Graphing – Part 1 - Ohlone Biotechnology Math Modules (2024)

• Introduction
• The Graphing System
• Elements of a Graph - Labeling
• Creating a Line of Best Fit
• Using a Line of Best Fit to Extrapolate Data
• Check for Understanding
• Solutions

Introduction

Take a look at the table below. We will use this table throughout this guide to help explain graphing. This table relates concentration (in mg/mL, or milligrams per milliliter) and absorbance in "absorbance units" (AU).Note: You might make a table like this when working with a spectrophotometer — a device we sometimes use in Biotech to measure the absorbance of light by solutions.

When looking at a table like this, it can be difficult to separate out a meaningful relationship from just looking at the numbers.Graphingis important because it helps us visualize data.

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The Graphing System

heck out the figures below this text — the figure on the left is what your typical graph looks like. You might have seen or created graphs like this in one of your math classes. It has anx-axisandy-axisthat range from negative to positive values. It also has an origin (0, 0).

In Biotechnology, and in "real-world application" graphs, we don't need the negative part of those axes. Check out the figure on the right again — it is simply a zoomed-in and cropped version of your typical graph. It contains x- and y-axes, and still has an origin, but does not include negative values on those axes. The graph on therightis the kind of graph you'll see and use:

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Elements of a Graph - Labeling

Rememberwhywe graph — to help visualize data. Graphing and plotting data isn't just about numbers, though. Numbers on a graph are meaningless unless you give them meaning bylabelingwhat they mean.

Looking back the figures above, you'll notice there are several parts labeled, such as the "x-axis," the "origin," the "horizontal axis label" and "vertical axis label," and so on. Let's go over four elements of labeling a graph, outlined below:

1. Horizontal Axis Label
   The label of thehorizontal axis— thex-axis— is important because it represents yourindependent variable. Following our table from the introduction of this guide, determine what the horizontal axis label is. "Concentration (mg/mL)" is our horizontal axis label. Why? Concentration (mg/mL) is our independent variable. In our experiment, we changed concentration but measured absorbance. This is one way to remember what the independent variable is.Helpful hint: your independent variable will likely have "nice numbers" that are rounded or whole.

2. Vertical Axis Label
   The label of thevertical axis, they-axis, represents yourdependent variable. Yourdependent variableis often what you will measure/record when taking data. In our scenario, our dependent variable is "Absorbance" (in absorbance units, AU). Think of this way: your measured "absorbance"depends onwhat the concentration is. Therefore, your dependent variabledepends onyour independent variable. Recall that our independent variable is concentration (mg/mL). Therefore, our vertical axis label — and dependent variable — is "Absorbance (AU)."

   **IMPORTANT: You must include units with your horizontal and vertical axis labels!**

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3. Label Axes as "X" and "Y"
   Label your axes as you would a typical graph, with x on the horizontal axis and y on the vertical axis.

   

4. Title Your Graph
   The title of your graph is up to you to decide. Your title should relate to data you're visualizing. For example, we might call our graph something like "Absorbance vs. Concentration." When deciding how to title your graph, remember this: your title should relate yourvertical axis Labelandhorizontal axis label.

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Elements of a Graph – Clear Scale and Size

When creating a graph, there are two more things to keep in mind:clear scaleandsize of the graph.

Each axis on the graphs you create need to have a scale that is clear and appropriate for your data. The table we've been referencing so far, under "Introduction," has been recreated here:

Remember that ourindependent variable— the variable we change — is Concentration (mg/mL). Ourdependent variable— the variable wemeasure— is Absorbance (AU).

Here, our values don't come in easy, rounded numbers like 0, 10, 20, 30, and so on. There aren't any quickly picked out places to mark on your graphing paper to divide up your graph. We must determine the scale ourselves. Creating a scale might take a few trial and errors, but if you follow these steps, your scale will be much easier to determine:

Creating a Clear Scale

1. First, look at the values for your independent variable. How many are there in the table? Use this to think about how many marks you want to divide your scale. We have six values for our independent variable. You might want to mark your graph with six lines, and end the graph there. Instead, add an extra mark on your graph instead to give yourself more room to plot data. Your graph might look something like this, with seven marks:

   

2. Now, look at your numbers again for the independent variable: 0.00, 3.25, 7.25, 15, 30, and 60.

   Do NOT fill in these numbers on the marks you have just created. A scale must have each "mark" an equal distance in numbers to the next. A good scale would look like this, with a numerical distance of 10 between each mark:

   

   Why did we decide to divide our graph this way? Remember that the last 3 numbers we have to work with are 15, 30, and 60. We must accommodate our largest values to fit the graph. Your horizontal axis (x) now has a clear scale! But we're not done yet — we still need to look at the vertical axis (y).

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3. How we create a clear scale for our vertical axis (y-axis) will be similar to what we did for our horizontal axis (x-axis). We also have six values: 0.00, 0.06, 0.121, 0.246, 0.474, and 0.905. Remember to consider your largest values — 0.474 and 0.905. Therefore, we can divide our scale into values of 0.1 — from 0.0 to 1. This is to make sure we include all our values, from our smallest (0.06) to largest (0.905). Check out the image below of a graph that's ready to have data plotted on it below for an ideal vertical axis scale.

   

Size of the Graph

The last element of a graph to cover is size. As you might notice, our graph above seems to have a good size based on the clear scale we created. But what if you were to draw this graph so it takes up a mere 3 by 3 inches of your lab notebook? What about a whopping 10 by 10 inches? Would those sizes still be appropriate? No.The general rule for sizing of a graph is that it takes up at least half of a page of your lab notebook.

Plotting Data on a Graph

You know how to label a graph. You also know how to create a clear scale, and what general size to draw your graph (no tiny graphs, please!). We can now move on to how to plot data on a graph.

We plot points on a graph based on acoordinate system— written as(x, y)wherexis the position along the horizontal axis and y is the position along the vertical axis.

Plotting a Point in the Coordinate System

To plot a point in the coordinate system, follow these steps:

From the origin (0, 0), move the indicated number of units along the horizontal axis (x-axis).

1. From that location, move the indicated number of units up the vertical axis (y-axis).
2. Place a small point on that spot, and
3. Label its coordinates.

Example: Plot the point (2,3) on a simplified graph:

1. Move 2 units to the right along the horizontal axis.
2. Move 3 units along the vertical axis.
3. Place a dot at that location.
4. Label the point with its coordinates.

Let's plot our first "real" data point — the first point on our graph Absorbance vs. Concentration. The table and graph for plotting data — complete with labels and a clear scale — is below for your use.

1. Our first point is (0.00, 0.00). This is the origin. Mark it with a dot.
2. Our second point is (3.25, 0.06). Follow the x-axis to 3.25 (blue arrow), then move up 0.06 (orange arrow) and place a dot. Label the coordinate where the dot is placed.
3. Continue plotting dots on your graph using the method of moving along the x- and y-axes. The first three points are shown to be plotted here. Continue plotting points until all points are plotted:

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Creating a Line of Best Fit

On our graph above, we can see the relationship between absorbance and concentration more clearly:

It is important that our line doesn't go througheverypoint — and we don't connect the dots either. The line shown above is called the line of best fit — the line that best fits our data. Just like measurement, graphing is approximate. It is impossible to be 100% accurate when gathering data, or drawing a graph. Because of this, we use a line of best fit to closely represent our data. The line of best fit is important inextrapolatingdata. Extrapolation is the process of drawing conclusions about data based on a relationship we have already visualized — such as the liner relationship between concentration and absorbance.

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Using a Line of Best Fit to Extrapolate Data

Let's say you wanted to approximate the absorbance of a solution at a concentration of 50 mg/mL. We didn't take any measurements of absorbance for that particular concentration, but we can take a good guess at its approximate absorbance using our graph. Let's visualize how to do this:

1. Locate the appropriate value on the x-axis, the axis representing concentration.
2. Trace up the line until you hit the line of best fit:

   

3. From that point, measure across the axis toward "Absorbance." This value is the approximate absorbance of the concentration of a solution that is 50 mg/mL:

   

We have now extrapolated that the absorbance of a solution of concentration 50 mg/mL is approximately 0.75 absorbance units (AU).

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Check for Understanding

Use the following table of information to create a graph. Label all parts of the graph (Hint: check "Elements of a Graph – Labeling" for a reminder of what parts to label).

Use your line of best fit to answer the following questions:

1. What is the approximate concentration of a solution with an absorbance of 0.3 AU?
2. What is the concentration of a solution with an absorbance of 0.08 AU?
3. What is the absorbance of a solution with a concentration of 30 mg/mL?
4. What is the absorbance of a solution with a concentration of 55 mg/mL?
5. What is the absorbance of a solution with a concentration of 18.5 mg/mL?

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Solutions

Answers to Check for Understanding

1. 37 mg/mL
2. 11 mg/mL
3. 0.24 AU
4. 0.44 AU
5. 0.15 AU

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