Finding Y Value For Function Y=8-2x When X=8

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#introduction In the realm of mathematics, functions play a crucial role in describing relationships between variables. One such function is the linear equation y = 8 - 2x, which represents a straight line when plotted on a graph. Understanding this function involves exploring how the value of y changes as the value of x varies. In this article, we delve into the function y = 8 - 2x, analyze its ordered pairs, and most importantly, determine the value of y when x equals 8. This exploration will not only enhance our understanding of linear functions but also demonstrate the practical application of algebraic concepts. The ability to evaluate functions for specific input values is a fundamental skill in mathematics, and mastering it opens doors to more complex mathematical concepts and problem-solving scenarios. This article aims to provide a comprehensive explanation and guide you through the process step by step, ensuring a clear understanding of how to work with linear functions and find specific values.

Understanding the Function y = 8 - 2x

The function y = 8 - 2x is a linear function, meaning it represents a straight line when graphed on a coordinate plane. The equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope m is -2, indicating that for every 1 unit increase in x, y decreases by 2 units. The y-intercept b is 8, meaning the line crosses the y-axis at the point (0, 8). Understanding these parameters helps us visualize the behavior of the function. A negative slope indicates that the line slopes downwards from left to right, and the y-intercept gives us a starting point on the graph. By knowing the slope and y-intercept, we can quickly sketch the line or predict the value of y for a given x. Linear functions are prevalent in various real-world applications, such as modeling relationships between distance and time, cost and quantity, and many other scenarios where a constant rate of change is involved. Therefore, a strong grasp of linear functions is essential for problem-solving in both mathematics and practical contexts. Furthermore, linear functions serve as a foundation for more advanced mathematical concepts, such as calculus and linear algebra, making their understanding crucial for further studies in mathematics and related fields.

Analyzing the Ordered Pairs

The table provided presents a series of ordered pairs (x, y) that satisfy the function y = 8 - 2x. Each pair represents a point on the line defined by the function. Let's examine these pairs:

  • (-3, 14): When x is -3, y is 14.
  • (-1, 10): When x is -1, y is 10.
  • (1, 6): When x is 1, y is 6.
  • (4, 0): When x is 4, y is 0.
  • (8, ?): This is the pair we need to solve for y when x is 8.
  • (10, -12): When x is 10, y is -12.

These ordered pairs demonstrate the linear relationship between x and y. As x increases, y decreases at a constant rate, which is consistent with the negative slope of the function. By plotting these points on a graph, we would see that they form a straight line. The table provides a discrete set of points that represent the continuous line defined by the function. Analyzing these points helps us understand the function's behavior and predict other values. For instance, we can observe the pattern that for every 1 unit increase in x, y decreases by 2 units, which is the slope of the line. This pattern allows us to fill in missing values or extrapolate beyond the given data. The ordered pairs serve as concrete examples of the abstract function, making the concept more tangible and easier to grasp. Understanding how to interpret and analyze ordered pairs is a fundamental skill in mathematics and is crucial for working with functions and graphs.

Determining the Value of y when x = 8

To find the value of y when x = 8, we substitute x with 8 in the function y = 8 - 2x:

y = 8 - 2(8)

Now, we perform the arithmetic:

y = 8 - 16

y = -8

Therefore, when x = 8, the value of y is -8. This means the ordered pair (8, -8) is also a solution to the function and lies on the line represented by the equation. This calculation demonstrates the direct application of the function to find specific values. By substituting the given value of x into the equation, we can easily determine the corresponding value of y. This process is fundamental in mathematics and is used extensively in various fields, including science, engineering, and economics. The ability to evaluate functions for specific input values is a key skill for problem-solving and data analysis. In this case, we found that when x is 8, y is -8, which completes the ordered pair in the table. This result confirms the linear relationship between x and y and provides another point on the line defined by the function. Understanding this process allows us to predict and calculate values for any given x, making the function a powerful tool for modeling and analysis.

Conclusion

In this exploration of the function y = 8 - 2x, we have successfully determined the value of y when x equals 8. By substituting x = 8 into the equation, we found that y = -8. This exercise highlights the importance of understanding and applying functions in mathematics. Linear functions, like the one we examined, are fundamental building blocks in mathematics and have wide-ranging applications in various fields. The ability to analyze ordered pairs and evaluate functions for specific values is a crucial skill for problem-solving and mathematical reasoning. Furthermore, this process reinforces the concept of a functional relationship, where the value of one variable depends on the value of another. Understanding this relationship is essential for interpreting data, making predictions, and modeling real-world phenomena. As we continue our mathematical journey, mastering these fundamental concepts will pave the way for tackling more complex problems and exploring advanced mathematical topics. The skills we have practiced in this article, such as substituting values into equations and performing arithmetic operations, are foundational for success in mathematics and related disciplines. Therefore, a thorough understanding of these concepts is crucial for both academic and practical applications. By consistently practicing and applying these skills, we can build a strong foundation in mathematics and confidently approach more challenging problems.