Transforming a square into a rectanglee by increasing or decreasing the length of the sides of the square leads to two equivalent experssions for the area of the rectangle. The classic handshake problem and variations on it offer an interesting context in which students can further explore quadratic functions. The equations of all these functions are similar to those students saw in Investigation 2 and provide an opportunity for them to discover that a single model can describe more than one context. Students also examine the rate of change between the two variables in a quadratic function. Classic projectile-motion problems extend students's understanding of quadratic equations and their graphs.
Students explore the pattern of change over time in the height of a ball thrown into the air.
They investigate quadratic equations describing the jumps of a frog, a flea, and a basketball player. Students then study the patterns of change in quadratic equations more closely. They discover that the differences between consecutive y values, called first differences, are constant for linear relationships and that second differences are constant for quadratic relationship. This classic problem reveals linear, quadratic, and cubic relationships. In their analysis of these questions, students compare quadratic relationships with cubic and linear relationships and discover that in cubic relationships, third differences are constant.