Relationships and Graphing Chapter 15

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

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

Graphing Straight Lines Linear Relationship means the graphed points will form a straight line when plotted Plot the following equation: Y = 2X 2 1 4 2 6 3 8 4

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

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

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

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

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

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.

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

25 1 50 2 75 3 100 4

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

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

Thresholds

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?

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

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

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

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

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

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

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

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

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?

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

Log plot of data 2.90309 1 2.60206 2 2.30103 3 4 1.69897 5 1.39794 6 1.09691 7 0.79588

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