Completing Table And Graph For P(kg) = (3/2)w + 1.2
Hey guys! Today, we're diving into a fun math problem involving completing a table and plotting a graph. This is super useful for understanding how different variables relate to each other. We're going to take the equation p(kg) = (3/2)w + 1.2 and break it down step by step. This equation represents a linear relationship, which we'll visualize on a graph. Let's get started!
Understanding the Equation
Before we jump into completing the table, let’s make sure we understand what the equation p(kg) = (3/2)w + 1.2 is telling us. In this equation, p(kg) represents a value that depends on w. The equation is in the form of a linear equation, y = mx + c, where m is the slope and c is the y-intercept. In our case, (3/2) or 1.5 is the slope, and 1.2 is the y-intercept. This means for every unit increase in w, p(kg) increases by 1.5, and when w is 0, p(kg) is 1.2. Understanding this relationship will help us fill in the missing values in the table and accurately plot the graph. A solid grasp of linear equations is crucial, guys, as they pop up everywhere in math and real-world applications. The beauty of a linear equation lies in its simplicity; it describes a straight line, and once you get the hang of identifying the slope and y-intercept, you can easily predict how the variables will behave. When dealing with such problems, always take a moment to identify the form of the equation – in this case, recognizing it as a linear equation immediately sets you on the right path. So, always remember, understanding the fundamentals makes tackling complex problems a breeze!
Calculating Missing Values for the Table
Okay, first things first, we need to complete the table. We have the equation p(kg) = (3/2)w + 1.2, and we're given values for w(kg): 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4. We also have some values for p(kg), but some are missing. Our mission is to find those missing values! To do this, we'll substitute each given w(kg) value into the equation and solve for p(kg). This is where the fun part begins, guys – plugging in numbers and seeing what we get! For instance, let’s start with w(kg) = 0.2. Plugging this into our equation gives us p(kg) = (3/2)(0.2) + 1.2. Let’s break it down: (3/2) times 0.2 is 0.3, and then we add 1.2, giving us p(kg) = 1.5. We'll repeat this process for each given w(kg) value. For w(kg) = 0.6, we get p(kg) = (3/2)(0.6) + 1.2, which simplifies to 0.9 + 1.2, resulting in p(kg) = 2.1. And so on! By carefully substituting each w(kg) value, we can systematically fill in the missing pieces of our table. This exercise is a fantastic way to practice your algebra skills, guys, and it's super rewarding when you see the table come together. Remember, accuracy is key here, so double-check your calculations as you go. With a little bit of patience and careful computation, we'll have our table fully populated and ready for the next step: plotting the graph!
Table Completion Step-by-Step
Let’s dive deeper into the table completion process, guys. We’ll walk through each calculation to make sure everything is crystal clear. Remember our equation: p(kg) = (3/2)w + 1.2. We’ve already tackled a couple, but let’s reinforce our understanding with more examples. When w(kg) = 0.4, we substitute that into the equation: p(kg) = (3/2)(0.4) + 1.2. First, (3/2) times 0.4 is 0.6. Adding 1.2 to that gives us p(kg) = 1.8. This value is already in the table, so we’re on the right track! Now, let’s move on to w(kg) = 0.8. Plugging this in, we get p(kg) = (3/2)(0.8) + 1.2. (3/2) times 0.8 is 1.2, and adding 1.2 gives us p(kg) = 2.4. So, the missing value for p(kg) when w(kg) is 0.8 is 2.4. Next up, w(kg) = 1.0. Our equation becomes p(kg) = (3/2)(1.0) + 1.2. (3/2) times 1.0 is 1.5, and adding 1.2 gives us p(kg) = 2.7. Again, this value is already provided in the table, confirming our calculations. Now for w(kg) = 1.2, we have p(kg) = (3/2)(1.2) + 1.2. (3/2) times 1.2 is 1.8, and adding 1.2 gives us p(kg) = 3.0. This is another missing value we’ve found! Finally, let’s calculate for w(kg) = 1.4. We get p(kg) = (3/2)(1.4) + 1.2. (3/2) times 1.4 is 2.1, and adding 1.2 gives us p(kg) = 3.3. With all these calculations, we’ve successfully filled in all the missing values in our table. This meticulous process demonstrates the importance of careful calculation in mathematics, guys. Each step builds on the previous one, and a single mistake can throw off the entire result. So, always take your time and double-check your work!
Setting Up the Graph
Alright, now that we've completed our table, it's time to plot the graph! This is where we visually represent the relationship between w(kg) and p(kg). We’re told to use a scale of 2cm to 0.2 units on the x-axis (which will represent w(kg)) and 2cm to 0.5 units on the y-axis (which will represent p(kg)). We also need to draw two perpendicular lines, OW and OR, which will serve as our axes. Setting up the graph correctly is crucial for an accurate representation, guys. The scale we choose affects how the line looks and how easy it is to read the graph. Choosing the right scale makes our data clear and understandable. So, let's start by drawing our axes, OW and OR, perpendicular to each other. OW will be our x-axis, representing w(kg), and OR will be our y-axis, representing p(kg). Now, we need to mark the units on each axis according to the given scale. On the x-axis, every 2cm represents 0.2 units. So, we’ll mark points at 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4. On the y-axis, every 2cm represents 0.5 units. We’ll mark points at 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5, ensuring we cover the range of our calculated p(kg) values. Remember, neatness counts here, guys! Use a ruler to make sure your axes are straight and your markings are evenly spaced. A well-prepared graph not only looks professional but also makes plotting points much easier. This careful setup is the foundation for our graph, and with it, we’re ready to plot our data points and visualize the relationship between w(kg) and p(kg). Let's get plotting!
Plotting Points and Drawing the Line
Okay, guys, the moment we've been waiting for – plotting the points on our graph! We've got our axes set up with the right scales, and we've got our completed table with the values of w(kg) and p(kg). Now, it's time to bring those numbers to life on our graph. Remember, each pair of values from our table represents a point on the graph. For example, when w(kg) = 0.2, we found that p(kg) = 1.5. So, we’ll find the point on the graph where w(kg) is 0.2 and p(kg) is 1.5, and we'll mark it with a small dot. We'll repeat this process for each pair of values in our table. When w(kg) = 0.4, p(kg) = 1.8, so we'll find and mark that point. For w(kg) = 0.6, p(kg) = 2.1, and so on. As we plot each point, we should start to see a pattern emerging. Since we know our equation is linear, we expect these points to form a straight line. Plotting accurately is key to seeing this linear relationship clearly, guys. Take your time and make sure each point is in the correct position. Once we've plotted all the points, the next step is to draw a line through them. Grab your ruler, guys, and carefully draw a straight line that passes through as many of the plotted points as possible. If all our calculations and plotting have been accurate, the points should line up beautifully! This line represents the equation p(kg) = (3/2)w + 1.2 on our graph. It visually shows how p(kg) changes as w(kg) changes. The line should start at 1.2 on the y-axis (our y-intercept) and rise steadily as we move to the right. This graphical representation makes it easy to see the relationship between the two variables at a glance. So, with our points plotted and our line drawn, we've successfully visualized our equation on a graph. High five!
Analyzing the Graph and Discussion
Alright, guys, we’ve plotted our graph, and it looks awesome! But the real magic happens when we start analyzing it. Our graph visually represents the equation p(kg) = (3/2)w + 1.2, and we can use it to understand the relationship between w(kg) and p(kg) in a deeper way. One of the first things we can look at is the slope of the line. Remember, the slope tells us how much p(kg) changes for every unit change in w(kg). In our equation, the slope is 3/2, or 1.5. This means that for every increase of 1 kg in w, p(kg) increases by 1.5 units. We can see this visually on our graph – the line rises 1.5 units on the y-axis for every 1 unit we move to the right on the x-axis. The y-intercept is another important feature of our graph. It's the point where the line crosses the y-axis, and it tells us the value of p(kg) when w(kg) is 0. In our equation, the y-intercept is 1.2. This means that when w(kg) is 0, p(kg) is 1.2. Again, we can see this on our graph – the line crosses the y-axis at the point 1.2. Beyond just reading the slope and y-intercept, we can also use the graph to estimate values that aren't explicitly in our table. For example, if we wanted to know the value of p(kg) when w(kg) = 0.7, we could find 0.7 on the x-axis, trace a line up to our plotted line, and then read the corresponding value on the y-axis. This gives us a visual estimation without having to do any calculations. Graphical analysis is a powerful tool in mathematics and science, guys. It allows us to see trends, make predictions, and understand relationships in a way that equations alone sometimes can't. So, take some time to really look at your graph and think about what it's telling you. What happens to p(kg) as w(kg) increases? Is the relationship linear, as we expected? How could we use this graph to solve real-world problems? These are the kinds of questions that can unlock the true potential of graphical analysis. Keep exploring!
In conclusion, guys, we've successfully completed the table, plotted the graph, and analyzed the relationship between w(kg) and p(kg) using the equation p(kg) = (3/2)w + 1.2. This exercise has shown us the power of linear equations and how they can be visually represented and understood through graphs. Awesome job!
