertically. This step is crucial in determining the overall magnitude and sign of the quadratic component of the expression. The multiplication combines the scaling effect of the constant term with the non-linear effect of the squared term. The resulting value, -600, represents a significant portion of the final result.

  5. The Constant Addition: Finally, we add 15,700 to the result of the previous steps (-600). This addition shifts the entire expression vertically. Adding 15,700 to -600 gives us 15,100. This final addition is crucial for determining the vertical position of the parabola. The constant addition represents the vertical translation of the quadratic function. It determines the y-coordinate of the vertex of the parabola, which in this case represents the maximum value of the expression.

By understanding each of these components, we gain a clear picture of how the expression works and how its value is derived. This detailed breakdown is essential for appreciating the expression's mathematical properties and its potential applications in various fields.

Evaluating the Expression

Having dissected the components, we can now evaluate the expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 step-by-step to arrive at the final numerical answer. This process not only provides the solution but also reinforces our understanding of the order of operations and the impact of each component on the overall result. The evaluation process is a practical application of the mathematical principles we have discussed. It demonstrates how the individual components interact to produce the final value.

  1. Solve the Parentheses: First, we address the expression within the parentheses: (40-60) = -20. This step is straightforward but crucial as it sets the stage for the subsequent operations. The negative result, -20, will play a significant role in the final outcome due to the squaring operation that follows.

  2. Square the Result: Next, we square the result from the parentheses: (-20)^2 = 400. Squaring -20 yields a positive value, 400. This is a fundamental property of squaring operations, where any number multiplied by itself results in a positive value. This step highlights the non-linear nature of the expression. The squaring operation is a key characteristic of quadratic expressions, leading to the parabolic shape when graphed.

  3. Multiply by the Constant: Now, we multiply the squared result by the constant -1.5: -1.5 * 400 = -600. This multiplication scales the squared term and introduces a negative sign, which is crucial for understanding the orientation of the parabola. The negative sign indicates that the parabola opens downwards. The constant multiplier plays a vital role in determining the steepness and direction of the parabola. The result, -600, is a significant component of the final value.

  4. Add the Constant Term: Finally, we add the constant term 15,700 to the result of the previous step: -600 + 15,700 = 15,100. This addition shifts the entire expression vertically, determining the final value. The addition of 15,700 is the final step in calculating the expression's value. This constant term represents the vertical translation of the parabola, directly influencing the maximum or minimum value of the expression.

Therefore, the final result of the expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 is 15,100. This step-by-step evaluation demonstrates the importance of following the order of operations and understanding the role of each component in the expression. The result, 15,100, represents the maximum value of the quadratic function defined by the expression.

Interpreting the Result and Connecting to Quadratic Functions

The result, 15,100, holds significant meaning when interpreted within the context of quadratic functions. The expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 is a specific instance of a quadratic function in vertex form, which is given by the general formula: f(x) = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. Understanding the vertex form is crucial for interpreting the expression's behavior and its graphical representation.

In our expression, we can map the components to the general form as follows:

  • a = -1.5
  • x can be generalized as a variable input
  • h = 60
  • k = 15,700

From this, we can infer several key characteristics of the quadratic function:

  1. Vertex: The vertex of the parabola is at the point (60, 15,700). This point represents either the maximum or minimum value of the function, depending on the sign of a. The vertex is a critical feature of a parabola, providing insights into the function's extreme values and symmetry. In this case, since a is negative (-1.5), the parabola opens downwards, and the vertex represents the maximum point.

  2. Maximum Value: The y-coordinate of the vertex, 15,700, represents the maximum value of the function. This is because the parabola opens downwards, and the vertex is the highest point on the curve. The fact that 15,700 is the maximum value is a direct consequence of the negative coefficient a. The maximum value is a crucial piece of information, indicating the upper bound of the function's output.

  3. Axis of Symmetry: The x-coordinate of the vertex, 60, defines the axis of symmetry for the parabola. This is a vertical line x = 60, about which the parabola is symmetric. The axis of symmetry is a line that divides the parabola into two mirror-image halves. The axis of symmetry is a fundamental property of parabolas, reflecting the symmetry inherent in quadratic functions.

  4. Impact of the -1.5: The coefficient -1.5 affects the shape of the parabola. The negative sign indicates that the parabola opens downwards, and the magnitude (1.5) influences the steepness of the curve. A larger magnitude would result in a steeper parabola, while a smaller magnitude would produce a flatter curve. The coefficient a plays a crucial role in determining the parabola's shape and orientation.

  5. Value at x=40: Evaluating the expression essentially calculates the value of the quadratic function at x = 40. Substituting x = 40 into the expression, we find the corresponding y-value, which is 15,100. This means that when x is 40, the function's value is 15,100. The evaluation demonstrates the function's behavior at a specific point, providing a concrete example of the function's input-output relationship.

In summary, the expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 represents a quadratic function with a maximum value of 15,700 at x = 60. The parabola opens downwards, and the function's value at x = 40 is 15,100. Understanding these characteristics allows us to appreciate the broader implications of the expression within the realm of quadratic functions. By connecting the specific expression to the general form of quadratic functions, we gain a deeper understanding of its mathematical properties and behavior. This interpretation extends beyond the numerical result, providing insights into the function's graphical representation and its broader significance in mathematical contexts.

Real-World Applications and Further Exploration

While the expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 appears purely mathematical, it has connections to various real-world applications and serves as a foundation for further exploration in mathematics and related fields. Quadratic functions, in general, are used extensively in modeling scenarios involving parabolic trajectories, optimization problems, and various physical phenomena. Recognizing the real-world applications of quadratic functions enhances our appreciation for their practical significance and versatility.

  1. Projectile Motion: One of the most common applications of quadratic functions is in modeling projectile motion. The path of a projectile, such as a ball thrown in the air, can be accurately represented by a parabola. The expression we analyzed could, in a simplified scenario, represent the height of a projectile at a given time or distance. Understanding projectile motion is crucial in fields such as physics, engineering, and sports.

  2. Optimization Problems: Quadratic functions are also widely used in optimization problems, where the goal is to find the maximum or minimum value of a function. In business, for example, a quadratic function might represent the profit earned from selling a certain number of products. Finding the vertex of the parabola would then correspond to finding the optimal quantity to maximize profit. Optimization problems are fundamental in economics, business, and various scientific disciplines.

  3. Engineering Applications: Engineers use quadratic functions in designing structures, bridges, and other constructions. The parabolic shape is inherently strong and stable, making it ideal for arches and suspension cables. The expression we examined could represent a component of a larger engineering model. Engineering applications highlight the practical importance of quadratic functions in creating efficient and safe structures.

  4. Curve Fitting: In data analysis and statistics, quadratic functions can be used to fit curves to data points. This is particularly useful when the relationship between variables is not linear but exhibits a curved trend. The expression could be part of a larger model used to analyze experimental data. Curve fitting is an essential tool in scientific research, allowing researchers to identify patterns and make predictions.

Further exploration of the expression and its related concepts could involve:

  • Graphing the Function: Plotting the graph of the quadratic function f(x) = -1.5(x-60)^2 + 15,700 provides a visual representation of its behavior. The graph would clearly show the vertex, axis of symmetry, and the downward-opening parabola. Graphing is a powerful tool for visualizing mathematical functions and understanding their properties.
  • Varying Parameters: Investigating how changing the coefficients (-1.5 and 15,700) and the value inside the parentheses (60) affects the shape and position of the parabola. This exploration would deepen the understanding of the role of each parameter in the quadratic function. Parameter variation is a valuable method for understanding the sensitivity and robustness of mathematical models.
  • Solving Related Equations: Exploring related equations, such as finding the roots of the quadratic equation -1.5(x-60)^2 + 15,700 = 0, would provide additional insights into the function's properties. Solving equations is a fundamental skill in mathematics, allowing us to find specific values that satisfy certain conditions.

In conclusion, the expression −1.5(40−60)2+15,700-1.5(40-60)^2+15,700 serves as a gateway to understanding quadratic functions and their applications. By dissecting the expression, evaluating it, and interpreting the result within the context of quadratic functions, we gain a deeper appreciation for its mathematical significance and real-world relevance. This exploration underscores the importance of mathematical literacy and its role in solving problems and understanding the world around us. The potential for further exploration highlights the rich and interconnected nature of mathematics and its ability to provide insights across various disciplines.