b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

When graphing roots, remember that for , if n is even, the domain includes all positive integers. Otherwise, negative values are included as well. When graphing roots of the form , remember the y-intercept is b (plug x = 0 into the equation and see for yourself).

If it helps your students, remind them that roots are actually (bippity-boppity-boo!) fraction-exponents. After all,  is just another way to write x^⅓. So as x increases, the value of x^⅓ increases too, but slowly.

When graphing piecewise functions, follow the same rules as for the above function, it's defined...well...in pieces. Usually, the function will have different equations for differing values of x. For example:

For x < 10, f(x) = 2x + 2

For 10 ≤ x < 15, f(x) = 22

For x ≥ 15, f(x) = 3x – 23

Step functions behave in a similar way to piecewise functions, only there's a gap in the function.

For x < 2, f(x) = 5
For x ≥ 2, f(x) = 10

Absolute value functions are similar to piecewise functions. They're essentially whatever function is contained in the absolute value, only not negative. Whatever parts of the function are underneath the x-axis are reflected back above it, like folding the graph in half along the x-axis. It's like having two different equations (since |x – 3| = |-x + 3)| that change every time the function is supposed to cross the x-axis. If all else fails, plugging in points is always a good skill to have.

Students might not understand how one function can have two different equations. Explain to them that it can be done so long as the two functions exist for different values of x. After all, if a magician can saw a person in half, why can't a function be sawed in half, too?