Plotting Ordered Pairs For A Linear Function F(x) = (3/2)x + 4

In the fascinating world of mathematics, functions serve as fundamental building blocks for modeling relationships and understanding how quantities change together. Among the diverse family of functions, linear functions hold a special place due to their simplicity and widespread applicability. In this article, we delve into the realm of linear functions, focusing on the specific function f(x) = (3/2)x + 4. Our goal is to explore how this function behaves by plotting ordered pairs for a given domain, thereby gaining a visual representation of its characteristics and properties. We will meticulously calculate the corresponding y-values for each x-value in the domain, and then plot these ordered pairs on a coordinate plane. This process will not only solidify our understanding of linear functions but also provide a practical illustration of the connection between algebraic expressions and their graphical representations.

Understanding the Function f(x) = (3/2)x + 4

To embark on our exploration, it's crucial to first grasp the essence of the function f(x) = (3/2)x + 4. This function embodies a linear relationship, characterized by a constant rate of change. The (3/2)x term signifies that for every unit increase in x, the value of f(x) increases by 3/2. This constant rate of change is the hallmark of a linear function, giving rise to its straight-line graph. The + 4 term, on the other hand, represents the y-intercept, the point where the line intersects the y-axis. In this case, the y-intercept is at the point (0, 4), indicating that when x is 0, f(x) is 4. Understanding these fundamental components allows us to predict the behavior of the function and visualize its graph even before plotting any points. The slope, represented by the coefficient of x (which is 3/2 in this case), dictates the steepness and direction of the line. A positive slope, as we have here, signifies that the line slopes upwards from left to right. The interplay between the slope and the y-intercept determines the unique characteristics of this linear function, shaping its graph and governing its behavior across the coordinate plane. As we delve into plotting specific points, these concepts will become even more tangible, allowing us to see firsthand how the algebraic representation translates into a visual depiction.

Defining the Domain: {-8, -4, -2, 0, 2, 4}

Before we can plot any points, we need to establish the domain of our function. The domain represents the set of all possible x-values that we will input into the function. In this particular case, the domain is explicitly defined as the set {-8, -4, -2, 0, 2, 4}. This means we will be evaluating the function f(x) = (3/2)x + 4 for each of these x-values individually. The choice of domain is crucial as it determines the portion of the function's graph that we will be visualizing. A different domain would yield a different set of ordered pairs and, consequently, a different segment of the line. By focusing on this specific set of x-values, we gain a focused perspective on the function's behavior within this range. Each element in the domain will be paired with a corresponding y-value, which we will calculate using the function's formula. These pairs, known as ordered pairs, will then serve as coordinates for plotting points on the coordinate plane. The domain acts as the foundation for our graphical representation, dictating the x-coordinates of the points we will be plotting. As we move forward, we will methodically evaluate the function for each x-value in the domain, unveiling the corresponding y-values and constructing the ordered pairs that will bring our graph to life. This step-by-step approach ensures that we capture the essence of the function's behavior within the specified domain.

Calculating Ordered Pairs

Now, let's embark on the process of calculating the ordered pairs. For each x-value in the domain {-8, -4, -2, 0, 2, 4}, we will substitute it into the function f(x) = (3/2)x + 4 and compute the corresponding y-value. This will generate a set of ordered pairs in the form (x, f(x)) or (x, y), which we can then plot on the coordinate plane.

1. For x = -8: f(-8) = (3/2)(-8) + 4 = -12 + 4 = -8. The ordered pair is (-8, -8).
2. For x = -4: f(-4) = (3/2)(-4) + 4 = -6 + 4 = -2. The ordered pair is (-4, -2).
3. For x = -2: f(-2) = (3/2)(-2) + 4 = -3 + 4 = 1. The ordered pair is (-2, 1).
4. For x = 0: f(0) = (3/2)(0) + 4 = 0 + 4 = 4. The ordered pair is (0, 4).
5. For x = 2: f(2) = (3/2)(2) + 4 = 3 + 4 = 7. The ordered pair is (2, 7).
6. For x = 4: f(4) = (3/2)(4) + 4 = 6 + 4 = 10. The ordered pair is (4, 10).

By meticulously substituting each x-value and performing the calculations, we have successfully derived a set of ordered pairs: (-8, -8), (-4, -2), (-2, 1), (0, 4), (2, 7), and (4, 10). These ordered pairs are the coordinates that will guide us in plotting the graph of the linear function. Each pair represents a unique point on the coordinate plane, and together, they will reveal the linear relationship defined by f(x) = (3/2)x + 4. With these ordered pairs in hand, we are now ready to visualize the function and gain a deeper understanding of its behavior.

Plotting the Ordered Pairs on a Coordinate Plane

With our ordered pairs calculated, the next step is to plot them on a coordinate plane. A coordinate plane is a two-dimensional space formed by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each ordered pair (x, y) corresponds to a unique point on this plane, where x represents the horizontal distance from the origin (the point where the axes intersect) and y represents the vertical distance. To plot a point, we locate its x-coordinate on the x-axis and its y-coordinate on the y-axis, and then mark the point where these two coordinates meet.

Let's plot the ordered pairs we calculated earlier:

• (-8, -8): Start at the origin, move 8 units to the left along the x-axis (since x is -8), and then 8 units down along the y-axis (since y is -8). Mark the point.
• (-4, -2): Move 4 units to the left along the x-axis and 2 units down along the y-axis. Mark the point.
• (-2, 1): Move 2 units to the left along the x-axis and 1 unit up along the y-axis. Mark the point.
• (0, 4): Stay at the origin along the x-axis (since x is 0) and move 4 units up along the y-axis. This point lies on the y-axis. Mark the point.
• (2, 7): Move 2 units to the right along the x-axis and 7 units up along the y-axis. Mark the point.
• (4, 10): Move 4 units to the right along the x-axis and 10 units up along the y-axis. Mark the point.

Once all the points are plotted, you'll notice a clear pattern: they all lie on a straight line. This is a characteristic feature of linear functions. By connecting these points with a straight line, we obtain the graphical representation of the function f(x) = (3/2)x + 4 within the specified domain. The line extends beyond the plotted points, representing the function's behavior for other x-values as well. Plotting the ordered pairs allows us to visually confirm the linear nature of the function and provides a concrete representation of the relationship between x and f(x).

Visualizing the Linear Relationship

After plotting the ordered pairs and connecting them with a straight line, we have a visual representation of the linear relationship defined by the function f(x) = (3/2)x + 4. The line serves as a powerful tool for understanding the function's behavior and properties. Several key aspects become apparent from the graph.

Firstly, the line's slope, which we previously identified as 3/2, is now visually evident. The line rises 3 units on the y-axis for every 2 units it moves along the x-axis, reflecting the constant rate of change inherent in linear functions. This steepness of the line directly corresponds to the slope, providing a tangible link between the algebraic representation (the slope value) and the geometric representation (the line's inclination). Secondly, the y-intercept, which we calculated as 4, is clearly visible as the point where the line crosses the y-axis. This point (0, 4) serves as an anchor for the line, indicating the function's value when x is zero. The y-intercept, along with the slope, uniquely determines the position and orientation of the line on the coordinate plane.

Furthermore, the graph allows us to extrapolate the function's values beyond the plotted points. By extending the line, we can estimate the corresponding y-values for x-values outside our initial domain. This ability to generalize from a set of points is a key advantage of graphical representations, providing insights into the function's overall behavior. The line also highlights the concept of linearity itself. The straightness of the line confirms that the function exhibits a constant rate of change, a defining characteristic of linear functions. Any deviation from a straight line would indicate a non-linear relationship. In essence, the graph transforms the abstract algebraic expression into a concrete visual form, making the function's properties and behavior readily accessible and intuitive. This visual understanding complements the algebraic analysis, providing a more complete and holistic comprehension of the function.

Conclusion

In this exploration, we embarked on a journey to understand the linear function f(x) = (3/2)x + 4 by plotting ordered pairs for a given domain. We meticulously calculated the y-values corresponding to each x-value in the domain {-8, -4, -2, 0, 2, 4}, generating a set of ordered pairs. These pairs then served as coordinates for plotting points on a coordinate plane. By connecting these points, we obtained a visual representation of the function as a straight line. This process not only solidified our understanding of linear functions but also highlighted the powerful connection between algebraic expressions and their graphical counterparts. The graph vividly illustrated the function's slope and y-intercept, revealing the constant rate of change and the function's value when x is zero. Moreover, the graphical representation allowed us to extrapolate beyond the plotted points, gaining insights into the function's overall behavior. This exercise underscores the importance of visualization in mathematics. By translating abstract algebraic concepts into concrete visual forms, we enhance our intuition and gain a deeper appreciation for the relationships between mathematical objects. The ability to plot functions and interpret their graphs is a fundamental skill that empowers us to model and understand real-world phenomena. As we continue our mathematical journey, the principles and techniques explored in this article will serve as a valuable foundation for tackling more complex functions and mathematical concepts. The interplay between algebra and geometry, as exemplified in this exercise, is a recurring theme in mathematics, and mastering this connection is key to unlocking its full potential.