Difficulty: Intermediate | Prerequisites: Familiarity with quadratic equations, coordinate geometry, and the concept of a function's graph. You should be comfortable with completing the square.
Parabolas turn up everywhere in maths, from modelling projectile trajectories to describing the shape of satellite dishes. Until now you have probably treated them as "U-shaped graphs" of quadratic functions. This topic reframes the parabola geometrically: every point on the curve is the same distance from a fixed point (the focus) and a fixed line (the directrix). Understanding this definition lets you derive the equation of any parabola, identify its key features from an equation, and move between algebraic and geometric representations, which is exactly what exam questions in this unit will ask you to do.
A parabola is the set of all points equidistant from a point (the focus) and a line (the directrix). The vertex sits exactly halfway between them. From the standard-form equation you can read off the vertex, focus, directrix, and axis of symmetry, and the sign of p tells you which way the parabola opens.
Parabola
The set of all points (x, y) in a plane that are equidistant from a fixed point and a fixed line. Think of it as a curve where every point "balances" its distance to the focus and the directrix.
Focus
A fixed point in the interior of the parabola that lies on the axis of symmetry. In simple terms, this is the point the parabola "wraps around."
Directrix
A fixed line perpendicular to the axis of symmetry. The parabola curves away from this line. Think of it as the boundary the curve is always measuring its distance from.
Vertex
The point on the parabola that lies exactly halfway between the focus and the directrix. This is the turning point of the curve, written as (h, k) in standard form.
Axis of symmetry
The line through the vertex and the focus that divides the parabola into two mirror-image halves. For a vertical parabola it is the vertical line x = h; for a horizontal parabola it is the horizontal line y = k.
Parameter p
The directed distance from the vertex to the focus. When p > 0 the parabola opens upward (vertical) or to the right (horizontal). When p < 0 it opens downward or to the left. The coefficient 1/(4p) in the standard form is what connects the algebra to the geometry.
There are two standard forms, depending on whether the parabola opens vertically or horizontally.
Vertical parabola (axis of symmetry is a vertical line, x = h):
y = 1/(4p) (x - h)^2 + k
Focus: (h, k + p)
Directrix: y = k - p
Opens up when p > 0, down when p < 0
Horizontal parabola (axis of symmetry is a horizontal line, y = k):
x = 1/(4p) (y - k)^2 + h
Focus: (h + p, k)
Directrix: x = h - p
Opens right when p > 0, left when p < 0
Decide whether the equation is solved for y (vertical) or solved for x (horizontal)
Match the equation to the appropriate standard form
Read off h, k, and p (the coefficient of the squared term equals 1/(4p), so solve for p)
Plug h, k, and p into the formulas for the focus, directrix, and axis of symmetry
Identify the vertex, focus, directrix, and axis of symmetry of x = (1/8)(y - 1)^2 + 4.
The equation is solved for x, so the parabola opens horizontally
Matching to x = 1/(4p)(y - k)^2 + h gives: 1/(4p) = 1/8, so 4p = 8, so p = 2. Also h = 4 and k = 1
Vertex: (h, k) = (4, 1)
Focus: (h + p, k) = (4 + 2, 1) = (6, 1)
Directrix: x = h - p = 4 - 2 = 2
Axis of symmetry: y = k = 1
Because p = 2 > 0, the parabola opens to the right
To graph, substitute y-values into the equation and solve for x. This is often easier than substituting x-values when the equation is solved for x.
Plot the vertex first
Mark the focus and draw the directrix as a dashed line
Use a table of values, substituting values for the variable that is squared
The curve wraps around the focus and away from the directrix
Check: every plotted point should look roughly equidistant from the focus and the directrix
Type | Standard form | Focus | Directrix | Axis of symmetry |
|---|---|---|---|---|
Vertical | y = 1/(4p)(x - h)^2 + k | (h, k + p) | y = k - p | x = h |
Horizontal | x = 1/(4p)(y - k)^2 + h | (h + p, k) | x = h - p | y = k |
Key relationship: the coefficient of the squared term is 1/(4p). To find p, set 1/(4p) equal to that coefficient and solve.
The vertex (h, k) is the midpoint between the focus and the closest point on the directrix.
Satellite dishes and car headlights are parabolic because any signal arriving parallel to the axis of symmetry reflects off the surface and passes through the focus. That is also why the receiver on a satellite dish sits at the focus point.
Suspension bridge cables hang in a parabolic shape when the load is distributed uniformly along the deck, which is why engineers model them with the same equations you are learning here.
Students often confuse h and k when the equation is for a horizontal parabola. In x = 1/(4p)(y - k)^2 + h, the number added outside the squared term is h (the x-coordinate of the vertex), not k.
The focus is inside the parabola, not on it. Some students plot the focus on the curve itself.
The sign of p determines direction, not the sign of h or k. A negative p means the parabola opens downward or to the left, regardless of where the vertex sits.
The directrix is a line, not a point. Students sometimes write the directrix as a coordinate pair when it should be an equation (y = ... or x = ...).
You will almost certainly be asked to identify vertex, focus, directrix, and axis of symmetry from a given equation. Have the steps automatic.
Expect at least one problem where you must find p from the coefficient 1/(4p). Practise solving 1/(4p) = given value until it is second nature.
Graphing problems may ask you to sketch the parabola and label all four features (vertex, focus, directrix, axis of symmetry). Partial marks often go for correctly labelling the directrix as a dashed line.
Some exams give a focus and directrix and ask you to write the equation, which is the reverse of the standard problem. Work from the vertex (midpoint of focus and directrix) and find p from the distance.
True or false: The vertex of a parabola is located at the focus.
Fill in the blank: If p is negative in a vertical parabola, the parabola opens ______.
True or false: The directrix is always perpendicular to the axis of symmetry.
Fill in the blank: The coefficient of the squared term in standard form equals ______.
True or false: In x = 1/(4p)(y - k)^2 + h, the axis of symmetry is x = h.
Answers: 1. False (the vertex is halfway between focus and directrix). 2. Downward. 3. True. 4. 1/(4p). 5. False (it is y = k).
Q: Given y = -(1/24)(x + 6)^2 - 4, identify the vertex, focus, directrix, and axis of symmetry.
A: Vertex: (-6, -4). Here 1/(4p) = -1/24, so 4p = -24, p = -6. Focus: (-6, -4 + (-6)) = (-6, -10). Directrix: y = -4 - (-6) = 2. Axis of symmetry: x = -6.
Q: Given x = -(1/4)(y + 5)^2 - 1, identify the vertex, focus, directrix, and axis of symmetry.
A: Vertex: (-1, -5). Here 1/(4p) = -1/4, so 4p = -4, p = -1. Focus: (-1 + (-1), -5) = (-2, -5). Directrix: x = -1 - (-1) = 0. Axis of symmetry: y = -5.
Q: Given y = (1/6)x^2 - 3, identify the vertex, focus, directrix, and axis of symmetry.
A: Vertex: (0, -3). Here 1/(4p) = 1/6, so 4p = 6, p = 1.5. Focus: (0, -3 + 1.5) = (0, -1.5). Directrix: y = -3 - 1.5 = -4.5. Axis of symmetry: x = 0.
Q: A parabola has its focus at (3, 2) and its directrix at x = 1. Write the equation in standard form.
A: The vertex is the midpoint between the focus and the directrix along the axis of symmetry: ((3 + 1)/2, 2) = (2, 2). The distance from vertex to focus is p = 3 - 2 = 1. The parabola opens horizontally (right), so x = 1/(4(1))(y - 2)^2 + 2 = (1/4)(y - 2)^2 + 2.
This connects to conic sections more broadly: ellipses and hyperbolas are defined by similar distance relationships, so the focus-directrix idea you learn here carries directly into those topics.
In calculus, the derivative of a quadratic gives you the slope of the tangent to a parabola at any point, which links this geometric view back to the algebraic differentiation you will study later.
Reflective properties of parabolas appear again in physics (optics and acoustics), where the focus is the point at which reflected waves converge.
Parabola, focus of a parabola, directrix, vertex, axis of symmetry, standard form of a parabola, conic sections, 1/(4p), horizontal parabola, vertical parabola, opens up, opens down, opens left, opens right, quadratic in standard form, reflective property, focal point, equidistant definition of parabola, graphing parabolas, Calculus 1, Topic 7.10, Penn State