Finding Y Value When X Equals 8 In Function Y=8-2x

by THE IDEN 51 views

In mathematics, functions play a crucial role in describing relationships between variables. One common way to represent a function is through ordered pairs, which illustrate the correspondence between input values (x) and output values (y). In this article, we will explore the function y = 8 - 2x and determine the value of y when x is equal to 8. This involves substituting the given value of x into the function and performing the necessary calculations. Understanding how to evaluate functions for specific input values is a fundamental skill in algebra and calculus, with applications ranging from simple problem-solving to complex mathematical modeling. This article aims to provide a clear, step-by-step explanation of the process, ensuring that readers can confidently apply this skill in various contexts. Let’s delve into the details and find the value of y when x equals 8 in the function y = 8 - 2x. This exploration will enhance your understanding of functions and their evaluations, a cornerstone of mathematical proficiency.

Understanding the Function

The function presented is a linear equation, specifically y = 8 - 2x. A linear equation is characterized by its straight-line graph on a coordinate plane, where the relationship between x and y is constant. In this equation:

  • 'y' is the dependent variable, which means its value depends on the value of 'x'.
  • 'x' is the independent variable, which can take any value.
  • '8' is the y-intercept, the point where the line crosses the y-axis (when x = 0).
  • '-2' is the slope, representing the rate of change of y with respect to x. A negative slope indicates that y decreases as x increases. Understanding these components is crucial for interpreting the behavior of the function. The slope tells us how steep the line is and in which direction it slants, while the y-intercept gives us a fixed point on the line. This linear equation can be used to predict y-values for any given x-value, which is a fundamental concept in algebra. This article will delve deeper into evaluating this function, showing how to substitute x-values to find corresponding y-values, a skill that is essential in various mathematical applications. Moreover, a solid grasp of linear functions forms the bedrock for understanding more complex mathematical concepts, including higher-degree polynomials and calculus. Thus, mastering the basics of linear equations is not just about solving immediate problems; it is about building a strong foundation for future mathematical studies.

Ordered Pairs and the Function

The table provided shows several ordered pairs (x, y) that satisfy the function y = 8 - 2x. An ordered pair is a set of two numbers written in the form (x, y), where the order is significant. Each ordered pair represents a point on the graph of the function. For example, the ordered pair (-3, 14) means that when x is -3, y is 14. Similarly, (-1, 10) means when x is -1, y is 10. These ordered pairs provide specific solutions to the equation, illustrating the relationship between x and y. By examining these pairs, we can verify that the function holds true for these values. For instance, substituting x = -3 into the equation gives y = 8 - 2(-3) = 8 + 6 = 14, confirming the first ordered pair. The table includes several such pairs, allowing us to see how changes in x affect y. This tabular representation is a practical way to present and analyze functional relationships, particularly in scenarios where patterns and trends are of interest. Understanding how to interpret and use ordered pairs is crucial for graphing functions, solving equations, and making predictions based on mathematical models. Each ordered pair acts as a specific data point that helps to define the line represented by the function. This approach is fundamental not only in mathematics but also in various fields such as statistics, economics, and engineering, where data analysis and interpretation are vital skills.

Determining the Value of y when x=8

To find the value of y when x is 8, we substitute x = 8 into the function y = 8 - 2x. This process involves replacing the variable x with the specific value we are interested in, which is 8 in this case. The substitution transforms the equation into a simple arithmetic problem, allowing us to calculate the corresponding y-value. This method is a fundamental technique in algebra and is used extensively to evaluate functions for different values of the independent variable. By substituting x = 8, we are essentially asking, “What is the output of the function when the input is 8?” This approach is not only applicable to linear functions but also to more complex functions, including polynomials, trigonometric functions, and exponential functions. Understanding how to substitute values into functions is a crucial skill for solving equations, graphing functions, and analyzing mathematical relationships. In this particular case, the substitution allows us to find a specific point on the line represented by the function y = 8 - 2x, thus contributing to our overall understanding of the function's behavior. This method also underscores the practical nature of algebra, demonstrating how abstract equations can be used to find concrete values and solutions.

Step-by-Step Calculation

  1. Substitute x = 8 into the equation: y = 8 - 2(8)

    This step involves replacing the variable x with the value 8 in the given function. It sets the stage for the arithmetic calculation that will yield the corresponding y-value. Substitution is a critical technique in algebra and is used across various mathematical contexts, from solving simple equations to evaluating complex functions. Understanding how to correctly substitute values is fundamental for any student of mathematics.

  2. Perform the multiplication: y = 8 - 16

    Here, we perform the multiplication operation, which follows the order of operations (PEMDAS/BODMAS). Multiplying -2 by 8 results in -16. This step simplifies the equation further, bringing us closer to the solution. Accuracy in performing arithmetic operations is crucial at this stage, as any error here will affect the final answer. The multiplication step highlights the importance of following established mathematical conventions to ensure correct results.

  3. Subtract: y = -8

    Finally, we subtract 16 from 8 to get -8. This step completes the calculation and provides the y-value corresponding to x = 8. The result, y = -8, is the solution we were seeking. It represents a specific point (8, -8) on the graph of the function y = 8 - 2x. This final calculation demonstrates the straightforward nature of evaluating linear functions once the substitution is done. The process of simplifying the equation through these steps showcases the logical progression inherent in mathematical problem-solving.

Result

The value of y when x = 8 in the function y = 8 - 2x is -8. This result indicates that the ordered pair (8, -8) is a solution to the equation. On the graph of the function, this point lies on the line defined by the equation y = 8 - 2x. The negative value of y suggests that this point is located in the fourth quadrant of the coordinate plane, where x values are positive and y values are negative. This specific solution is crucial for understanding the behavior of the function at x = 8. It provides a concrete example of how the independent variable x affects the dependent variable y. Moreover, this result can be used in various applications, such as plotting the graph of the function or solving related problems. The calculation underscores the importance of precise substitution and arithmetic operations in determining function values. Overall, the result y = -8 not only answers the immediate question but also enhances our understanding of the functional relationship between x and y as defined by the equation y = 8 - 2x.

In conclusion, evaluating the function y = 8 - 2x at x = 8 reveals that the value of y is -8. This process involves substituting the given x-value into the equation and performing the necessary arithmetic operations. This example demonstrates the fundamental principle of evaluating functions, a critical skill in mathematics and its applications. Understanding how to find the value of a function for a specific input is essential for solving equations, graphing functions, and analyzing mathematical relationships. The result, y = -8, provides a specific solution to the equation and a point (8, -8) on the graph of the function. This methodical approach to problem-solving highlights the logical and precise nature of mathematical calculations. By mastering these basic techniques, students can build a strong foundation for more advanced mathematical concepts. The ability to evaluate functions is not only crucial in algebra but also in calculus, statistics, and various other fields where mathematical models are used. Therefore, a solid understanding of this skill is invaluable for both academic and practical purposes. This detailed exploration of the function y = 8 - 2x serves as a clear example of how mathematical concepts can be applied to find specific solutions, enhancing overall mathematical proficiency.