Science 122 Lab 3 Graphs & Proportion

Science 122

Laboratory

Laboratory 3
Graphs and Proportion

Work through all of the exercises, then take the quiz and submit it to be graded. The grade for the lab is based entirely on the quiz, so be sure that you understand the exercise and the learning outcomes.

Background Information on Lab 3

Computer Notation

Contents

Introduction

Outcomes

Supplies

Procedure 1

Procedure 2

Procedure 3

Questions


1. INTRODUCTION


A graph is a pictorial representation of ordered pairs of numbers. The reader can quickly determine relationships between quantities that the ordered pairs represent.

In many of the experiments performed in our laboratory exercises, we will study quantities that change in value. A change in one quantity may cause another quantity to change. We say one quantity is a FUNCTION of the other. For example, the area of a circle is a FUNCTION of the radius. That is, the area depends on the size of the radius in a regular and predictable way. When we increase the size of the radius the area of the circle increases accordingly. In this example the radius is called the INDEPENDENT VARIABLE and the area is the DEPENDENT VARIABLE.

A graph is plotted as a picture of ordered pairs of numbers where one number is related to another by a specific relationship, sometimes incorrectly called a formula. To draw a graph we use graph paper and a set of ordered pairs of numbers. Spreadsheet programs such as Excel have the capability to draw graphs and determine the relationship between the ordered pairs.

The ordered pairs of numbers might be calculated or determined by measurements,

In this exercise, you will determine the ordered pairs and plot graphs using the mathematical relationship for the circumference of a circle as a function of the diameter of the circle, and the area of a circle as a function of its radius.

The pairs of points, representing different values of the variables, will be plotted on graph paper or with computer software. The plotted points will connected by a smooth line that may be straight or it may be curved depending on the type of relationship that the ordered pairs represent. The general name "CURVE" is used in reference to all graphs whether the line is actually curved or straight.

Key Points to Note

The constant of proportion of a linear relationship is the same number as the slope of the graph that plots the relationship.

Definition: constant of proportion - the constant value of the ratio of two proportional quantities x and y; usually written y/x = k or y = kx, where k is the constant of proportion.

See "Direct Proportion"

In this exercise you will explore the connections between numbers, symbolic notation (otherwise known as equations) and graphs.


2. LEARNING OUTCOMES


After completing this exercise you should be able to do the following: 

  1. Define and explain the term "graph
  2. Plot a proper graph of a linear relationship.
  3. Label and name a properly drawn graph.
  4. Plot a graph of a nonlinear relationship using a mathematical equation to obtain ordered pairs of numbers.
  5. Distinguish between dependent and independent variables.
  6. Define slope, determine the slope of a straight line, and equate the slope with the constant of proportion.
  7. State the symbolic relationship for a straight line
  8. State the general form of the relationship for a parabola.
  9. Replot a parabola as a straight line.


3. EQUIPMENT AND SUPPLIES


See web sites for computer graphing


4. PROCEDURE 1: Circumference vs. Diameter


In this procedure you will see the relationship between "ratio", "constant of proportion", and "slope" of a linear graph.

The number known as "pi" occurs frequently in calculations involving circles and circular motion. It is an irrational number that represents the constant ratio between the circumference and diameter of a circle.

Relationship 4.1

The number 'pi' () is the ratio of the circumference of a circle to its diameter. It is the same for all circles because all circles are the same shape. The value of pi is approximately 3.14.

C= x D

where C = the circumference,

D = the diameter,

= 3.14


4.1. COMPLETE DATA TABLE 1.1.

Multiply each of the values in the diameter column (independent) by to calculate the value in the circumference (dependent) column.

1. Round off the calculated circumferences to two decimal places.

2. Complete the ratio (C/D) column by dividing the number in the circumference (C) column by the number in the same row of the diameter (D) column.

Data Table 1.1

Diameter D)
(Independent)
Circumference (C)
(Dependent)
Ratio C/D

1 cm

   

2 cm

   

3 cm

   

4 cm

   

5 cm

   

6 cm

   

7 cm

   


4.2 PLOT A GRAPH [graph 1] of table 1.1

Plot the circumference vs. the diameter using the data in table 1.1.

Click here for graph paper to print or use with drawing software.

Look here to see how to properly name, label and plot a graph.

Here are links to some web pages on linear proportions and computer graphing.


4.3 CALCULATE THE SLOPE OF THE TABLE 1.1 data.

Look here to see how to calculate the slope.

Point of interest

A linear relationship (also called a direct proportion) has the general form y = mx+b, where

"y" is the dependent variable

"m" is the slope of the graph

"x" is the independent variable

"b" is the intercept (where the line crosses the y-axis; this number is not significant for this exercise)


4.4 MEASURE THE DIAMETER AND CIRCUMFERENCE OF FIVE DIFFERENT CIRCLES.

Measure the circles! Do not use the formula to calculate the circumference.

All measurements must be in centimeters..

Use anything circular that you can find in the house or office. Coins, cans, lids, medicine containers, pots and pans, anything that looks circular.

Make note of what circular objects you measure and include it when you submit the lab exercise.

Record the measurements of diameter and circumference in table 1.2, then calculate the ratios of C/D for each ordered pair of data as in table 1.1.


(Use a length of string to measure the circumference.)

Data Table 1.2

The data in this table represent actual measurements of the diameters and circumferences of circles of various sizes.

D (cm)
(Independent)
C (cm)
(Dependent)
Ratio
C/D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


4.4. PLOT A GRAPH [graph 2] of table 1.2

Plot the circumference vs. the diameter using the information in data table 1.2.

Click here for graph paper to print or use with drawing software.

Look here to see how to properly name, label and plot a graph.

Here are links to some web pages on linear proportions and computer graphing.

Data 1.2 will probably not fall on a straight line because these are 'real' circles, not 'perfect' circles, and there is no 'perfect' measurement. Measurements always contain errors and we will learn more about measurement and errors in lab 5.

If the table 1.2 data does not fit a straight line, use a ruler to draw a straight line that comes closest to all of the points. This is known as a 'best fit' line, or 'regression line'.Here is information about how to draw the best fit line.

Excel or other spreadsheet programs will draw the best fit line ('trendline') and calculate the slope. See the links above for tutorials on using Excel to draw graphs.


4.5 CALCULATE THE SLOPE OF THE BEST FIT LINE FROM THE TABLE 1.2 DATA. Use the line and not the points since any one of the points may not be 'perfect' due to measurement error.

Look here to see how to calculate the slope. Use the best fit line to calculate the slope.

As you look at the two graphs (graph 1 & graph 2), think about how you would expect that the two slopes would compare. Include a brief statement about this in your summary.


5. PROCEDURE 2: Area vs. Radius


In this procedure you will produce a nonlinear graph where one variable is related to the second power of the other. The graph of this relationship will not be a straight line. This is known as a "direct square" proportion and the graph will curve upward.

The relationship between the area of a circle and the radius of the circle can be stated as follows: The area of a circle is proportional to the square of the radius. Written in symbolic notation: A = R2

RELATIONSHIP 5.1

Area = x (radius) squared OR A = R2

the area A is in square centimeters (cm2)

the radius R in centimeters

= 3.14.


5.2. CALCULATE THE AREAS OF CIRCLES

Use relationship 5.1 to calculate the areas of the circles in table 2.1. Multiply by the square of the radius (R) for each circle in table 2.1. Record the calculated values in the Area (A) column of table 2.1


5.3. CALCULATE THE RATIO of Area to Radius

1. Round off the calculated circumferences to two decimal places.

2. Complete the ratio (A/R) column by dividing the number in the Area (A) column by the number in the same row of the radius (R) column.

Data Table 2.1

radius (R) radius squared (R2) area (A)

Area/radius
(A/R)

1

     

2

     

3

     

4

     

5

     

6

     

7

     


5.4. PLOT THE GRAPH [graph 3] of table 2.1

Plot area (dependent) vs. radius (independent). Prepare and label the graph completely and correctly as in Procedure 1.

No straight line can be drawn that passes close to all of the points. A nonlinear relationship graphs as a curved line. This one curves upwards because it is a second power (square) relationship.

Note hat the slope increases as the values get larger.

The slope of a curved graph does not have useful information since it changes. This means that the relationship (ratio) between the two variables is not constant.

Note this change in the A/R column in Table 2.1

The square relationship (also called a direct square proportion) has the symbolic form y = kx2 (also written as y/x2 = k). In this case it is A = R2 (also written as A/ R2= )

We would state the relationship as: The area is proportional to the square (second power) of the radius.

This is the general relationship for a curve called a parabola, one of the conic sections.


6. PROCEDURE 3


6.1. COMPLETE TABLE 2.2.

It is identical to table 2.1 except for the ratio column. Instead of calculating A/R calculate A/R2 and record in the appropriate column.

Data Table 2.2

radius (R) radius squared (R2) area (A)

area/radius squared
(A/R2)

1

     

2

     

3

     

4

     

5

     

6

     

7

     


6.2. PLOT THE GRAPH [graph 4] of table 2.2

Plot area area (A) (dependent) vs. radius squared (R2) (independent). Prepare and label the graph completely and correctly as in Procedure 1.

In this procedure we have straightened out the parabola of procedure 2 and made the squared proportion into a linear one by plotting the graph of x2 as the independent variable instead of x as in procedure 1. For this graph the independent variable is R2 , the dependent variable is A, and the slope should equal since that is the constant of proportion that relates the two variables (A/ R2= )


6.3 Calculate the slope of the line on the graph of table 2.2


Point of interest

Data from an experiment in the laboratory may yield a curve whose relationship is not obvious. The type of relationship can often be determined when the data are replotted in order to obtain a straight line.


When you have finished you should have completed 4 data tables, plotted 4 graphs, and calculated 3 slopes.


7. QUESTIONS

7.1. Discuss the relationship between the following:


7.2. Name the dependent and independent variables in each graph.


7.3. What did you measure to get data for table 1.2?


7.4. Based on the data in table 1.1, what is the diameter of a coconut tree with a 50 cm circumference?


7.5. What would be the UNITS (not the numbers) for the slope of a graph that plots seconds as the independent variable and meters as the dependent variable?


7.6. What is the connection between the slope of the line in Procedure 1 and the ratio of circumference to diameter?


7.7. How do the slopes of the two graphs in procedure 1 compare? What would you expect the slope of graph 2 (table 1.2) to be?


7.8. Is the relationship between the circumference and the diameter of a circle a linear relationship? Explain your answer.


7.9. If the value for x increased faster than the values for y, which way (toward or away from) would the graph curve with respect to the "y" axis?


7.10. State in words the relationship between the area of a circle and its radius.