The Investigation below poses questions to generate interest in various mathematical topics from the text and encourages students to formulate and investigate their own conjectures. One use of the investigations is for term papers in which students report on their conjectures and the patterns they find. Click on the Read Me file below to open the investigation in a Word file:
Read Me - Graphs of Functions Instructions (Word Format)
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GSP file--Investigation 2.2: Graphs of Functions
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Graphing Calculator Investigation 2.2
Graphs of Functions A graphing calculator is a powerful devise for exploring the graphs of functions, discovering patterns, and forming generalizations. To enter the following equations into the calculator, the variable y is used for f(x). Before graphing, set the range of the variables x and y as follows:
X MIN = 0, X MAX = 20, Y MIN = 0, Y MAX = 20. Starting Points for Investigations 1. Graph the following equations and observe how their graphs change: y = 1x, y = 2x, y = 3x, y = 4x How are the graphs similar and how are they different? What general statement can be made for the graph of y = kx, for any whole number k? 2. What happens if the value of k in y = kx is a fraction? Graph the following equations and make a generalization about what happens as the fractions decrease. y = (1/2)x y = (1/3)x y = (1/4)x y = (1/5)x 3. Try to predict what will happen to the graph of y = x, if whole numbers are added to x. For example, y = x + 3 y = x + 7 y = x + 10 y = x + 12 Graph these equations and describe the graph of y = x + b, for any whole number b. 4. The above investigations can be carried out for other functions. For example, try graphing the following equations for k = 1, 2, 3, 4, . . . or k = 1/2, 1/3, 1/4, . . . .
y = kx2 y = x2 + k y = kx3 y = x3 + k