Graph of y = x² + 2x + 1
Interactive graph of y = x² + 2x + 1 (perfect square trinomial). Explore this shifted parabola, its vertex form, and factored form.
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Understanding the Function
The function y = x² + 2x + 1 is a perfect square trinomial that factors as y = (x + 1)². This means the parabola has the same shape as y = x² but is shifted one unit to the left, placing its vertex at (-1, 0). The vertex sits exactly on the x-axis, making x = -1 a double root.
Completing the square transforms the standard form ax² + bx + c into vertex form a(x - h)² + k, revealing the vertex at (h, k). For this equation, the process yields (x + 1)² + 0, confirming the vertex at (-1, 0). This technique is essential for graphing any quadratic and for deriving the quadratic formula.
Key properties: vertex at (-1, 0), axis of symmetry at x = -1, y-intercept at (0, 1), opens upward, minimum value of 0. The discriminant b² - 4ac = 4 - 4 = 0 confirms exactly one real root (a repeated root at x = -1).