Vertex Form Calculator

Vertex form is a useful way to write a quadratic function because it reveals the parabola's turning point immediately. The form is y = a(x - h) squared + k, where the vertex is the point (h, k). The coefficient a controls the direction and width of the parabola. A positive a opens upward, a negative a opens downward, and a larger absolute value makes the graph narrower. Standard form writes the same quadratic as y = ax squared + bx + c. In this form, c is the y-intercept, but the vertex is not immediately visible. To convert standard form to vertex form, calculate h as negative b divided by 2a. Then evaluate the function at h to find k, or use k = c - b squared divided by 4a. These formulas are equivalent to completing the square. To convert from vertex form back to standard form, expand the square. Multiplying a by (x - h) squared gives ax squared minus 2ahx plus ah squared. Adding k means the standard coefficients are b = -2ah and c = ah squared + k. The leading coefficient a stays unchanged during either conversion because both expressions describe exactly the same parabola. The vertex is a minimum when a is positive and a maximum when a is negative. The vertical line x = h is the axis of symmetry. Points the same horizontal distance to either side of that line have equal y-values. This symmetry makes vertex form especially useful for graphing, optimization, projectile motion, revenue models, and any problem asking for a maximum or minimum. Coefficient a cannot be zero because that would remove the squared term and produce a linear equation rather than a quadratic. The other coefficients and vertex coordinates may be positive, negative, zero, or decimal values. When reading vertex form, pay attention to the subtraction sign: an expression such as (x - 3) has h = 3, while (x + 3) has h = -3. This calculator performs the algebra numerically and displays the converted equation together with the vertex. Results are rounded to six decimal places when needed. The output keeps all terms visible, including coefficients of one and zero, to make the relationship between entered values and calculated coefficients explicit. You can use it to check completing-the-square work, prepare a graph, or move between equation forms required by an algebra problem.