A 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).
In Spanish
parábola
Also called
U-shaped curve, graph of a quadratic, parabolic curve
The focus-directrix rule: from P(3, 2.25), the distance to the focus and the distance to the directrix are both 3.25.
Formal definition and notation
A parabola is the set of all points in a plane that are equidistant from a fixed point, the focus, and a fixed line not through it, the directrix. A parabola whose axis is vertical is the graph of y = ax2 + bx + c with a ≠ 0; with vertex at the origin it is y = 14px2, where the focus is (0, p) and the directrix is y = −p.
Symbol
Read it as
Meaning
y = ax2 + bx + c
y equals a x squared plus b x plus c
a parabola that opens up (a > 0) or down (a < 0)
x2 = 4py
x squared equals 4 p y
vertex at the origin, focus (0, p), directrix y = −p
x = ay2
x equals a y squared
a parabola opening sideways (not a function of x)
Key facts
Every quadratic function graphs as a parabola, and every parabola that opens up or down is the graph of a quadratic function.
A parabola is symmetric: points at equal distances left and right of the axis have the same height.
The vertex lies exactly halfway between the focus and the directrix.
Satellite dishes and car headlights are parabolic because rays parallel to the axis reflect through the focus.
The path of a thrown ball, ignoring air resistance, is a parabola.
Parts of a parabola
Parts of a parabola
For y = ax2 + bx + c: axis x = −b2a
vertex
the turning point on the axis; the lowest or highest point
axis of symmetry
the line through the vertex that splits the parabola into mirror halves
focus
the point inside the curve, |p| units from the vertex along the axis
directrix
the line outside the curve, |p| units from the vertex on the other side
In y = 14px2, a small p makes a narrow parabola and a large p makes a wide one.
Parabola examples, step by step
Example 1: Which way does it open?
Warm-up
Does the parabola y = −3x2 + 2x + 7 open up or down, and where does it cross the y-axis?
The coefficient of x2 is −3, which is negative, so it opens down.
At x = 0, y = 7, so it crosses the y-axis at (0, 7).
Answer: It opens down and crosses the y-axis at (0, 7).
Example 2: Use symmetry to find a point
Core
A parabola has vertex (1, −4) and passes through (4, 5). Name another point on it.
The axis of symmetry is the vertical line through the vertex: x = 1.
The point (4, 5) is 4 − 1 = 3 units right of the axis.
Its mirror image is 3 units left: x = 1 − 3 = −2, at the same height 5.
Answer: The point (-2, 5) is also on the parabola.
Example 3: Focus of a satellite dish
Test-style
A dish’s cross-section is y = 116x2, in inches. Where should the receiver go so it sits at the focus?
Match y = 14px2: 14p = 116.
So 4p = 16 and p = 4.
The focus is (0, p) = (0, 4).
Answer: The receiver goes at (0, 4), 4 inches above the vertex.
Common mistakes
✗ MistakeEvery U-shaped curve is a parabola, including a semicircle or a catenary (hanging chain).
✓ InsteadA parabola must fit y = ax2 + bx + c (or the focus-directrix rule). A semicircle is part of x2 + y2 = r2, and its sides rise much more steeply.
Shape alone can fool you. Check the equation or the distances.
✗ MistakeThe parabola x = y2 is the graph of a function y = f(x).
✓ InsteadIt opens sideways: x = 4 gives y = 2 and y = −2, so it fails the vertical line test.
Only parabolas that open up or down are functions of x.
✗ MistakeFor y = 18x2, the focus is (0, 8).
✓ Instead4p = 8, so p = 2 and the focus is (0, 2).
The quadratic function is the rule, such as f(x) = x2 − 1. The parabola is the shape it draws on a graph.
Parabolavs.Hyperbola
A hyperbola has two separate branches, as in xy = 1. A parabola is one connected curve with one vertex.
Quick check: 3 questions
Question 1 of 3
What is the vertex of the parabola y = (x + 2)2 − 5?
Show answer and explanation
Answer: C. In vertex form a(x − h)2 + k, x + 2 means h = −2, and k = −5. (2, −5) misreads the inside sign, (−2, 5) flips the outside sign, and (5, −2) swaps the coordinates.
Question 2 of 3
A parabola has vertex (3, 1) and passes through (0, 10). Which point must also be on it?
Show answer and explanation
Answer: A. The axis is x = 3. (0, 10) is 3 units left, so its mirror is 3 units right at (6, 10). (3, 10) is on the axis, (0, −10) reflects over the x-axis, and (−3, 10) reflects over the y-axis instead.
Question 3 of 3
Which equation describes a parabola that opens down?
Show answer and explanation
Answer: C.6x − x2 has x2 coefficient −1, so it opens down. The other three have a positive x2 coefficient; the negative constant or x-term does not change the direction.
Parabola: frequently asked questions
What is the difference between a parabola and a quadratic?
A quadratic is an expression, equation or function with an x2 term, such as y = x2 + 2x. A parabola is a curve, the shape you get when you graph a quadratic function. Parabolas also exist as geometric objects defined by a focus and a directrix, including ones that open sideways, which are not quadratic functions of x.
How do you find the vertex of a parabola?
For y = ax2 + bx + c, the vertex has x-coordinate −b2a; substitute it to get the y-coordinate. For vertex form y = a(x − h)2 + k, read the vertex directly as (h, k). On a graph, it is the lowest point of a parabola that opens up or the highest point of one that opens down.
Where do parabolas appear in real life?
The path of a ball, a jet of water or any object thrown near Earth’s surface is close to a parabola. Satellite dishes, solar cookers and flashlight reflectors use parabolic shapes because they focus parallel rays to a single point, the focus. Some bridge arches are also built as parabolas.
VertexA 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 6OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8Quadratic 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, PrecalculusHeightHeight 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 6Y-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 7CoefficientA 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 8Axis 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 2InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4SideA 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 2Vertical Line TestThe vertical line test is a way to tell from a graph whether it shows a function: if any vertical line crosses the graph more than once, it is not a function. If every vertical line crosses it at most once, each input x has only one output y, so the graph is a function.Functions · Grade 8, Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5SubstituteSubstitute 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 8GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7Quadratic 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 2Vertex 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 2Equation of a CircleThe equation of a circle with center (h, k) and radius r is (x − h)2 + (y − k)2 = r2. It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x2 + y2 = r2.Geometry · Geometry, Algebra 2, PrecalculusX-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 2