Vertex Form Calculator

A quadratic function can be expressed in two primary ways: standard form as f(x) = ax² + bx + c, and vertex form as f(x) = a(x - h)² + k. In these expressions, the point (h, k) designates the exact coordinates of the parabola's vertex. To convert a standard equation into vertex form, the horizontal coordinate of the vertex is computed using h = -b / (2a), while the vertical coordinate is evaluated via k = (4ac - b²) / (4a). Conversely, expanding a vertex form equation yields standard coefficients through b = -2ah and c = ah² + k. Roots, or x-intercepts, are subsequently determined by solving for when f(x) equals zero.

Let us convert a standard quadratic equation into vertex form. Consider the function f(x) = 2x² - 8x + 6, where a = 2, b = -8, and c = 6. First, find the horizontal coordinate of the vertex: h = -(-8) / (2 * 2) = 8 / 4 = 2. Next, calculate the vertical coordinate: k = (4 * 2 * 6 - (-8)²) / (4 * 2) = (48 - 64) / 8 = -16 / 8 = -2. Substituting these values into the vertex form f(x) = a(x - h)² + k, we obtain f(x) = 2(x - 2)² - 2. Finally, to find the roots, set the expression to zero and solve for x, yielding x-intercepts at x = 1 and x = 3.

Always verify that your leading coefficient 'a' remains consistent when converting between standard and vertex forms, as it dictates both the vertical stretch and the opening direction of the parabola. If 'a' is positive, the vertex represents a minimum point; if negative, it represents a maximum point. Double-check your arithmetic for negative signs when calculating h, as forgetting to distribute a negative sign is the most frequent source of calculation errors.