Business Calculus Graphing

 2.1 & 2.2 Graphing: Polynomials and Radicals Facts about graphs: 1.Polynomials are smooth and continuous. 2.Radicals are continuous, but not necessarily smooth. We will look for the locations where a graph changes direction and changes concavity.

 Inflection points We will use the first derivative number line to illustrate the direction of the function. 1.Find all critical values and endpoints. 2.Draw the 1 st derivative number line, using test values to determine the + or – for each interval.  2.1 Intervals of Increase and Decrease: An inflection point is a point on the graph of a function where the graph changes concavity.  2.2 Intervals of Concavity:

It is important to note two things: 1.This is a list of possible inflection points. The graph might or might not change concavity at these values. 2.We are not testing extrema here! We are looking for changes in concavity, to give us the shape of the graph.  A list of possible inflection points is found by: 1. 2.

A function f has an inflection point at x = a if: 1. a is in the domain of f and 2. f changes concavity on either side of x = a. We will use a 2 nd derivative number line to illustrate the concavity of the graph. Draw the number line, place all possible inflection points and endpoints on the number line. Test each interval with test values plugged into the 2 nd derivative to determine the concavity in each interval.

 Combined Number Line: Draw a third number line which will incorporate all the information found thus far. Place all critical values, possible inflection points, and endpoints on this number line. Insert the + and – information from the first and second derivative number lines. Draw the shape of the graph in each interval.

y ’ + − y ” − + + − y  Combined Number Line:

 Graphing for polynomials and radicals 1.Determine the domain of the function. 2.Find all critical values and endpoints, and draw the first derivative number line. 3.Find all possible inflection points, include any endpoints, and draw the second derivative number line. 4.Draw the combined number line. 5.Find the y values of all important points, and sketch the graph.

 2.3 Graphing Rational Functions Facts about graphs: Rational functions are smooth, but are often not continuous. The rational functions we will graph will have the possibility of vertical asymptotes, open holes, horizontal or slant asymptotes.

 Vertical Asymptotes vs. Open Holes A rational function may have as many vertical asymptotes, and as many open holes as the function requires. Both vertical asymptotes and open holes of a graph come from an input which forces the denominator = 0. For x = c which comes from setting the denominator = 0: An input x = c will create a vertical asymptote if An input x = c will create an open hole if. In this case, the hole will occur at the point (c, L).

A rational function may have either one horizontal asymptote or one slant asymptote, not both. To find the horizontal or slant asymptote, find. If, the graph has an horizontal asymptote at y = L. If, the graph has a slant asymptote (for our examples).  Horizontal Asymptotes vs. Slant Asymptotes

 Graphing for rational functions 1.Determine the domain of the function. 2.Find all vertical asymptotes and open holes. 3.Find any horizontal or slant asymptotes. 4.Find all critical values and endpoints, and draw the first derivative number line. 5. Find all possible inflection points, include any endpoints, and draw the second derivative number line. 6. Draw the combined number line. 7. Find the y values of all important points, and sketch the graph.