The vertex of a parabola is its turning point: the lowest point when the parabola opens up, or the highest point when it opens down. For y = ax2 + bx + c, the vertex has x-coordinate x = −b2a; substitute that value to get the y-coordinate. The vertex sits on the axis of symmetry.
In Spanish
vértice de una parábola
Also called
turning point, vertex of a quadratic, maximum or minimum point, extreme point
The green parabola turns at its minimum (3, −4), exactly halfway between its roots 1 and 5. The blue one opens down, so its vertex (1, 4) is a maximum.
Formal definition and notation
For the quadratic functionf(x) = ax2 + bx + c with a ≠ 0, the vertex is the point ( −b2a, f( −b2a ) ). It is the only point where the graph changes from decreasing to increasing (if a > 0, a minimum) or from increasing to decreasing (if a < 0, a maximum). The vertical line through it, x = −b2a, is the parabola‘s axis of symmetry.
Symbol
Read it as
Meaning
(h, k)
h, k
the usual name for the vertex coordinates
−b2a
negative b over 2 a
the x-coordinate of the vertex
f( −b2a )
f of negative b over 2 a
the y-coordinate: the maximum or minimum value
Key facts
a > 0: the parabola opens up and the vertex is a minimum. a < 0: it opens down and the vertex is a maximum.
The vertex lies on the axis of symmetry x = −b2a; points at equal distances left and right of it have equal y-values.
The x-coordinate of the vertex is the average of the two x-intercepts: roots 1 and 5 give x = 3.
A parabola has exactly one vertex. It is the range boundary: for y = (x − 1)2 + 2 the range is y ≥ 2.
In word problems the vertex answers “maximum height“, “minimum cost” or “greatest area” questions.
In calculus, the vertex is where the derivative equals 0.
Vertex formula
Vertex formula
h = −b2a, k = f(h)
a, b
the coefficients of x2 and x in y = ax2 + bx + c
h
the x-coordinate of the vertex
k
the y-coordinate, found by substituting h
In vertex form y = a(x − h)2 + k the vertex (h, k) can be read directly. If you know the x-intercepts, h is their average.
The y-intercept is where the graph crosses the y-axis, (0, c). It is the vertex only when b = 0, as in y = x2 + 3.
Vertex of a Parabolavs.Focus of a parabola
The focus is a point inside the curve, not on it. The vertex lies on the parabola, halfway between the focus and the directrix.
Quick check: 3 questions
Question 1 of 3
What is the vertex of y = x2 + 4x − 1?
Show answer and explanation
Answer: B.x = −42 = −2 and y = 4 − 8 − 1 = −5. (2, 11) drops the negative sign, (−2, −1) uses the constant as y, and (4, −1) uses b and c.
Question 2 of 3
The vertex of y = 3 − x2 is
Show answer and explanation
Answer: D. The coefficient of x2 is −1, so the parabola opens down and the vertex is a maximum. With b = 0, x = 0 and y = 3. (3, 0) swaps the coordinates.
Question 3 of 3
A parabola crosses the x-axis at x = −1 and x = 7. What is the x-coordinate of its vertex?
Show answer and explanation
Answer: C. The vertex is on the axis of symmetry, halfway between the roots: −1 + 72 = 3. 4 is half of 8, the distance between them; 6 adds them; 8 is the distance.
Vertex of a Parabola: frequently asked questions
How do you find the vertex of a parabola?
From y = ax2 + bx + c, compute x = −b2a, then substitute it into the equation to get y. From vertex formy = a(x − h)2 + k, just read (h, k). From a graph, find the turning point. From two x-intercepts, average them to get x, then evaluate.
Is the vertex always the maximum or the minimum?
Yes. Every parabola that is the graph of a quadratic function has exactly one vertex, and it is either the lowest point, when a > 0, or the highest point, when a < 0. The y-coordinate of the vertex is the minimum or maximum value of the function, and it sets the boundary of the range.
What is the axis of symmetry and how is it related to the vertex?
The axis of symmetry is the vertical line that splits the parabola into two mirror halves. It always passes through the vertex, so its equation is x = h, the x-coordinate of the vertex. For y = x2 − 6x + 5, the vertex is (3, −4) and the axis of symmetry is x = 3.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.C.8.a, CCSS HSF-IF.C.7.a
Cite this entry
Effortless Math Team. “Vertex of a Parabola.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/vertex-of-a-parabola/
Flashcard
TermVertex of a Parabolax = −b2a
DefinitionThe turning point of a parabola: its lowest point if it opens up, highest if it opens down. For y = x2 − 6x + 5, the vertex is (3, −4).
Quadratic FunctionA quadratic function is a function that can be written f(x) = ax2 + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola, which opens up when a > 0 and down when a < 0. Its highest or lowest point is the vertex.Functions · Algebra 1, Algebra 2, PrecalculusParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryAxis of SymmetryThe axis of symmetry of a parabola is the vertical line through its vertex that splits it into two mirror-image halves. For y = ax2 + bx + c, it is the line x = −b2a. Points at equal distances on either side of this line have the same y-value.Functions · Algebra 1, Algebra 2AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7X-InterceptAn x-intercept is a point where a graph crosses or touches the x-axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Algebra · Grade 8, Algebra 1, Algebra 2RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6DerivativeA derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · CalculusSubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1Vertex FormVertex form is a way to write a quadratic function as y = a(x − h)2 + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)2 + 1 has its vertex at (3, 1).Algebra · Algebra 1, Algebra 2Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2Domain and RangeDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Functions · Grade 8, Algebra 1, Algebra 2Critical PointA critical point of a function is an input x = c in its domain where the derivative is 0 or does not exist. These are the only places a smooth graph can turn around, so every local maximum or minimum happens at a critical point. For f(x) = x2 − 4x, f′(x) = 2x − 4 = 0 at x = 2.Calculus · Calculus