Finding Ordered Pairs For The Function Y = 16 + 0.5x

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In the realm of mathematics, understanding functions and their representations is paramount. Among the various ways to represent a function, ordered pairs play a crucial role. Ordered pairs, denoted as (x, y), provide a direct mapping between the input variable 'x' and the corresponding output variable 'y'. In this article, we delve into the function y = 16 + 0.5x and explore how to identify ordered pairs that satisfy this function. We will dissect the concept of ordered pairs, their significance in representing functions, and how to verify if a given ordered pair belongs to a specific function. Furthermore, we will address the specific problem of finding the missing ordered pair for the function y = 16 + 0.5x, providing a step-by-step solution and highlighting the underlying mathematical principles. This comprehensive exploration aims to solidify your understanding of functions and ordered pairs, equipping you with the skills to solve similar problems confidently. Grasping the concept of functions and how they manifest in ordered pairs is fundamental to your mathematical journey, opening doors to more advanced topics and applications.

H2: Unveiling the Significance of Ordered Pairs in Function Representation

Ordered pairs serve as the fundamental building blocks for representing functions graphically and analytically. In essence, an ordered pair (x, y) signifies a specific point on the coordinate plane, where 'x' represents the horizontal coordinate (abscissa) and 'y' represents the vertical coordinate (ordinate). For a function, each 'x' value is associated with a unique 'y' value, and this relationship is precisely captured by the set of all ordered pairs that satisfy the function's equation. This set of ordered pairs constitutes the function's graph, a visual representation that allows us to understand the function's behavior and characteristics. For example, a linear function will have a straight-line graph, while a quadratic function will have a parabolic graph. By plotting ordered pairs, we can visually discern the function's increasing or decreasing nature, its intercepts with the axes, and any points of maxima or minima. Beyond graphical representation, ordered pairs also enable us to analyze functions algebraically. By substituting specific 'x' values into the function's equation, we can calculate the corresponding 'y' values and generate ordered pairs. These ordered pairs can then be used to determine the function's properties, such as its domain (the set of all possible 'x' values) and range (the set of all possible 'y' values). Understanding the interplay between ordered pairs, function equations, and graphical representations is crucial for a comprehensive grasp of mathematical functions.

H2: Dissecting the Function y = 16 + 0.5x

The function under consideration, y = 16 + 0.5x, is a linear function. Linear functions are characterized by their straight-line graphs and a constant rate of change. In this particular function, the coefficient of 'x' (0.5) represents the slope of the line, indicating the rate at which 'y' changes with respect to 'x'. The constant term (16) represents the y-intercept, the point where the line crosses the y-axis. To further understand this function, let's analyze its components. The term '0.5x' signifies that for every unit increase in 'x', the value of 'y' increases by 0.5 units. This positive slope indicates that the function is increasing; as 'x' increases, 'y' also increases. The y-intercept of 16 tells us that when 'x' is 0, 'y' is 16. This point (0, 16) lies on the graph of the function. To find other points on the graph, we can substitute different values of 'x' into the equation and calculate the corresponding 'y' values. For instance, if we substitute x = 2, we get y = 16 + 0.5(2) = 17. This gives us the ordered pair (2, 17). By generating a series of such ordered pairs, we can accurately plot the graph of the function and visually confirm its linear nature. Analyzing the equation y = 16 + 0.5x allows us to predict the function's behavior and identify specific points that lie on its graph.

H2: Problem Analysis: Identifying the Missing Ordered Pair

The core question revolves around identifying the missing ordered pair that satisfies the function y = 16 + 0.5x from a given set of options. This task requires us to verify which of the provided ordered pairs adheres to the function's equation. In simpler terms, we need to check if substituting the 'x' and 'y' values from each ordered pair into the equation results in a true statement. The provided options are (0, 18), (5, 19.5), (8, 20), and (10, 21.5). To determine the correct ordered pair, we will systematically substitute the 'x' and 'y' values from each option into the equation y = 16 + 0.5x. If the equation holds true, then the ordered pair belongs to the function. If the equation does not hold true, then the ordered pair does not belong to the function. This process of substitution and verification is a fundamental technique for working with functions and ordered pairs. It allows us to confirm whether a point lies on the graph of a function and to identify solutions that satisfy the function's equation. By carefully applying this method, we can confidently pinpoint the missing ordered pair in the given problem.

H2: Step-by-Step Solution: Finding the Correct Ordered Pair

Let's embark on a step-by-step solution to identify the correct ordered pair that satisfies the equation y = 16 + 0.5x. We will examine each of the given options meticulously:

  1. (0, 18): Substitute x = 0 and y = 18 into the equation: 18 = 16 + 0.5(0). This simplifies to 18 = 16, which is false. Therefore, (0, 18) is not a solution.
  2. (5, 19.5): Substitute x = 5 and y = 19.5 into the equation: 19.5 = 16 + 0.5(5). This simplifies to 19.5 = 16 + 2.5, which further simplifies to 19.5 = 19.5. This is true. Therefore, (5, 19.5) is a solution.
  3. (8, 20): Substitute x = 8 and y = 20 into the equation: 20 = 16 + 0.5(8). This simplifies to 20 = 16 + 4, which further simplifies to 20 = 20. This is true. Therefore, (8, 20) is also a solution.
  4. (10, 21.5): Substitute x = 10 and y = 21.5 into the equation: 21.5 = 16 + 0.5(10). This simplifies to 21.5 = 16 + 5, which further simplifies to 21.5 = 21. This is false. Therefore, (10, 21.5) is not a solution.

Based on our analysis, both (5, 19.5) and (8, 20) satisfy the equation y = 16 + 0.5x. If the question implies only one missing ordered pair, there might be an error in the question itself or in the provided options. However, if multiple solutions are possible, then both (5, 19.5) and (8, 20) are valid missing ordered pairs. This step-by-step approach demonstrates the importance of systematic substitution and verification when working with functions and ordered pairs.

H2: Conclusion: Mastering Functions and Ordered Pairs

In conclusion, this exploration has illuminated the significance of ordered pairs in representing functions, specifically focusing on the linear function y = 16 + 0.5x. We have demonstrated how ordered pairs act as coordinates on a graph, mapping input values ('x') to corresponding output values ('y'). The process of identifying ordered pairs that satisfy a function's equation involves substituting the 'x' and 'y' values into the equation and verifying if the equation holds true. Through this systematic approach, we successfully identified the ordered pairs (5, 19.5) and (8, 20) as solutions for the given function. This exercise underscores the fundamental connection between functions, equations, and their graphical representations. A solid understanding of functions and ordered pairs is crucial for further studies in mathematics and related fields. Mastering these concepts allows you to analyze and interpret mathematical relationships effectively, solve problems confidently, and build a strong foundation for more advanced topics. By grasping the principles discussed in this article, you have taken a significant step towards mathematical proficiency.