1. Graphing, Plotting and Modeling Data

2. A plot is an important way to graphically analyze or determine relationships among data.
Graphs must be constructed properly to avoid misinterpretation by others.
Graphs must tell us something meaningful.

3. Take your time selecting the scale for each axis.
The entire look and ease of understanding a graph depend on the scale of the axis.
The scale must be broad enough to include all of the data that is to be plotted.
The scale does not have to go beyond the values of data that are to be plotted.

4. Follow these 8 steps to create a graph.
Use a ruler to draw the x and y directions for your graph.
(1)
Label each axis with the independent (x) and dependent (y) variables.
(2)
Mark each axis with the scale values using an interval that makes it easy to plot each data point.
(3)
(4)
Include units of measure for each scale.

5. Follow these 8 steps to create a graph.
Use dots to plot the data as x, y ordered pairs on your graph.
(5)
Use a symbol, usually a circle, to enclose the data point.
(6)
Create descriptive title for the graph and place it at the top of the graph.
(7)
Place a conclusion state about your plot below your graph.
(8)

6. A simple graphing activity.
Rose Blossom Promoter 6 1 2 3 4 5
A simple graphing activity.
Dependent Variable y
Plot the following data.
Number of roses 1 2 3 4 5 6
day
Number of roses 1 2 3 4 5
day
Independent Variable x
The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter
(1)
Use a ruler to draw the x and y directions for your graph.
(2)
Label each axis with the independent (x) and dependent (y) variables.
(3)
Mark each axis with the scale values using an interval that makes it easy to plot each data point.
(4)
Include units of measure for each scale.
(5)
Use dots to plot the data as x, y ordered pairs on your graph.
(6)
Use a symbol, usually a circle, to enclose the data point.
(7)
Create title for the graph and place it at the top of the graph.
(8)
Place a conclusion state about your plot below your graph.

7. When do you connect the data points together?
Rose Peddle Blossom Promoter 6 5
When do you connect the data points together? 4
Dependent Variable
Number of roses 3 y 2 1 1 2 3 4 5
It depends on where the data points came from!
day
Independent Variable x
If the numbers came from an equation, you can connect the data points.
If the numbers came from an experiment, try to draw the simplest curve that has the smallest distance possible away from the data points.

8. Fundamental calculation for plotting data
Slope – a value that indicates the vertical to horizontal change in the data between two data points.
Area – a value that indicates the area under the curve between two selected x values.
Slope –
like the slope of a hill, a graph’s slope indicates how much the data is going up or down between these two data points.
If you ask a carpenter, it is the rise divided by the run.
If you ask a mathematician it is the difference between two Y values divided by the corresponding difference between two X value.
Slope (pitch) of a roof Y 2 Y 1
Rise
Slope of a line = =
Run X 2 X 1

9. Fundamental calculation for plots
Graphic Analysis y 2 y 1
Fundamental calculation for plots
Slope = x 2 x 1 x y 6
5 , 6
Using the two ordered pairs; 5
1 , 2
Rise= (6-2) 4
Dependent Variable
Number of roses 3 =
Run
Rise
(5-1)
(6-2) 4 y
Slope =? = 1 2
Run = (5-1) 1
Using the two ordered pairs;
2 , 3
4 , 5 1 2 3 4 5
day
Independent Variable
Run
Rise =
(4-2)
(5-3) 2 1
Run
Rise =
(4-2)
(5-3) 2 1
Slope = x
If the slope value is positive, the data points are going up hill.
If the slope value is negative, the data points are going down hill.
If the slope value is zero, the data points are horizontal.

10. Fundamental calculations for plots
Graphic Analysis
Rose Blossom Promoter
Fundamental calculations for plots 6 5 4
Dependent Variable
Number of roses 3 y
A line is a curve that has the same value for the slope calculation when any data point ordered pair is used in the calculation. 2 1 1 2 3 4 5
day
Independent Variable x
If you have experimental data, draw the simplest curve so that all the data points are as close to the curve as possible.
Whenever possible, a straight line is the curve of choice.

11. Almost ! Which pairs did we miss? 1 2 2 3 4 5 5 6
Graphic Analysis
Fundamental calculation for plots
day
Number of roses
(5 , 6)
Rise
(6-2) 4
Slope = = = = 1 1 2
(1 , 2)
Run
(5-1) 4 2 3 4 5
(4 , 5)
Rise
(5-3) 2
Slope = = = = 1
(2 , 3) 5 6
Run
(4-2) 2
Do these ordered pairs define a line?
(2 , 3)
(1 , 2)
Run
Rise =
(2-1)
(3-2) 1
Slope =
Did we do all of the possible slope calculations?
(5 , 6)
(4 , 5)
Run
Rise =
(5-4)
(6-5) 1
Slope =
Almost ! No
(1 , 2)
(4 , 5)
Run
Rise =
(4-1)
(5-2) 3 1
Slope =
Which pairs did we miss?

12. One more graphing exercise.
Independent Variable
Dependent Variable x 1 2 3 4 5 6 y
Number of roses
day
Rose Blossom Promoter
The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter
Graphic Analysis
One more graphing exercise.
The following data is from a 2nd rose experiment. Plot the following data.
day
Number of roses 1 2 2 3 3 5 4 5 5 6
Use a ruler to draw the x and y directions for your graph.
(1)
Label each axis with the independent (x) and dependent (y) variables.
(2)
Mark each axis with the scale values using an interval that makes it easy to plot each data point.
(3)
Include units of measure for each scale.
(4)
Use dots to plot the data as x, y ordered pairs on your graph.
(5)
Use a symbol, usually a circle, to enclose the data point.
Create title for the graph and place it at the top of the graph.
(6)
(7)
Place a conclusion state about your plot below your graph.
(8)

13. Slope Calculation Good news!
Independent Variable
Dependent Variable x 1 2 3 4 5 6 y
Number of roses
day
Rose Peddle Blossom Promoter
The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter
Graphic Analysis
Slope Calculation
Good news!
You have already decided that the plot is a line so you only have to do one slope calculation.
Bad news!
You have to pick two ordered pairs from this line to use in the slope calculation.
Good news! = x 1 y 2
Slope
You can pick any two orders pair that are easy to use in the slope formula.

14. Fundamental calculation for plots
Graphic Analysis
Fundamental calculation for plots
Slope – a value that indicates the vertical to horizontal change in the data between two data points.
Area – a value that indicates the area under the curve between two selected x values.
Area –
The area under a plot is a number that is directly related to the sum of all of the (x,y) pair products of points on the curve between the first x and last x value.
Since there are an infinite number of x,y pairs on a curve between two points on the curve, it is usually easier to find the area under the curve than to add the x times y product for every possible (x,y) pair on the plot.

15. Simple Area Calculation Example
Rose Peddle Blossom Promoter Experiment # 2
Graphic Analysis
Simple Area Calculation Example 6 5
Number of roses 4 y 3 2 1 1 2 3 4 5
day x
Calculate the area under the curve between x = 1
and x = 5

16. Area 1 Area 2 6 5 Number of roses 4 y 3 2 1 1 2 3 4 5 day x
Rose Peddle Blossom Promoter Experiment # 2
Area Calculation Example 6
Area 1 5
Number of roses 4 y 3 2
Area 2 1 1 2 3 4 5
day x
Total Area =
Area 1 +
Area 2

17. Area 1 is the area of a triangle
Rose Peddle Blossom Promoter Experiment # 2
Area Calculation Example 6
Area 1
Area 1 is the area of a triangle 5
Number of roses 4 y 3 1 2 h x b A 1 = 2
Area 2 1
Area 2 is the area of a rectangle 1 2 3 4 5
day x x L w A 2 =
Total Area =
Area 1 +
Area 2
Total Area = A 1 + 2

18. Area 1 Area 2 w = = = 1 2 h b A 1 = 1 2 (5-1) (6-2) A 1 = 16 A 1 = A 2
Rose Peddle Blossom Promoter Experiment # 2
Area Calculation Example 1 2 h x b A 1 = 6
Area 1 1 2
(5-1) (6-2) 5 A 1 =
Number of roses 4 y 16 A 1 = 3 2
Area 2 1 1 2 3 4 5
day A 2 = x L w x A 2 =
(5-1)
(2-0)
Total Area =
16 + 8 = 24
# of roses day A 2 =
(4)
(2)
Total Area = A 1 + 2 A 2 = 8

19. Area 1 Area 2 w = = = 1 2 h b A 1 = 1 2 (5-1) (6-2) A 1 = 16 A 1 = A 2
Rose Peddle Blossom Promoter Experiment # 2
Area Calculation Example 1 2 h x b A 1 = 6
Area 1 1 2
(5-1) (6-2) 5 A 1 =
Number of roses 4 y 16 A 1 = 3
If the starting numbers have units, the answer units are the product of those units 2
Area 2 1 1 2 3 4 5
day A 2 = x L w x A 2 =
(5-1)
(2-0)
Total Area =
16 + 8 = 24
# of roses day A 2 =
(4)
(2) A 1 + 2
Total Area = A 2 = 8

20. Fundamental modeling activities using data plots
Interpolation;
Predicting what the dependent variable (y) value will be for an independent variable (x) value that is not one of the original x values but is between any two of the original x values.
Extrapolation;
Predicting what the dependent variable (y) value will be for an independent variable (x) value that is not one of the original x values nor is it between any two of the original x values.

21. Interpolation Example 6 5
Rose Peddle Blossom Promoter Experiment # 2
Interpolation
Example 6 5
How many rose blossoms are expected on day 3?
4.1
Number of roses 4
Dependent Variable y 3 2
Rose Blossoms
4.1 1
How many rose blossoms actually showed up on day 3? 1 2 3 4 5
day
Independent Variable x
Thus, interpolation of the plot indicates we can expect 4.1 or 4 rose blossoms on day 3.
5 Rose
blossoms

22. Extrapolation Example 7.1 7 6 5
Rose Peddle Blossom Promoter Experiment # 2
7.1 7
Extrapolation
Example 6 5
How many rose blossoms are expected on day 6?
Number of roses 4
Dependent Variable y 3 2
Rose Blossoms
7.1 1
How many rose blossoms actually showed up on day 6? 1 2 3 4 5
day
Independent Variable x
Thus, extrapolation of the plot indicates we can expect 7.1 or 7 rose blossoms on day 7.
We don’t know!

23. plot these ordered pairs.
Interpolation
Extrapolation
Technicians, engineers and scientists often:
collect data,
arrange data as ordered pairs,
plot these ordered pairs.
Then, they interpolate or extrapolate as a modeling tool to determine y values they did not do experiments for.
Is it a better idea to interpolate a data plot or to extrapolate a data plot?
Why ?