Graphing Polynomial Functions

Pre-Calculus · Axiom Academy

EXAMPLE Graphing Polynomial Functions Sketch a polynomial step by step: zeros, multiplicity, end behavior, y-intercept, then the curve. Graph the polynomial function f(x) = x^4 - 5x^2 + 4 . Find its zeros and their multiplicities, determine the end behavior, locate the y-intercept, and use these features to sketch the curve. Nicely done. You sketched a degree-4 polynomial from its structure alone, no point-by-point plotting. The recipe: End behavior: set by the degree's parity and the leading coefficient's sign. Even degree with a positive lead sends both arms to . Zeros: factor completely, then set each factor to zero. Here (x+2)(x+1)(x-1)(x-2) gives x = -2, -1, 1, 2 — the x-intercepts. Multiplicity: odd multiplicity means the graph crosses the axis; even multiplicity means it touches and bounces. All four zeros here are multiplicity 1, so it crosses at every one. Y-intercept: evaluate f(0) . Here f(0) = 4 , giving the point (0, 4) . Sketch: connect the end behavior, zeros, and y-intercept with one smooth continuous curve. This same four-feature pass — end behavior, zeros, multiplicity, y-intercept — sketches any polynomial. Try it on functions of different degree and leading sign to build the instinct.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.