Precalculus I Class 03/13

During the preceding class we linearized temperature vs. clock time data by using a log(T) vs. t transformation.  

Here we will illustrate the general process we will use in this course to linearize data.  

The y vs. x table shown below is built using the y = 5 x^3 function.  Usually when we linearize data we don't know what the best function is (as was the case with the temperature data), but for this illustration we'll start with a known function.

• Our first step is to create log(x) and log(y) tables, which is easily done using calculator or spreadsheet.

• We will then create graphs of log(y) vs. log(x) and log(y) vs. x.  A third type of graph we might sometimes use is y vs. log(x), though for this illustration we'll stick to the first two.  

A more accurate table is shown here:

 Our graphs will be set up as illustrated below.

Graph of log(y) vs. x is not very linear, increasing at a decreasing rate:

The graph of log y vs. log x is pretty much linear, increasing at what appears to be a constant rate.

The straight line through these data points has an equation of form log(y) = m * log(x) + b, where m is slope and b is the vertical intercept.

• The specific function that fits our graph has slope 3 and vertical intercept .699.  

• The best-fit function might not be given by your software in appropriate form.  Excel, for example, will tell you that the resulting function is y = 3 x + .699.  However we know that the graph is not y vs. x but log(y) vs. log(x) so we write this function as log(y) = 3 log(x) + .699.

• In the graph below we have changed Excel's y = 3 x + .699 to the more appropriate log y = 3 log x + .699.

Our results correlate with the power function we used to generate the data.  We haven't yet developed the tools to see precisely how these relationships exist.  However for future reference: 

• We note that the slope 3 matches the power of the power function y = 5 x^3 we used to generate the data.

• Also check out the fact that 10^.699 = 5, which matches the coefficient of our power function.

For reasons that we'll see later, a power function y = A x^p will result in a log y vs. log x graph whose slope is p and whose y-intercept b has the property that 10^b = A.

We will see in upcoming classes why the following general rules for linearization hold:

We now investigate the idea of inverse functions. 

• We start with a table for y = x^3.

• To get a table for the inverse function we reverse the numbers in the two columns.

Our original function is y = x^3.  We obtain the formula for the inverse function as follows:

• We first reverse the roles of x and y in the formula y = x^3, obtaining x = y^3.

• We then solve the reversed equation for y.  We do this as follows:

x = y^3.  We take the 1/3 power of both sides, obtaining

x^(1/3) = (y^3)^(1/3).  We use laws of exponents to simplify the right-hand side and get

x^(1/3) = y.  We switch the two sides to obtain

y = x^(1/3).

We sketch a graph of the function and the inverse function and note its significant characteristics:

• The graph of the inverse function is obtained by reversing the coordinates of the points on the graph of the original function.

• The result is that every point on the graph of the original function can be joined to the corresponding point on the graph of the inverse function by a line segment which (provided the graph is drawn with equal scales for the x and y axes) is perpendicular to the line y = x and which is bisected by this line.

• For example, (3, 27) on the graph of y = x^3 corresponds to the point (27, 3) on the graph of the inverse function y = x^(1/3).  If we join (3, 27) to (27, 3) by a line segment that segment passes through the graph of y = x at a right angle.  This segment is bisected (i.e., cut into two equal parts) by the y = x graph.

The main characteristic of a graph of a function and its inverse is that the two functions are symmetric with respect to reflection about the line y = x.

An accurate plot of these functions is shown below:

'Zooming in' on the domain 0 <= x <= 3:

The log function y = log(x) is defined as the inverse of the y = 10^x function.

• Note that if we switch coordinates in the expression y = 10^x we get x = 10^y.  The equation y = log(x) is therefore equivalent to the equation x = 10^y.

A table of y = 10^x and the 'inverted' table y = log(x) is shown below.

A rough graph of these functions is depicted below.  The graph in the lower-left-hand corner is an enlarged graph of these functions near the origin.

The figure below is a more accurate graph showing y = 10^x, y = log(x) and y = x.  Note the symmetry of these graphs with respect to reflection thru the line y = x.

A closeup of the same graphs near the origin shows the following important characteristics:

• The graph of y = 10^x passes through (0, 1), is asymptotic to the negative x axis and increases at an increasing rate.

• The graph of y = log(x) passes through (1, 0), is asymptotic to the negative y axis and increases at a decreasing rate.

A closeup of the same graphs near the origin shows the following important characteristics:

• The graph of y = e^x passes through (0, 1), is asymptotic to the negative x axis and increases at an increasing rate.

• The graph of y = ln(x) passes through (1, 0), is asymptotic to the negative y axis and increases at a decreasing rate.

A comparison between the graphs of y = 10^x and y = log(x), and the graphs of y = e^x and y = ln(x), reveals the following:

• The x and y intercepts (0, 1) and (1, 0) are the same for the pair y = 10^x and y = log(x), and for the pair y = e^x and y = ln(x).

• The graphs of y = e^x and y = ln(x) change more gradually than those of y = 10^x and y = log(x), approaching their asymptotes more slowly and changing curvature more gradually.