Intent Students continue finding equations for graphs—in this case, for nonlinear graphs.

Mathematics This is the third of four activities designed to help build intuition about the graphs of functions with certain types of rules. Students are being introduced to certain families of elementary functions and their symbolic, graphical, and tabular representations. In this activity, the focus is on quadratic functions of the form y = a(x + b)² + c and square-root functions of the form y=ax+b+c size 12{y=a sqrt {x+b} +c} {}. The square-root function is presented graphically as a quadratic function rotated 90 degrees. This affords the opportunity to discuss that, in this case, x is a function of y.

Progression Students will work on the activity individually and then share their results in class.

Approximate Time 15 minutes for activity (at home or in class) 15 minutes in class

Classroom Organization Individuals, followed by whole-class discussion

Doing the Activity As in Graphs in Search of Equations I, making In-Out tables for these graphs will help students look for the patterns that will, in turn, assist them in finding equations.

Discussing and Debriefing the Activity Ask students to share both their answers and their solution methods. Students will likely not have trouble with graphs a or b, but for graph b they should be careful not to place the negative sign inside parentheses. That is, they should not write y = (-x)². You may want to remind students that graph c does not represent a function, because each positive x-value has two corresponding y-values. In particular, be sure students see that the upper half of graph c is the graph of the equation y=x size 12{y= sqrt {x} } {}. This example is especially important, because the data for the pendulum experiments relating period to length should more or less fit an equation of the form y=cx size 12{y=c sqrt {x} } {} (but don’t tell students this, as it is their job to figure out what sort of function describes a pendulum’s period). If students try to use a calculator to draw graph c, they will have to graph the functions y=x size 12{y= sqrt {x} } {} and y=-x size 12{y"=-" sqrt {x} } {} simultaneously in the same window.