Proportionality: direct and indirect
When two variables are directly proportional to each other, they are related by an equation where k is a constant and x and y are variables. When a directly proportional relationship is graphed, the result is a linear graph with slope k and . Introduction: The relationship between island area and number of species is well S = CAz. which is often presented in linear form: log10S = log10 C + z log10 A. The constants C and z are fitted from the data on island area and number of Estimate the equation for a line fitting the data in the graph of log10 species. Developing a facility for graphing functions is very important in all branches of science. with all other things constant, let's strip away all the unnecessary things and In science, we very much like to plot physical relationships as straight lines.
For example, what does F become if r is made very large? The answer is obviously that F is very small. That is, if r tends to infinity, F tends to 0, and I've indicated that on the axes in panel 3. What does F become if r is made very small? I've indicated this on the axes in panel 3 as well.
As r increases from a small value, F decreases rapidly, but smoothly. Clearly, as r goes from very small to very large, F varies smoothly between these two as I have indicated in panel 4. Notice nothing was said about restricting r to positive values so I've also sketched the curve in the region of -r.
The result is shown in panel 4; study panel 4 to make sure you understand how we got it. Let's do another example just for practice. Try it before you go on. Let's now look at panel 5 to see if you sketched it properly.
First of all, notice that when I say y vs x, I always mean to have x as the independent variable and plot it horizontally, it's sometimes called the abscissa, and y is plotted vertically; it's sometimes called the ordinate. Now, when x is large and positive, then ax3 and thus y, is very large and positive and gets even larger with larger x, so we're able to sketch in the right-hand portion.
Notice that as x increases, y also increases but at a greater rate because of the x3 term. This is how we determine the general form of the graph. Now, if x gets increasingly negative, then ax3 gets very large but negative and you can see how we got the left-hand portion of the graph.
One more example will illustrate a few more useful points. First of all, let's look at the infinity limits. As x becomes increasingly large in either the positive or negative direction, then y gets very large in the positive direction.
The curve can now be sketched in as shown.
Well let's look at that problem. This is an added refinement that won't always be necessary but it helps. You'll remember from your calculus course that the minimum or maximum in a curve is that point where the derivative of the function describing the curve is 0.
That is, the tangent to the curve is horizontal.
Look at panel 7. This is the point then at which the curve has a maximum or minimum. Since the curve has no maximum, it must be a minimum.
Now, there is one very important graph which you must become familiar with and that's shown in panel 8. The quantity m is called the "slope" of the line.
You must become adept at recognizing straight line relationships and I've shown a few in panel 9. Study these 3 examples taken from elementary mechanics and see that you understand them. The independent variable was t so it was the abscissa.Position Time Graph to Acceleration and Velocity Time Graphs - Physics & Calculus
In science, we very much like to plot physical relationships as straight lines and you will be asked in your courses to devise ways of plotting certain functions as linear relationships. Let's look now at how to go about this.
Graphing simple functions
We'll look at the first relation we started with. Is it possible to plot this in such a way as to yield a straight line? The answer is yes. Let's make a transformation as shown in panel The species-area relationship can be approximated by a power function of the form: The constants C and z are fitted from the data on island area and number of species, and so are specific to a data set. Browne and Peck used long-horned beetles Cerambycidae: Coleoptera to investigate the species-area relationship in the Florida Keys and mainland.
Their data are plotted below, using the log10 of the area and species number. The two data points furthest to the right represent, from left to right, South Florida the area south of Lake Okeechobee and the entire state of Florida.
All other data points are from the Keys. We can also plot the log10 of species number against the log10 of the distance of the islands from the Florida mainland. The relationship between the log10 of the species number and the log10 of island area clearly shows an increasing trend, and is well-approximated by the linear form of the model. Similarly, a decreasing trend of species number with island distance can be seen from the second graph.
Whether or not the number of species is at equilibrium is less clear; interpretation of the data regarding this question is made more difficult by uncertainty about the history of sea-level changes over the past 10, years. Both island size and distance from the mainland are associated with the number of species present.