Vertex Form Calculator
Find the vertex of any quadratic function in standard form ax² + bx + c. Use this when graphing a parabola or solving optimization problems that require the minimum or maximum value.
About this calculator
A quadratic function f(x) = ax² + bx + c forms a parabola whose peak or trough is called the vertex. The x-coordinate of the vertex is found using the formula x = −b / (2a). Once x is known, substitute it back into the original equation to get the y-coordinate: y = a(x_v)² + b(x_v) + c. The vertex form of the parabola is then f(x) = a(x − x_v)² + y_v. If a > 0 the parabola opens upward and the vertex is a minimum; if a < 0 it opens downward and the vertex is a maximum. This is essential in physics (projectile height), economics (profit maximization), and engineering (structural curves).
How to use
Take the quadratic f(x) = 2x² − 8x + 3. Enter a = 2, b = −8, c = 3. Compute the x-coordinate of the vertex: x = −(−8) / (2 × 2) = 8 / 4 = 2. Now find y by substituting x = 2: y = 2(2²) − 8(2) + 3 = 8 − 16 + 3 = −5. The vertex is at (2, −5), and the vertex form is f(x) = 2(x − 2)² − 5. Because a = 2 > 0, this is a minimum point.
Frequently asked questions
What is the difference between standard form and vertex form of a quadratic equation?
Standard form is written as f(x) = ax² + bx + c, which makes it easy to identify coefficients and the y-intercept (c). Vertex form is written as f(x) = a(x − h)² + k, where (h, k) is the vertex of the parabola. Converting between the two forms is useful depending on what information you need: standard form suits finding roots via the quadratic formula, while vertex form instantly reveals the maximum or minimum value of the function.
How do I find the maximum or minimum value of a quadratic function using the vertex?
The vertex y-coordinate gives the maximum or minimum value of the function. If the coefficient a is positive, the parabola opens upward and the vertex y-value is the minimum. If a is negative, the parabola opens downward and the vertex y-value is the maximum. Simply compute x_v = −b / (2a), then evaluate f(x_v) to get that extreme value. This technique is widely used in profit maximization, projectile motion, and any scenario modeled by a quadratic.
Why does the vertex formula use −b divided by 2a to find the x-coordinate?
The formula x = −b / (2a) comes from completing the square on the standard form ax² + bx + c, or equivalently from setting the derivative f′(x) = 2ax + b equal to zero and solving for x. Both approaches yield the same result. It represents the axis of symmetry of the parabola — the vertical line that splits it into two mirror-image halves. The vertex always lies on this axis, so its x-coordinate equals −b / (2a).