Parabola

Q: What is the difference between a parabola and a hyperbola?

A parabola is a single U-shaped curve produced by slicing a cone parallel to its side; it has one branch and one focus. A hyperbola consists of two separate branches and has two foci. Algebraically, a parabola comes from a quadratic equation in one variable, while a hyperbola involves two squared variables with opposite signs, such as $\tfrac{x^2}{a^2} - \tfrac{y^2}{b^2} = 1$.

Q: How do you find the focus of a parabola from its equation?

First convert the equation to vertex form $y = a(x - h)^2 + k$. The focus lies on the axis of symmetry at a distance $\tfrac{1}{4a}$ from the vertex. For an upward- or downward-opening parabola, the focus is at $(h,\, k + \tfrac{1}{4a})$. If $a > 0$ the focus is above the vertex; if $a < 0$ it is below.

Q: Is every U-shaped curve a parabola?

No. A curve is a parabola only if every point on it is equidistant from a fixed focus and a fixed directrix, which produces a quadratic equation. Many U-shaped curves — such as catenaries (the shape of a hanging chain) or the curves $y = x^4$ — look similar but are not parabolas.

Parabola

A u-shaped curve with certain specific properties. Formally, a parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

Note: It is a common error to call any u-shaped curve a parabola. A parabola must satisfy the conditions listed above, and a parabola always has a quadratic equation.

Two parabola graphs: vertical (vertex (h,k), y=a(x-h)²+k) and horizontal (vertex (h,k), x=a(y-k)²+h), with orientation notes...

Diagram of a parabola with vertex (h,k), focus point p above vertex. Formulas: vertical 4p(y-k)=(x-h)², horizontal 4p(x-h)=(y-k)².

Parabola diagram showing point P on curve, focus inside curve, directrix below. L1 (to focus) = L2 (to directrix).

Example:

Graph of upward-opening parabola with vertex at (3,-2), focus above vertex, vertical axis of symmetry at x=3, and horizontal...

This is a graph of the parabola with all its major features labeled: axis of symmetry, focus, vertex, and directrix.

Key Formula

y = a(x - h)^2 + k

Where:

$a$ = Controls how wide or narrow the parabola is and whether it opens upward (a > 0) or downward (a < 0)
$h$ = The x-coordinate of the vertex
$k$ = The y-coordinate of the vertex
$(h, k)$ = The vertex — the lowest point (if a > 0) or highest point (if a < 0) on the parabola

Worked Example

Problem: Find the vertex, axis of symmetry, focus, and directrix of the parabola

y = 2x^2 - 8x + 6

Step 1: Rewrite in vertex form by completing the square. Factor the leading coefficient from the first two terms.

y = 2(x^2 - 4x) + 6

Step 2: Complete the square inside the parentheses. Half of

-4

-2

, and

(-2)^2 = 4

. Add and subtract 4 inside.

y = 2(x^2 - 4x + 4 - 4) + 6 = 2(x - 2)^2 - 8 + 6 = 2(x - 2)^2 - 2

Step 3: Read off the vertex from the vertex form

y = 2(x - 2)^2 - 2

\text{Vertex} = (h, k) = (2, -2)

Step 4: The axis of symmetry is the vertical line through the vertex.

x = 2

Step 5: For a vertical parabola in the form

y = a(x - h)^2 + k

, the focus is at

(h,\, k + \tfrac{1}{4a})

and the directrix is

y = k - \tfrac{1}{4a}

. Here

a = 2

, so

\tfrac{1}{4a} = \tfrac{1}{8}

\text{Focus} = \left(2,\, -2 + \tfrac{1}{8}\right) = \left(2,\, -\tfrac{15}{8}\right), \qquad \text{Directrix: } y = -2 - \tfrac{1}{8} = -\tfrac{17}{8}

Answer: Vertex

(2, -2)

, axis of symmetry

x = 2

, focus

(2, -\tfrac{15}{8})

, directrix

y = -\tfrac{17}{8}

Another Example

This example starts from the geometric definition (focus and directrix) instead of an algebraic equation, showing how to build the equation from the parabola's defining features.

Problem: A parabola has its focus at

(0, 3)

and its directrix at

y = -3

. Write its equation.

Step 1: The vertex is the midpoint between the focus and the directrix. The focus is at

y = 3

and the directrix is

y = -3

, so the vertex is at

y = 0

\text{Vertex} = \left(0,\, \frac{3 + (-3)}{2}\right) = (0, 0)

Step 2: The distance from the vertex to the focus is

p = 3

. Since the focus is above the vertex, the parabola opens upward.

p = 3

Step 3: Use the standard form

x^2 = 4py

for a parabola with vertex at the origin opening upward.

x^2 = 4(3)y = 12y

Step 4: Solve for

y

to write it in the more familiar form.

y = \frac{x^2}{12}

Answer: The equation is

x^2 = 12y

, or equivalently

y = \dfrac{x^2}{12}

Frequently Asked Questions

What is the difference between a parabola and a hyperbola?

A parabola is a single U-shaped curve produced by slicing a cone parallel to its side; it has one branch and one focus. A hyperbola consists of two separate branches and has two foci. Algebraically, a parabola comes from a quadratic equation in one variable, while a hyperbola involves two squared variables with opposite signs, such as

\tfrac{x^2}{a^2} - \tfrac{y^2}{b^2} = 1

How do you find the focus of a parabola from its equation?

First convert the equation to vertex form

y = a(x - h)^2 + k

. The focus lies on the axis of symmetry at a distance

\tfrac{1}{4a}

from the vertex. For an upward- or downward-opening parabola, the focus is at

(h,\, k + \tfrac{1}{4a})

. If

a > 0

the focus is above the vertex; if

a < 0

it is below.

Is every U-shaped curve a parabola?

No. A curve is a parabola only if every point on it is equidistant from a fixed focus and a fixed directrix, which produces a quadratic equation. Many U-shaped curves — such as catenaries (the shape of a hanging chain) or the curves

y = x^4

— look similar but are not parabolas.

Parabola vs. Circle

	Parabola	Circle
Definition	Set of points equidistant from a focus and a directrix	Set of points equidistant from a single center point
Equation form	$y = ax^2 + bx + c$ (one squared variable)	$(x-h)^2 + (y-k)^2 = r^2$ (two squared variables, same coefficient)
Conic section type	Eccentricity $e = 1$ ; slice parallel to cone's side	Eccentricity $e = 0$ ; slice perpendicular to cone's axis
Shape	Open curve extending to infinity	Closed curve
Key features	Vertex, focus, directrix, axis of symmetry	Center, radius

Why It Matters

Parabolas appear throughout algebra, precalculus, and physics. You use them when graphing quadratic functions, solving projectile motion problems (the path of a thrown ball is a parabola), and analyzing satellite dishes or headlight reflectors that rely on the parabola's reflective property. Mastering the vertex form and focus–directrix relationship is essential for standardized tests and for understanding conic sections in later courses.

Common Mistakes

Mistake: Calling any U-shaped curve a parabola.

Correction: A true parabola must satisfy the focus–directrix distance condition and have a quadratic equation. Curves like

y = x^4

y = |x|

, or a catenary are U-shaped but are not parabolas.

Mistake: Confusing the sign of

a

with the direction the parabola opens.

Correction: When

a > 0

the parabola opens upward (vertex is a minimum). When

a < 0

it opens downward (vertex is a maximum). Students sometimes reverse this or forget it applies only to parabolas written with

y

isolated.

Related Terms

Vertex of a Parabola — The turning point of a parabola
Focus of a Parabola — Fixed point defining the parabola
Directrix of a Parabola — Fixed line defining the parabola
Axis of Symmetry of a Parabola — Line through vertex and focus
Quadratic Equation — Algebraic equation whose graph is a parabola
Locus — Set of points satisfying a geometric condition
Area of a Parabolic Segment — Area enclosed between a parabola and a chord
Focal Radius — Distance from a point on the parabola to the focus