Polynomial Graphing Calculator

A polynomial function is expressed generally as P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀. This calculator evaluates your defined degree (DegP) and coefficients (such as A0, A1, A2, A3, and A4) across a specified interval running from IntStart to IntEnd. It computes the absolute values of coefficients (like a_{3_abs} = abs(a₃)) to assist with bound estimations, and systematically calculates output values for y to trace smooth continuous curves.

Let us graph a cubic polynomial defined by P(x) = 2x³ - 4x² - 2x + 4. Here, our degree is 3, with coefficients A3 = 2, A2 = -4, A1 = -2, and A0 = 4. Suppose we set our evaluation interval from IntStart = -3 to IntEnd = 3. First, the calculator determines absolute values, such as a_{3_abs} = 2 and a_{2_abs} = 4. Next, it evaluates the polynomial at key points: at x = -3, P(-3) = 2(-27) - 4(9) - 2(-3) + 4 = -80. At x = 1, P(1) = 2(1) - 4(1) - 2(1) + 4 = 0, confirming a root at x = 1. By plotting multiple calculated coordinate pairs across the interval, the calculator generates the complete continuous curve of the function.

When analyzing polynomials, always verify your leading coefficient and degree first, as they strictly dictate the global end behavior. Choose an interval that captures all local extrema and real roots; if a graph looks cut off, widen your IntStart and IntEnd boundaries. Additionally, tracking absolute values of coefficients can help you anticipate the steepness and vertical stretch of the resulting curve.