Quadratic Vertex CalculatorFree online math calculator — instant results, no sign-up
Calculate the vertex of a quadratic function.
About the Quadratic Vertex Calculator
The Quadratic Vertex Calculator finds the turning point of a quadratic function written as y = ax² + bx + c. Enter the quadratic, linear, and constant coefficients to calculate the vertex’s x- and y-coordinates. With the default coefficients 1, -4, and 3, the vertex is (2, -1).
- Quadratic coefficient
- The coefficient a multiplies x² and defaults to 1; it is unitless in pure algebra or has units of y per x-unit squared in an applied model.
- Linear coefficient
- The coefficient b multiplies x and defaults to -4; it is unitless in pure algebra or has units of y per x-unit in an applied model.
- Constant
- The constant c is the value of y when x = 0 and defaults to 3; it is unitless in pure algebra or has the same units as y in an applied model.
Formula and method
For y = ax² + bx + c with a ≠ 0, the vertex is (h, k), where h = -b/(2a) and k = ah² + bh + c. With a = 1, b = -4, and c = 3: h = -(-4)/(2 × 1) = 2; k = 1 × 2² + (-4) × 2 + 3 = -1.
- xThe independent variable, representing the horizontal coordinate.
- yThe function value, representing the vertical coordinate.
- aThe nonzero quadratic coefficient, which determines the parabola’s opening direction and width.
- bThe linear coefficient, which combines with a to determine the vertex’s horizontal position.
- cThe constant term, equal to the y-intercept.
- hThe vertex’s x-coordinate, displayed as Vertex x.
- kThe vertex’s y-coordinate, displayed as Vertex y.
The calculation first finds the parabola’s axis of symmetry using h = -b/(2a). It then substitutes h into the original quadratic to calculate k = ah² + bh + c. These values give the vertex (h, k); no root-finding is needed.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Quadratic coefficient1
- Linear coefficient-4
- Constant3
Results
- Vertex x2
- Vertex y-1
How to use the Quadratic Vertex Calculator
- 1Write your quadratic in the form y = ax² + bx + c and identify each coefficient, including its sign.
- 2Enter the Quadratic coefficient a; for the worked example, keep the default value 1.
- 3Enter the Linear coefficient b as -4 and the Constant c as 3 for the worked example.
- 4Calculate the vertex using the entered coefficients.
- 5Read Vertex x = 2 and Vertex y = -1 together as the point (2, -1).
Reading your result
The Vertex x row gives the horizontal location of the turning point and the axis of symmetry: x = 2 for the default inputs. The Vertex y row gives the function value at that location: -1. Because the default quadratic coefficient is positive, (2, -1) is the parabola’s minimum point. A negative quadratic coefficient would make the vertex a maximum instead.
Limitations
The formulas require a nonzero quadratic coefficient; when a = 0, the function is not quadratic and has no unique parabolic vertex. The calculation assumes real, finite coefficients in standard form and consistent units if the function models a physical quantity. Decimal arithmetic or display rounding may affect the final digits. The vertex is the minimum or maximum over all real x; if your model restricts x to an interval, the vertex may lie outside that interval.
Common questions
Why can’t the Quadratic coefficient be zero?+
The vertex formula divides by 2a, so a = 0 makes it undefined. It also removes the x² term, leaving a linear or constant function rather than a parabola.
Is the default vertex a minimum or a maximum?+
It is a minimum because a = 1 is positive, so the parabola opens upward. For y = x² - 4x + 3, the minimum value is -1 at x = 2.
How do the vertex outputs relate to vertex form?+
The outputs h and k let you write the same function as y = a(x - h)² + k. With the default inputs and outputs, this becomes y = (x - 2)² - 1.
Does changing only the Constant move Vertex x?+
No. Vertex x depends only on a and b through -b/(2a). Changing c shifts Vertex y by the same amount without changing the horizontal location of the vertex.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.