P210 Graphing Problem Solving Lab Objective: Learn how to present different types of data using graphs, both manually and using Microsoft Excel 7.0 Step one is to manually graph each of the following using graphing paper. Step 2 is to graph them using Excel 2007. Refer to the document titled Creating Graphs Excel7.doc in the "Course Documents" tab of Blackboard for the P210S lab course for training in the use of Excel graphs. Note: Use the lab computers during your lab session to experiment with doing graphs via Excel 7.0 Graphing Basics: • • • • • • Graphs or plots are ways to present information Most graphs involve the relationship between two sets of data, e.g. temperature and time One set of data is the independent variable, the other is the dependent variable, i.e. it depends on the independent variable. The dependent variable is always plotted on the vertical axis, while the independent variable is on the horizontal axis Start with an “x-y” (Cartesian) coordinate system X is horizontal axis Y is vertical axis Individual points on the graph are loc the x & y values (x,y) y = mx+b is the equation of a line on a graph • • • • x is independent variable y is dependent variable b is the offset m is the slope, the change in y/change in x (rise over run) SLOPE = m y f yi x f xi y x y=mx+b 25 20 y axis • • 15 10 5 0 1 2 3 4 5 6 x axis 7 8 9 10 1 A. Linear relationships: A linear relationship is a straight line. It means that when X changes by a certain amount there is a corresponding change in Y. If an apple always costs a dollar, every time you buy another apple, the amount of money you spend goes up by $1.00. The relationship between cost and the number of apples is linear. A linear relationship can be either positive or negative. A linear graph has a constant slope. The following questions refer to linear relationships. 1. Draw a manual graph and create an Excel graph titled "Linear #1" using the following numbers. 2. Calculate the slope of the line as X increases from 1 to 2, 3 to 4 and 5 to 6. X Y 1 2 3 4 5 6 2 4 6 8 10 12 3. Draw a manual graph and create an Excel graph titled "Linear #2" using the following numbers. 4. Calculate the slope of the line as X increases from 1 to 2, 3 to 4 and 5 to 6. X Y 1 2 3 4 5 6 15 12 9 6 3 0 2 5. You are now going to graph the circumference of a circle as a function of the diameter of the circle. Another way of stating this is to graph circumference vs. diameter. The Y (dependent) variable is always listed first in these statements. Discuss how you think this graph should look before you actually do the graph – any ideas what the slope this graph might represent. Graph the data points below and calculate the slope – both manually and using Excel. Diameter 2 3 4 5 6 7 Circumference 6.3 9.4 12.6 15.7 18.8 22.0 3 B. Non Linear relationships: A non-linear relationship is not a straight line. Sometimes Y changes a lot when X changes, sometimes Y changes a little bit. Even the direction that Y changes can change. As a spring is stretched, initially the distance stretched increases at a constant rate with the force applied (linear). However, as the spring approaches its limit, an increasing force is required to stretch the spring, which is a non-linear relationship. If an arrow is shot into the sky, initially it rises and eventually it falls back to earth. At first as the arrow moves farther from me (X is the horizontal direction the arrow has moved); it rises many feet into the air for every foot across the ground. (Y is the vertical distance in the air.) Then gradually, the number of feet that the arrow rises vertically for each foot it moves away from me horizontally declines. (The arrow is reaching the top of its arc.) For an instant, the arrow neither gains nor loses height. Then the arrow falls a little bit for every foot it travels away from me, but the distance it falls increases for every foot it travels away from me. The relationship between the distance that the arrow travels from me and the height of the arrow in the air is non-linear. Initially, the relationship is positive with large slope, and then the slope gets smaller. At the peak, the slope is zero. When the arrow begins to fall the relationship is negative with a flat slope, but the slope gets steeper and steeper as the arrow approaches the ground more quickly. 4 6. Draw a manual graph and create an Excel graph titled "Non-Linear #1" using the following numbers. Use a "Scatter Plot with smooth lines and markers". 7. Calculate the slope of the line as X increases from 3 to 6, 9 to 12, 15 to 18, 18 to 21, 27 to 30, 33 to 36 and 39 to 42. Note how the steepness of the line changes as the slope changes. X 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 43 Y 0 20 35 45 50 53 55 56 55 53 50 45 35 20 5 0 5 C. Analysis of Graphs 8. Answer the following questions about the graph shown below: a. Is the relationship between X and Y linear or non-linear? b. Is the relationship between X and Y positive or negative? c. As X gets larger, does the change in Y become larger or smaller? d. Briefly describe in words how Y changes as X changes, that is describe the direction that Y changes, how fast Y change relative to X, and describe the changes in the rate at which Y changes as X changes. 6 D. Fitting Data When collecting and plotting experimental data, it is common to compare the gathered data with an expected relationship. An example would be measuring the intensity of sound at varying distances from the source of the sound. We would expect the intensity to drop as the distance is increased, but may not be sure if this is a linear relationship or not. The data can be plotted, but a "line fit" would be required to see if the relationship between the dependent and independent variables is linear or not. Plot the following data in Excel 2007, using a "scatter plot" chart style and then apply a linear fit to the data points. Under Trend line options, select to print the formula of the linear fit on the graph. Here are your data points: x y 0.99 2.03 3.04 3.89 5.08 149 304 449 600 678 7
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