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Polynomial Graph Plotter

Understanding Polynomial Graphs

Polynomial graphs reveal important characteristics about the function's behavior. The shape of the graph depends on the polynomial's degree and coefficients. Here's how to interpret and plot them:

  1. Identify Key Features

    Look for roots (x-intercepts), y-intercepts, turning points, and end behavior. The degree determines the maximum number of roots and turning points.

  2. Determine End Behavior

    The leading term (highest degree) dictates how the graph behaves as x approaches ±∞. Odd-degree polynomials have opposite end behaviors, while even-degree polynomials have the same behavior at both ends.

  3. Find Roots and Multiplicity

    Roots are where the polynomial equals zero. The multiplicity of a root affects how the graph behaves at that point (crosses or touches the x-axis).

  4. Locate Turning Points

    A polynomial of degree n has at most n-1 turning points. These are where the graph changes direction from increasing to decreasing or vice versa.

  5. Plot Key Points

    Calculate and plot the roots, y-intercept, and a few additional points to determine the graph's shape between key features.

Example: Cubic Polynomial (x³ - 2x² - x + 2)

Key Features: - Roots: x = -1, x = 1, x = 2 (found by factoring) - y-intercept: (0, 2) - Turning points: Approximately at x ≈ -0.33 and x ≈ 1.67 - End behavior: Falls to left, rises to right (odd degree, positive leading coefficient) Graph Behavior: - Crosses x-axis at all three roots (multiplicity 1 for each) - Changes direction at two turning points - Approaches -∞ as x → -∞ and +∞ as x → +∞