Graphing Linear Relationships

1. Graphing Linear Relationships

2. What are the objectives of Unit 3? • To learn how to critically analyze data using graphs. • To learn how to use graphing techniques to arrive at mathematical equations that describes the relationship between two variables. • To learn to connect the mathematical slope of a line to a physical property

3. Graphs compensate for shortcomings of data tables. • Difficult to determine relationship between variables. • Impractical to find equation

4. The most important purposes of a graph are • Quick and efficient organization of large amounts of data. • Demonstration of relationship between independent and dependent variables. • Calculation of a mathematical equation that describes the relationship in the data.

5. Scientists who used graphs to make important discoveries. • Galileo’s falling body experiments showed all objects, no matter what they weigh fall at the same rate. • Schwabe’s sunspot data showed that sunspots are at a maximum every 11 years.

6. Dependent Axis and Independent Axis • In any graph there are at least 2 axes • Vertical axis is called the y-axis • Plot the dependent variable on the y-axis. • May refer to it as y.

7. Horizontal axis is called the x-axis. • Plot the independent variable here except when time is a variable • Always plot time on the x-axis • May refer to the variable as x

8. (x,y) is the pair of data. Plotting this point allows you tosee the relationship between the variables. • Linear Data – data points form a straight line.

9. Best Fit Line Approximates the linear trend of the data. • The line passes in between the data points. • The line does not have to pass through all of the points. • Points should be relatively close to one another. • Random error causes about ½ the points to be above the line and ½ to be below the line. Use a straight edge to approximate. • Can be computed by mathematical equation or computer program like Excel.

10. Best Fit Curve • Used when data points form a curve. • Should pass between points but must remain smooth.

12. Types of Linear Relationships Direct Linear Relationships • As the independent variable increases, the dependent variable increases. • As the independent variable decreases, the dependent variable decreases. • Example of both is ice cream cone - Melting vs. Temperature: Both variables move in the same direction. • Have positive slopes and appear to go uphill. • Equation is y=mx+b

13. Direct Linear Relationship

14. Direct Linear Proportion • A direct linear relationship that passes through the origin (0,0). • y-intercept is 0. • Equation is y = mx.

15. Direct Linear Proportion

16. Indirect Linear Relationships • Indirect means as one variable increases, the other decreases. • Have negative slopes and look like downhill lines. • Indirect linear proportions are extremely unusual in the physical sciences since science deals with real situations.

17. Indirect Linear Relationship

18. Indirect Linear Proportion • Data starts at (0,0) and decreases into the negative region of the graph.

19. Horizontal Line • 1. Indicates there is no measurable relationship between the variables. • 2. The independent variable is an irrelevant variable. • Mass and Arc Size in the pendulum lab

20. Indirect Linear Proportion

21. Let’s think a minute • What would a vertical line tell you? • The values of the replicates are changing. • A variable has not been controlled. • You’ve made a mistake in your procedure.

22. Calculating the Equation of a Plotted Line • In Algebra the general equation for a line is y =mx + b • x is the independent variable • y is the dependent variable • m is the slope • b is the y-intercept • We will determine a specific equation for a specific set of data where x and y will be replaced with more descriptive variables.

23. Calculating the Slope • The slope of a line describes how quickly the dependent variable changes with respect to the independent variable. • Large positive slopes - steep uphill • Small positive slopes – gradual uphill • Large negative slopes – steep downhill • Small negative slopes – gradual downhill

24. To calculate slope • Pick two points on the line (not data points) • Pick points where the line perfectly crosses a corner of the grid. Move from left to right. • Choose points as far apart as possible • Use

25. Significant Figures in Graphing • The significant figures should be determined by the accuracy of the data. • Use the place value of the independent (x) and dependent variables (y) in the data table when selecting points off the graph

26. Calculating the y- intercept • The y-intercept is the value of the dependent variable at the point where the best fit line crosses the y-axis. • Estimate this value from the graph. • Express with units of the dependent variable. • If the y-intercept is not shown on the graph, plug in values of a point and solve for b in the equation for the line.

27. Putting it all Together • Change the variables from x and y to more appropriate variable. • 2. Example pressure = P, temperature = T • a. clarity • b. real situation • If there is not a standard variable, define a variable

28. Using the Equation • Why use the equation? • So we do not have to redo the experiment each time we need a new data point.

29. Melissa Hubert Example p 65

30. P 66 continued • Select data points (2.0°C, 5.00atm) and (68.0°C,16.00atm) • Calculating Slope: • Slope = 0.167atm/°C • Equation of the line is: • y = (0.167atm/°C )x + b

31. Now we need b • y = (0.167 atm/ºC) x + b • 5.00 atm = (0.167 atm/ºC) 2.0ºC + b • 5.00 atm = 0.34 atm + b • b = 5.00 atm – 0.34 atm = • 4.66 atm = 4.66 atm • Plugging in the value of the y-intercept, the equation of the best-fit line becomes: • y = (0.167 atm/ºC) x + 4.66 atm

32. Now change the variables • P = (0.167atm/˚C)T + 4.66 atm

33. Finding Equations PWYR p69

34. PWYR p69 • 1. Which variable is being changed by the experimenter? • Mass • Explain why you chose this variable? • It is the independent variable and is plotted on the x axis. • 2. Find the equation of the best-fit line in the graph. • (0g,0Kj) and (84g,250Kj) E = energy E = (3.0Kj/g) M

35. P 69 continued • 3. How many kilojoules (kJ) of energy would be released if 4,129 grams of ammonium dichromate decomposed? • E = (3.0Kj/g) M • E = (3.0 Kj/g) 4,129g = 12,387Kj = 12,000Kj • 4. If researchers needed to limit the energy released in the chemical reaction to 975 kJ, what is the maximum mass of ammonium dichromate they can use? • 975Kj = (3.0 Kj/g) M • M=975Kj/3.0 Kj/g = 325 g = 320 g

36. Derived and Empirical Equations • Derived Equations can be found by graphing data or can be formulated by combining 2 or more already known equations. • C=∏d • d=2r where r = radius • Substitute 2r for d in the first equation • C=2∏r

37. Empirical Equations • Empirical Equations can be formulated by observing patterns in nature and creating an equation. • Johanne Kepler observed the patterns of the planets orbiting the sun and concluded they had something in common. • If the periods of their revolutions were squared and divided by their average distances from the sun cubed, the result for all the planets was a constant.

38. Extrapolation • The extension of a best fit line beyond the plotted points. • Allows you to make predictions. • Use it cautiously and make sure the following conditions are met. • The plotted data is accurate. • The data points were plotted correctly. • The relationship between the 2 variables continues. • Small error can result in huge error over large distances.

39. Interpolation • To go between data points. • Can have a great deal of confidence that the interpolated data is as good as the experimental data.