1. Relationships and Graphing
Chapter 15

2. Graphing
Origin is the point that vertical (Y axis) and horizontal (X axis) lines intersect
X and Y are divided into equal subdivisions
To the right of the origin on X axis are + values; to the left are – values
Below the origin on the Y axis are – values
Above the origin on the Y axis are + values

3. Continued
For any value of a variable on the x axis, there is a corresponding value of a variable on the y axis
The two values are called coordinates because they are associated with each other
The abscissa is the distance of a point on the X axis, the ordinate is the distance of a point on the Y axis

5. Graphing Straight Lines
Linear Relationship means the graphed points will form a straight line when plotted
Plot the following equation:
Y = 2X

6. What is the difference between Y = 2X, Y = 3X, Y = 4X, etc

7. Slope For Y = 2X (Y = MX) M (or 2) is the slope
For Y = 3X, what is the slope
Slope is rise over run
Run is the amount by which the X coordinate increases
Rise is the amount by which the Y coordinate increases

8. Continued Slope = Rise = DY = Y2-Y1 Run DX X2-X1
Uphill left to right is positive slope
Downhill left to right is negative slope

9. Y intercept
The point at which the line intercepts the Y axis (X=0) is the y intercept
For Y = 2X, the y intercept is 0
Or Y = 2X + 0
Graph A = 2C + 4

11. General Equation for a Line
Y = MX + A (in my time it was B)
Where M is slope and A is y intercept
Finding the equation for a line
Find Y intercept (where X = 0
Find the slope of the line
Put the values in the equation
Y = _______ (X) + _________
Slope y intercept

12. Application of Graphing Linear Relationships
Quantitative analysis is the determination of how much of a particular material is present in a sample.
Standard Curve is a graph of the relationship between the concentration of a material of interest and the response of a particular instrument.

13. Construction of Standard Curves
A series of standards are prepared containing known concentrations of the material of interest.
Each sample is analyzed using the instrument of choice and the response measured and recorded
A curve is plotted using the measured response Vs the concentration
This curve will then be used to determine the concentration of samples of interest

14. 25 1 50 2 75 3
100 4

15. Best fit for a straight line
Using a ruler, draw the line that appears to be the closest to all points
Using the Least Squares Method

16. Using graphs to display results
Independent variable is a variable that the investigator controls
Dependent variable is a variable that changes in response to the independent variable
Concepts to consider in graphing
Thresholds
A threshold is a point on a graph where there is a change ion the relationship of variables

17. Thresholds

18. Continued Best Fit line
In previous graph, middle points are close to forming a line.
To determine the equation
Connect points and calculate slope
Extend the line straight through until in intersects the y axis
What does this tell us?
Look at the data in table 10.2
Have students plot the data
What do the threshold limits tell us?

19. Continued
Prediction
Equations and graphs are tools that can be used to make predictions
Can we determine the yield of fruit at enzyme quantities that were not plotted?
Limit to predictions is that you cannot make predictions beyond the range of the data

20. Continued Using graphs to summarize data
A picture is worth a thousand words
Relationship between two variables is easily shown by graphs
What type of graph do we get if the variables are not related to each other?
Is this important to note?
Scattered data or flat lines are seen
I hormone experiment, it would mean that the particular hormone studied is not effective in increasing fruit production

21. Graphs and exponential relationships
Many systems do not follow linear relationships. Two examples of this are Nuclear decay and bacterial growth
The bacteria have a generation time of 1 hour. How many are present in 3 hours
One bacterial cell divides into 2, 2 into 4, 4 into 8.
The mathematical relationship is Y = 2x G

22. Counting Bacteria General equation for Bacterial Population Growth is:
Y = 2x(No)
Where No = number of bacteria initially
Y is the number of cells after x generations
X is the number of generations
Graph is exponential

23. Exponential graph of Bacteria growth
100 1
200 2
400 3
800 4
1600 5
3200 6
6400 7
12800

24. Semilog paper Linear plots are easier to work with
Converting exponential graphs to linear graphs often involves logarithms
Semilog paper is a type of graph paper that substitutes for calculating logs
Has normal, linear divisions on the x axis and wide to narrow spacing on the y axis

26. Exercise
Plot the previous data on regular graph paper and semilog paper
Compare the results of these graphs

27. Decay of Radioisotopes
This is based on half-life
Half-life of a radioactive substance is the length of time it takes for the amount of radioactivity to decay to half its original level
8 units  4 units  2 units 1 unit  0.5
In 4 half-lives go from 8 to 0.5 units

28. Continued Equation for radioactive decay N = 1/2tNo
Where N is the amount remaining
T is the number of half-lives
No is the amount initially
If a sample has 800 disintegrations per second now, how many will it have at 6 hours if the half-life is 1 hour?

29. Calculation
800 1
400 2
200 3
100 4 50 5 25 6
12.5

30. Log plot of data 1 2 3 4 5 6 7

31. Comparison of equations for Bacterial growth and radioactive decay
Growth of Organisms: N =2t(No)
Decline in radioactivity: N =1/2t(No)
Differ in the base number