Plotting Voltage Vs Current Graph A Step By Step Guide

Hey guys! Today, we're diving deep into the fascinating world of electricity and exploring the relationship between voltage and current. We're going to tackle a common type of question you might encounter in your math or physics class: plotting a graph to visualize this relationship. So, buckle up and get ready to learn!

Plotting the Graph Voltage (V) vs. Current (I)

So, you've got this question asking you to plot a graph with voltage (V) on the y-axis and current (I) on the x-axis. No sweat! Let's break down exactly what that means and how to do it like a pro. Understanding the fundamental relationship between voltage and current is crucial in electrical circuits. Voltage, often described as electrical potential difference, is the driving force that pushes electrons through a circuit. Think of it like the pressure in a water pipe – the higher the pressure, the more water flows. Current, on the other hand, is the rate of flow of electrical charge. It's the number of electrons zipping past a certain point in a circuit per unit of time. The relationship between these two is governed by Ohm's Law, a cornerstone of electrical engineering. Ohm's Law, in its simplest form, states that voltage (V) is directly proportional to current (I) when the resistance (R) is constant. Mathematically, this is expressed as V = IR. This equation tells us a lot. First, it tells us that if we increase the voltage in a circuit (keeping the resistance constant), the current will increase proportionally. Double the voltage, double the current. Similarly, if we decrease the voltage, the current will decrease. Second, it reveals that resistance plays a vital role in controlling the current flow. Higher resistance means lower current for the same voltage, and vice versa. Now, let’s talk about graphing this relationship. When we plot voltage on the y-axis and current on the x-axis, and the resistance is constant, we get a straight line. This straight line visually represents the direct proportionality between voltage and current as described by Ohm's Law. The slope of this line is equal to the resistance (R). A steeper slope indicates a higher resistance, meaning a greater voltage is required to drive the same amount of current. A shallower slope signifies lower resistance, where even a small voltage can produce a significant current. To accurately plot this graph, you'll need a set of data points – pairs of voltage and current values. These data points can come from an experiment, a simulation, or a problem statement. Once you have your data, you can choose appropriate scales for your x and y axes. The scales should be chosen to accommodate the full range of your data points while also making the graph easy to read. After setting up your axes, you plot each data point individually. Each point represents a specific voltage and current combination. Finally, after plotting all your points, you draw a line of best fit through them. Since the relationship is linear (if the resistance is constant), this line should be straight. The closer your points are to forming a straight line, the more closely your data follows Ohm's Law. If the points deviate significantly from a straight line, it might indicate that the resistance isn't constant or that there are other factors affecting the circuit.

Getting Started The Fundamentals

Before we jump into plotting, let's make sure we're all on the same page with the basics. A graph, at its core, is a visual representation of the relationship between two or more variables. In our case, we're focusing on two key players: voltage and current. Voltage, often measured in volts (V), is like the electrical pressure that pushes electrons through a circuit. Think of it as the force that drives the flow of electricity. The concept of voltage is fundamental to understanding how electrical circuits work. It represents the electrical potential difference between two points in a circuit. This difference in potential is what drives the flow of electric charge, much like a difference in height drives the flow of water. A higher voltage means a greater “push” on the electrons, leading to a stronger current. It's the energy per unit charge available to drive electrons through the circuit. In practical terms, voltage is what powers our devices. From the small voltage powering your smartphone to the high voltage powering an electric train, it’s the driving force behind the electricity that makes our modern world function. Without voltage, there would be no current flow, and electrical circuits would be inactive. Understanding voltage also means recognizing its role in safety. High voltages can be dangerous because they can force a large current through the human body, potentially causing serious harm. This is why electrical safety is a critical aspect of electrical engineering and electronics. Safety measures are designed to minimize the risk of exposure to high voltages, protecting both people and equipment. Voltage can be generated in various ways, including batteries, generators, and power supplies. Each of these sources provides a different means of creating the electrical potential difference needed to drive current. Batteries, for instance, use chemical reactions to generate voltage, while generators convert mechanical energy into electrical energy. Power supplies, on the other hand, typically transform AC voltage from the grid into a stable DC voltage suitable for electronic devices. Understanding the sources of voltage and how they operate is crucial for designing and troubleshooting electrical systems. Voltage also plays a crucial role in energy transmission. Power companies use high voltages to transmit electricity over long distances because it reduces energy loss due to resistance. The voltage is then stepped down to lower levels for distribution to homes and businesses. This complex system of voltage transformation is essential for efficiently delivering electricity from power plants to consumers. Voltage measurement is a fundamental skill in electronics and electrical work. Instruments like voltmeters are used to measure voltage in circuits, allowing technicians and engineers to diagnose problems and ensure proper operation. Accurate voltage measurement is essential for safety and for the effective functioning of electrical systems. In summary, voltage is the fundamental electrical potential difference that drives the flow of current. It’s a crucial concept for understanding circuits, powering devices, ensuring safety, and transmitting energy. A solid grasp of voltage is essential for anyone working with electricity or electronics, and it forms the foundation for more advanced studies in electrical engineering and physics. Then there's current, measured in amperes (A), which is the rate at which electrons flow through a circuit. Think of it as the amount of electricity flowing. Current, in the realm of electricity, is the lifeblood that courses through circuits, powering our devices and lighting up our world. It’s the flow of electric charge, typically electrons, through a conductive material. Understanding current is essential for comprehending how electrical circuits function and for designing safe and efficient electrical systems. Current is measured in amperes (A), often shortened to amps. One ampere is defined as the flow of one coulomb of electric charge per second. To put this in perspective, a single amp represents a substantial number of electrons flowing past a point in a circuit every second. This flow of electrons is what allows electrical devices to operate. The concept of current is closely linked to voltage and resistance, as described by Ohm's Law (V = IR). Voltage, as we discussed, is the electrical potential difference that drives the flow of current, while resistance opposes this flow. The relationship between these three quantities determines the amount of current that will flow in a circuit. A higher voltage will drive more current, while a higher resistance will restrict current flow. Current can be either direct current (DC) or alternating current (AC). Direct current flows in one direction only, like the current from a battery. Alternating current, on the other hand, periodically reverses direction, which is the type of current used in most household electrical systems. Understanding the difference between DC and AC is crucial for various applications, from designing electronic devices to managing power grids. The magnitude of current flowing through a circuit is a critical factor in electrical safety. High currents can generate significant heat, which can lead to fires or damage to equipment. Additionally, high currents passing through the human body can cause severe injury or even death. This is why circuit breakers and fuses are used to protect circuits from excessive current. These devices are designed to interrupt the current flow if it exceeds a safe level. Current distribution in a circuit is also a key consideration in circuit design. The way current divides and flows through different paths in a circuit depends on the resistances in those paths. Kirchhoff's Current Law states that the total current entering a junction in a circuit must equal the total current leaving that junction. This principle is fundamental to analyzing and designing complex circuits. Current measurement is a vital skill for electrical technicians and engineers. Ammeters are used to measure current in circuits, allowing professionals to diagnose problems and ensure that circuits are operating within safe limits. Accurate current measurement is essential for both safety and the effective functioning of electrical systems. In summary, current is the flow of electric charge that powers our devices and systems. It is closely related to voltage and resistance, and understanding these relationships is essential for working with electricity. Current is a critical factor in both safety and performance, and its behavior in circuits is governed by fundamental laws of physics.

Step-by-Step Guide to Plotting Your Graph

Alright, let's get practical! Here's how you can plot your graph like a pro:

1. Gather Your Data: First things first, you'll need your data points. This will be a set of pairs, each showing a corresponding voltage (V) and current (I) value. For example, you might have points like (1A, 2V), (2A, 4V), (3A, 6V), and so on. Accurate data collection is the foundation of any meaningful graph. Without precise and reliable data, the resulting graph will not accurately represent the relationship between the variables being studied. This is particularly important in scientific and engineering contexts, where decisions and conclusions are often based on graphical analysis. The process of gathering data can vary depending on the context. In a laboratory setting, it might involve conducting experiments and carefully recording measurements using calibrated instruments. For instance, when studying the relationship between voltage and current in an electrical circuit, a multimeter would be used to measure these values at different points in the circuit. The accuracy of the measurements will depend on the quality of the instrument and the care taken by the person conducting the experiment. In other situations, data might be collected through surveys, observations, or simulations. For example, market research might involve surveying customers to gather data on their preferences and purchasing habits. In such cases, the methods used to collect data, such as the design of the survey questions or the selection of participants, can significantly impact the quality of the data. Regardless of the method used, it's essential to follow a systematic approach to data collection. This includes defining the variables being measured, identifying the sources of data, and establishing a clear procedure for recording the data. It's also crucial to minimize errors and biases in the data collection process. This might involve using standardized procedures, training data collectors, and implementing quality control measures. Data validation is another critical step in ensuring data accuracy. This involves checking the data for inconsistencies, errors, and outliers. For example, if a data point falls far outside the expected range, it might indicate a measurement error or a data entry mistake. These errors should be investigated and corrected before the data is used for further analysis. The format in which data is recorded is also important. Data should be organized in a clear and consistent manner, making it easy to analyze and interpret. This often involves using spreadsheets or databases to store the data, with each variable represented in a separate column. Data should also be labeled clearly, with units of measurement specified. In addition to the data itself, it's also important to record any relevant contextual information. This might include the date and time of data collection, the conditions under which the data was collected, and any notes or observations that could help in interpreting the data. For example, in a scientific experiment, it would be important to record the ambient temperature, the calibration status of the instruments, and any unexpected events that occurred during the experiment. Finally, ethical considerations are crucial in data collection, particularly when dealing with human subjects. Informed consent should be obtained from participants, and data should be handled in a way that protects their privacy and confidentiality. Data should also be stored securely to prevent unauthorized access or loss. In summary, accurate data collection is a multi-faceted process that involves careful planning, systematic execution, and rigorous quality control. By following best practices in data collection, we can ensure that our graphs and analyses are based on reliable information, leading to sound conclusions and informed decisions.
2. Set Up Your Axes: Grab your graph paper (or a digital graphing tool). The y-axis (the vertical one) will represent voltage (V), and the x-axis (the horizontal one) will represent current (I). Make sure to label your axes clearly! Setting up graph axes correctly is a fundamental step in creating meaningful and informative graphs. The axes provide the framework for plotting data points and visually representing the relationship between variables. The choices made during this setup process can significantly impact the clarity and interpretability of the graph. The first decision to make is which variable to plot on which axis. Conventionally, the independent variable (the one that is manipulated or controlled) is plotted on the x-axis, while the dependent variable (the one that is measured or observed) is plotted on the y-axis. This convention helps to illustrate cause-and-effect relationships, where changes in the independent variable are thought to influence changes in the dependent variable. For example, in an experiment studying the effect of temperature on reaction rate, temperature would be plotted on the x-axis and reaction rate on the y-axis. Once the axes have been assigned, the next step is to determine appropriate scales for each axis. The scales should be chosen to accommodate the full range of data values while also making the graph easy to read and interpret. This often involves selecting a suitable starting point, ending point, and interval size for each axis. The starting point does not necessarily need to be zero, especially if the data values do not include zero or if starting at zero would compress the data too much. The ending point should be chosen to be slightly larger than the maximum data value, providing some buffer space around the data points. The interval size should be chosen to create a clear and uncluttered graph. Too many intervals can make the graph appear crowded, while too few intervals can obscure important details. It’s often helpful to choose interval sizes that are multiples of 1, 2, 5, or 10, as these are easy to work with and interpret. Labeling the axes is crucial for making the graph understandable. Each axis should be clearly labeled with the name of the variable being plotted and the units of measurement. For example, if the x-axis represents time in seconds, it should be labeled “Time (s)”. The labels should be placed close to the axes and should be large enough to be easily read. In addition to labeling the axes, it’s also important to add tick marks along the axes to indicate the scale values. Tick marks should be evenly spaced and should correspond to the chosen interval size. The values at the tick marks should also be labeled, providing a clear reference for the data points. The choice of scale can also affect the visual impression of the graph. For example, using different scales on the x and y axes can distort the apparent relationship between the variables. It’s important to be mindful of this and to choose scales that accurately represent the data. In some cases, it might be necessary to use a logarithmic scale for one or both axes. Logarithmic scales are useful for displaying data that spans a wide range of values, as they compress the higher values and expand the lower values. For example, logarithmic scales are often used in graphs of exponential growth or decay. Finally, the overall layout of the axes should be considered. The axes should be drawn with clear, solid lines, and the graph area should be large enough to accommodate all the data points comfortably. The origin (the point where the x and y axes intersect) should be clearly marked, and the axes should extend beyond the data points to provide context. In summary, setting up graph axes correctly is essential for creating clear, accurate, and informative graphs. The choices made during this process, including the assignment of variables to axes, the selection of scales, and the labeling of axes and tick marks, can significantly impact the interpretability of the graph. By following these guidelines, you can create graphs that effectively communicate the relationships in your data.
3. Choose Your Scale: Look at your data and figure out the range of values for both voltage and current. You'll want to choose a scale for each axis that allows you to plot all your points comfortably. This means selecting appropriate intervals so your data isn't squished into one corner or stretched out too thin. Choosing the right scale for your graph is a critical step in presenting data effectively. The scale determines how the data points are positioned on the graph and can significantly impact the visual impression and interpretability of the results. A well-chosen scale will allow the data to be displayed clearly and accurately, while a poorly chosen scale can distort the data and mislead the viewer. The first consideration in choosing a scale is the range of data values. The scale should be large enough to accommodate all the data points, from the smallest to the largest value. If the data range is wide, it might be necessary to use a larger scale or a logarithmic scale to prevent the data points from being compressed into a small area of the graph. Conversely, if the data range is narrow, a smaller scale can be used to expand the data points and reveal finer details. The choice of scale should also take into account the desired level of precision. If the data values are known to high precision, the scale should be fine enough to display these details. This might involve using smaller intervals or more tick marks on the axes. However, it’s important to avoid using a scale that is too fine, as this can make the graph appear cluttered and difficult to read. The intervals on the scale should be chosen to be easy to work with and interpret. Intervals that are multiples of 1, 2, 5, or 10 are generally preferred, as they make it easier to read the values at the tick marks. Using intervals that are fractions or decimals can make the graph more difficult to interpret. The starting point of the scale is another important consideration. While it’s common to start the scale at zero, this is not always necessary or appropriate. If the data values do not include zero, or if starting at zero would compress the data too much, it might be better to start the scale at a value slightly below the smallest data point. This will allow the data points to be spread out more evenly across the graph. The aspect ratio of the graph, which is the ratio of the height to the width, can also influence the choice of scale. A graph with a poor aspect ratio can distort the visual impression of the data. For example, a graph that is too tall and narrow can exaggerate the vertical differences between data points, while a graph that is too wide and short can minimize these differences. It’s generally recommended to choose an aspect ratio that is close to 1:1, as this will provide a balanced representation of the data. In some cases, it might be necessary to use different scales for different parts of the graph. For example, if the data values change rapidly over a certain range, a finer scale might be used in that range to reveal the details of the change. This can be achieved by using a segmented axis or a scale break. However, it’s important to use these techniques carefully, as they can potentially mislead the viewer if not clearly labeled and explained. Software tools for creating graphs often provide options for automatically scaling the axes. While these tools can be helpful, it’s important to review the automatically generated scales and adjust them if necessary to ensure that the graph is clear, accurate, and informative. The final scale should be clearly labeled on the graph, with tick marks and values indicated along each axis. The labels should be large enough to be easily read and should include the units of measurement. In summary, choosing the right scale for your graph is essential for effectively communicating your data. The scale should be chosen to accommodate the data range, provide the desired level of precision, and create a clear and easy-to-interpret graph. By considering these factors, you can create graphs that accurately represent your data and effectively convey your message.
4. Plot Your Points: Now comes the fun part! For each data pair, find the corresponding point on the graph where the current and voltage values intersect. Mark this point clearly, usually with a dot or an 'x'. Plotting points on a graph is a fundamental skill in data visualization. It involves translating pairs of numerical values into visual markers on a two-dimensional plane defined by the graph's axes. This process allows us to represent relationships between variables and to identify patterns, trends, and anomalies in the data. The first step in plotting points is to understand the coordinate system of the graph. Most graphs use a Cartesian coordinate system, which consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. The point where the axes intersect is called the origin, and it represents the value (0, 0). Each point on the graph is defined by its coordinates, which are a pair of numbers (x, y) that indicate the point's position relative to the origin. The x-coordinate represents the point's horizontal distance from the origin, and the y-coordinate represents the point's vertical distance from the origin. To plot a point, you first locate the x-coordinate on the x-axis and draw a vertical line at that point. Then, you locate the y-coordinate on the y-axis and draw a horizontal line at that point. The intersection of these two lines is the location of the point on the graph. The point is then marked with a symbol, such as a dot, a cross, or a circle. The choice of symbol can depend on the number of points being plotted and the desired clarity of the graph. For example, if a large number of points are being plotted, smaller symbols might be used to avoid cluttering the graph. The accuracy of plotting points is crucial for the overall accuracy of the graph. This involves carefully reading the values from the data and accurately locating the corresponding positions on the axes. It’s helpful to use a ruler or a straightedge to draw the vertical and horizontal lines, ensuring that they are perpendicular to the axes. It’s also important to use a sharp pencil or pen to mark the points clearly and precisely. The density of points on a graph can provide valuable information about the data. A cluster of points in a particular region of the graph might indicate a strong relationship between the variables, while scattered points might suggest a weaker relationship or the presence of outliers. Outliers are data points that fall far outside the general pattern of the data, and they can sometimes indicate errors in the data collection or measurement process. When plotting points, it’s important to maintain consistency in the use of symbols and colors. If multiple sets of data are being plotted on the same graph, different symbols or colors can be used to distinguish between them. A legend should be included to identify the meaning of each symbol or color. The order in which points are plotted does not affect the final appearance of the graph, but it can sometimes be helpful to plot the points in a logical order, such as from left to right or from smallest to largest x-coordinate. This can make it easier to check the accuracy of the plotting process. Software tools for creating graphs often provide features for automatically plotting points from a data table. These tools can greatly simplify the plotting process and reduce the risk of errors. However, it’s still important to review the plotted points to ensure that they are correctly positioned and that the graph is clear and informative. In summary, plotting points on a graph is a fundamental skill in data visualization. It involves translating pairs of numerical values into visual markers on a two-dimensional plane. By carefully plotting points and using consistent symbols and colors, we can create graphs that accurately represent our data and effectively communicate our findings.
5. Draw Your Line (or Curve): Once you've plotted all your points, take a look at the overall pattern. In many cases, especially with a direct relationship like Ohm's Law, the points will form a roughly straight line. Draw a line of best fit that passes as close as possible to all the points. This line represents the general trend in your data. If the relationship isn't linear, you might see a curve instead, in which case you'd draw a smooth curve that best fits the points. Drawing a line or curve of best fit is a crucial step in data analysis and visualization. It involves creating a visual representation that best approximates the relationship between variables in a dataset. This line or curve helps to highlight the underlying trend in the data, making it easier to interpret and draw meaningful conclusions. The first step in drawing a line or curve of best fit is to plot the data points on a graph. The graph provides a visual representation of the data, allowing you to see the general pattern or trend. The next step is to visually assess the data points to determine the type of relationship that exists between the variables. This might be a linear relationship, where the points tend to fall along a straight line, or a non-linear relationship, where the points follow a curved pattern. For a linear relationship, the line of best fit is a straight line that minimizes the distance between the line and the data points. There are several methods for determining the line of best fit, including visual estimation, the median-median method, and the least-squares regression method. Visual estimation involves drawing a line that appears to pass as close as possible to all the data points. This method is quick and easy but can be subjective and less accurate than other methods. The median-median method involves dividing the data points into three groups, finding the median point for each group, and then drawing a line through the median points. This method is more objective than visual estimation but can be less accurate than the least-squares regression method. The least-squares regression method is the most common and accurate method for determining the line of best fit. It involves using mathematical formulas to calculate the slope and y-intercept of the line that minimizes the sum of the squared distances between the line and the data points. This method can be implemented using statistical software or calculators. For a non-linear relationship, the curve of best fit is a curved line that minimizes the distance between the curve and the data points. There are various types of curves that can be used, including quadratic, exponential, and logarithmic curves. The choice of curve depends on the nature of the relationship between the variables. The curve of best fit can be determined using visual estimation or mathematical methods. Visual estimation involves drawing a curve that appears to pass as close as possible to all the data points. This method is subjective and can be less accurate than mathematical methods. Mathematical methods involve using regression analysis to fit a curve to the data points. This can be done using statistical software or calculators. Once the line or curve of best fit has been drawn, it’s important to assess how well it represents the data. This can be done by visually inspecting the graph and looking for points that fall far away from the line or curve. These points might be outliers or might indicate that the relationship between the variables is not well-represented by the chosen line or curve. Statistical measures, such as the correlation coefficient for linear relationships or the coefficient of determination for non-linear relationships, can also be used to assess the goodness of fit. A high value for these measures indicates that the line or curve fits the data well, while a low value suggests a poor fit. The line or curve of best fit can be used to make predictions about the relationship between the variables. For example, if you have a linear relationship, you can use the equation of the line to predict the value of one variable given the value of the other variable. These predictions are most accurate within the range of the data points used to create the line or curve. Extrapolating beyond this range can lead to inaccurate predictions. In summary, drawing a line or curve of best fit is a powerful tool for analyzing and visualizing data. It involves creating a visual representation that best approximates the relationship between variables, highlighting the underlying trend in the data. By using appropriate methods for determining the line or curve and assessing its goodness of fit, you can gain valuable insights from your data and make informed predictions.

Analyzing Your Graph The Big Picture

Once you've plotted your graph, take a step back and analyze it. What does the shape of the graph tell you? If it's a straight line, it confirms the direct relationship between voltage and current. The steeper the line, the greater the resistance in the circuit (since a larger voltage is needed to produce the same current). Analyzing graphs effectively is a crucial skill in various fields, including science, engineering, economics, and data analysis. Graphs are visual representations of data, and the ability to interpret them allows us to extract meaningful information, identify trends, and make informed decisions. The first step in analyzing a graph is to understand its basic components. A typical graph consists of two axes: the horizontal x-axis and the vertical y-axis. Each axis represents a variable, and the values of these variables are plotted along the axes. The graph also includes data points, which are the individual measurements or observations that are being represented. The relationship between the variables is often depicted by a line or curve that connects the data points or represents a trend. Once the basic components of the graph are understood, the next step is to examine the overall pattern or trend in the data. This involves looking at the shape of the line or curve and identifying any significant features, such as peaks, valleys, or inflection points. The trend can be linear, where the data points tend to fall along a straight line, or non-linear, where the data points follow a curved pattern. The slope of a linear trend indicates the rate of change between the variables, while the shape of a non-linear trend can reveal more complex relationships. Identifying the type of trend is essential for making accurate predictions and understanding the underlying process that generated the data. In addition to the overall trend, it’s also important to look for any outliers or anomalies in the data. Outliers are data points that fall far away from the general pattern, and they can sometimes indicate errors in the data collection or measurement process. Anomalies are unexpected or unusual features in the graph that deviate from the expected trend. These could be caused by external factors, measurement errors, or genuine variations in the system being studied. Outliers and anomalies should be investigated further to determine their cause and impact on the analysis. The axes of the graph provide important information about the variables being plotted. The scales of the axes indicate the range of values being represented, and the labels on the axes identify the variables and their units of measurement. It’s crucial to pay attention to the scales of the axes, as they can influence the visual impression of the graph. For example, using different scales on the x and y axes can distort the apparent relationship between the variables. Logarithmic scales are often used when the data spans a wide range of values, as they compress the higher values and expand the lower values. Comparing different graphs can reveal patterns and relationships that might not be apparent from a single graph. This involves looking for similarities and differences in the trends, shapes, and scales of the graphs. For example, comparing graphs of different time periods can reveal how a trend has changed over time. Comparing graphs of different variables can reveal correlations or dependencies between the variables. To effectively analyze graphs, it’s also important to consider the context in which the data was collected. This includes understanding the purpose of the study, the methods used to collect the data, and any potential limitations or biases in the data. The context can help to interpret the graph and draw meaningful conclusions. Software tools for creating and analyzing graphs often provide features for calculating statistical measures, such as means, medians, and standard deviations. These measures can help to quantify the trends and patterns in the data. Regression analysis can be used to fit mathematical models to the data, allowing for more precise predictions and insights. In summary, analyzing graphs effectively involves understanding the basic components of the graph, identifying the overall trend, looking for outliers and anomalies, paying attention to the axes, comparing different graphs, considering the context of the data, and using statistical measures and tools. By developing these skills, you can extract valuable information from graphs and make informed decisions.

Wrapping Up

So, there you have it! Plotting a graph of voltage versus current is a powerful way to visualize and understand their relationship. By following these steps, you'll be able to tackle similar questions with confidence. Remember, practice makes perfect, so keep plotting those graphs!

If you guys have any questions, feel free to ask in the comments below. Happy graphing!