A 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.
In Spanish
función cuadrática
Also called
quadratic, second-degree polynomial function, degree 2 function
One quadratic function, five key features: direction, vertex, axis of symmetry, zeros and y-intercept.
Formal definition and notation
A quadratic function is a polynomial function of degree 2, f(x) = ax2 + bx + c, where a, b, c are real numbers and a ≠ 0. Its domain is all real numbers. Its graph is a parabola with axis of symmetryx = −b2a and vertex(−b2a, f(−b2a)); the vertex gives the minimum value if a > 0 and the maximum value if a < 0.
The sign of a sets the direction: up (a smile, minimum) if a > 0, down (a frown, maximum) if a < 0.
The size of a sets the width: |a| > 1 is narrower than y = x2, |a| < 1 is wider.
f(0) = c, so the graph crosses the y-axis at (0, c).
A quadratic function has 0, 1 or 2 real zeros, the x-intercepts of its parabola.
In a quadratic function, the second differences in a table with equal x-steps are constant (the first differences are not).
Vertex of a quadratic function
Vertex of a quadratic function
x = −b2a, then y = f(−b2a)
a, b
the coefficients of x2 and x in standard form
x = −b2a
the axis of symmetry and the x-coordinate of the vertex
f(−b2a)
the minimum or maximum value of the function
For f(x) = x2 − 4x + 3: x = 42 = 2 and f(2) = −1, so the vertex is (2, −1).
Try it: table of values
Interactive
This machine computes f(x) = (x − 2)2 − 1, which equals x2 − 4x + 3. Run inputs 0 to 4 and watch the outputs fall to −1 and rise again, mirrored around x = 2.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Quadratic Function examples, step by step
Example 1: Read the features
Warm-up
For f(x) = −2x2 + 8, does the graph open up or down, and what is its y-intercept?
a = −2 < 0, so the parabola opens down.
c = 8, so f(0) = 8: the y-intercept is (0, 8).
With b = 0, the vertex is also (0, 8), the maximum.
Answer: It opens down, and the y-intercept is (0, 8).
Example 2: Find the vertex
Core
Find the vertex of g(x) = 3x2 + 12x + 5 and say whether it is a maximum or a minimum.
a = 3, b = 12: x = −122(3) = −2.
g(−2) = 3(4) + 12(−2) + 5 = 12 − 24 + 5 = −7.
a > 0, so the parabola opens up and the vertex is the lowest point.
Answer: The vertex is (-2, -7), a minimum.
Example 3: Maximum in context
Test-style
A store’s daily profit is P(x) = −5x2 + 200x − 1500 dollars when it sells an item for x dollars. What price gives the greatest profit, and what is that profit?
a = −5 < 0, so the vertex is a maximum.
Price: x = −2002(−5) = 20 dollars.
Profit: P(20) = −5(400) + 4000 − 1500 = 500.
Answer: A price of 20 dollars gives the maximum profit of 500 dollars.
Common mistakes
✗ MistakeFor f(x) = x2 + 6x + 1, the axis is x = 62 = 3.
✓ Insteadx = −b2a = −62 = −3.
The formula has a minus sign. Positive b with positive a puts the vertex left of the y-axis.
✗ Mistakef(x) = 4 − x2 opens up because the 4 is positive.
✓ InsteadRewrite as −x2 + 4: a = −1, so it opens down.
Only the coefficient of x2 decides the direction, wherever it is written.
✗ MistakeThe minimum of f(x) = (x − 5)2 + 2 is at x = 2.
✓ InsteadThe vertex is (5, 2): the minimum value 2 happens at x = 5.
In vertex form, h (inside, with the opposite sign) is the x-value and k is the minimum or maximum value.
A quadratic function is a rule with a whole graph: f(x) = x2 − 4x + 3. A quadratic equation asks when it equals a number: x2 − 4x + 3 = 0 has solutions 1 and 3.
A linear function has degree 1 and a straight-line graph with constant rate of change. A quadratic has an x2 term, a curved graph, and a rate of change that keeps changing.
Quadratic Functionvs.Exponential function
In x2 the variable is the base; in 2x it is the exponent. Exponentials grow much faster and have no vertex.
Quick check: 3 questions
Question 1 of 3
Which is a quadratic function?
Show answer and explanation
Answer: B.7 − x2 has degree 2 with a = −1 ≠ 0. 2x + 7 is linear, 2x is exponential, and x3 − x is cubic.
Question 2 of 3
What is the axis of symmetry of f(x) = 2x2 − 8x + 1?
Show answer and explanation
Answer: C.x = −−82(2) = 84 = 2. x = −2 dropped a negative sign, x = 4 divided by a instead of 2a, and x = 8 used −b alone.
Question 3 of 3
A quadratic function has vertex (3, 10) and opens down. Which statement is true?
Show answer and explanation
Answer: D. Opening down makes the vertex the highest point, so the maximum value is its y-coordinate, 10. 10 is not a minimum, 3 is where the maximum occurs, and a downward parabola that starts at height 10 must cross the x-axis twice.
Quadratic Function: frequently asked questions
How do you know if a function is quadratic?
Simplify it and look at the highest power of x. If it is exactly 2 and the x2 coefficient is not 0, the function is quadratic. From a table with equal steps in x, the first differences of the outputs change, but the second differences are constant and not zero. Its graph is always a parabola.
What are the three forms of a quadratic function?
Standard form ax2 + bx + c shows the y-intercept c. Vertex form a(x − h)2 + k shows the vertex (h, k). Factored form a(x − r)(x − s) shows the zeros r and s. All three describe the same function, and the value of a is the same in each.
What is the difference between a quadratic function and a parabola?
The quadratic function is the rule, such as f(x) = x2 − 4x + 3. The parabola is the shape of its graph. Every quadratic function graphs as a parabola that opens up or down, but some parabolas, such as x = y2, open sideways and are not graphs of functions of x.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.C.7.a, CCSS HSF-IF.C.8.a
Cite this entry
Effortless Math Team. “Quadratic Function.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/quadratic-function/
Flashcard
TermQuadratic Functionf(x) = ax2 + bx + c
DefinitionA degree-2 function with a ≠ 0. Its graph is a parabola with a vertex. x2 − 4x + 3 has vertex (2, −1).
PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2ParabolaA 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 2VertexA 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 6Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4Y-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 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 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 2DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Quadratic 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 2Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Standard Form of a QuadraticThe standard form of a quadratic is ax2 + bx + c, with the terms in descending powers of x and a ≠ 0. As an equation it is ax2 + bx + c = 0; as a function, y = ax2 + bx + c. It shows a, b and c directly, which the quadratic formula needs, and c is the y-intercept.Algebra · Algebra 1, Algebra 2