Sketching functions

Solve f(x) = 0 for x-intercepts and evaluate f(0) for a y-intercept only when zero lies in the domain. Use symmetry and the leading term to establish overall shape. Label exact coordinates when you can find them.

A graphing calculator supplies a view of a chosen window. It may miss a root, an asymptote or a small feature. For piecewise functions, treat each branch only on its stated interval and distinguish included endpoints from excluded endpoints.

Worked example

Sketch y = (x − 1)² − 4.

1. The vertex is (1,−4) and the parabola opens upwards.
2. Set (x − 1)² = 4 to find x = −1 and x = 3.
3. At x = 0, y = −3.

Answer: Upward parabola through (−1,0), (3,0), (0,−3), with vertex (1,−4)

Revise first: Completing the square.