Equation of a Parabola

Definition, worked examples, and an interactive grapher — vertex, focus, and directrix

Definition and Equation of a Parabola with a Vertical Axis

A parabola is the set of all points \( M(x,y) \) in a plane such that the distance from \( M \) to a fixed point \( F \) (the focus) equals the distance from \( M \) to a fixed line (the directrix).

Consider a parabola with vertex \( V(0,0) \) at the origin and focus \( F(0, p) \) on the y-axis (\( p > 0 \)), so the directrix is \( y = -p \).

The distance from \( M(x,y) \) to the focus is:

\[ MF = \sqrt{(x - 0)^2 + (y - p)^2} \]

The distance from \( M(x,y) \) to the directrix \( y = -p \) is:

\[ MD = y + p \]

By definition these distances are equal. Squaring both sides and simplifying:

\[ x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2 \] \[ x^2 = 4py \]

\[ y = \frac{1}{4p} x^2 \]

Diagram showing a point on the parabola at equal distance from the focus and directrix
Figure 1. Geometric definition of the parabola.

Worked Examples — Vertical Axis

For a parabola with a vertical axis and vertex \( V(h,k) \), the standard form is:

\[ y = \frac{1}{4p} (x - h)^2 + k \]

Example 1 — Find p given a point on the parabola

The point \( (4, 2) \) lies on a parabola with vertex at the origin and a vertical axis of symmetry. Find the focus and the equation of the directrix.

View Solution

For a vertical parabola with vertex at the origin:

\[ y = \frac{1}{4p} x^2 \]

Substitute \( x = 4 \) and \( y = 2 \):

\[ 2 = \frac{16}{4p} \]

Solving: \( p = 2 \).

Hence the focus is \( F(0, 2) \) and the directrix is the horizontal line \( y = -2 \).

Graph of y = x²/8, the parabola for Example 1
Figure 2. \( y = \dfrac{x^2}{8} \).

Example 2 — Complete the square to find the vertex

Find the vertex, focus, and directrix of \( y = \dfrac{1}{16} x^2 - \dfrac{1}{4} x + \dfrac{9}{4} \).

View Solution

Factor \( \dfrac{1}{16} \) out of the \( x \)-terms:

\[ y = \frac{1}{16}(x^2 - 4x) + \frac{9}{4} \]

Complete the square inside the parentheses:

\[ y = \frac{1}{16}\bigl((x - 2)^2 - 4\bigr) + \frac{9}{4} \]

Distribute and combine the constants:

\[ y = \frac{1}{16}(x - 2)^2 - \frac{1}{4} + \frac{9}{4} \] \[ y = \frac{1}{16}(x - 2)^2 + 2 \]

Compare with \( y = \dfrac{1}{4p}(x - h)^2 + k \):

  • \( \dfrac{1}{4p} = \dfrac{1}{16} \Rightarrow p = 4 \)
  • \( h = 2, \quad k = 2 \)

Vertex: \( V(2, 2) \)
Focus: \( F(2, 6) \)
Directrix: \( y = -2 \)

Graph of y = (x−2)²/16 + 2, the parabola for Example 2
Figure 3. \( y = \dfrac{(x-2)^2}{16} + 2 \).

Equation of a Parabola with a Horizontal Axis

For a horizontal axis of symmetry and vertex \( V(h,k) \):

\[ x = \frac{1}{4p} (y - k)^2 + h \]

Example 3 — A horizontal parabola

Find the vertex, focus, and directrix of \( x = \dfrac{1}{4} y^2 - y + 11 \).

View Solution

Factor \( \dfrac{1}{4} \) out of the \( y \)-terms:

\[ x = \frac{1}{4}(y^2 - 4y) + 11 \]

Complete the square:

\[ x = \frac{1}{4}\bigl((y - 2)^2 - 4\bigr) + 11 \] \[ x = \frac{1}{4}(y - 2)^2 + 10 \]

Compare with \( x = \dfrac{1}{4p}(y - k)^2 + h \):

  • \( \dfrac{1}{4p} = \dfrac{1}{4} \Rightarrow p = 1 \)
  • \( h = 10, \quad k = 2 \)

Vertex: \( V(10, 2) \)
Focus: \( F(11, 2) \)
Directrix: \( x = 9 \)

Graph of x = (y−2)²/4 + 10, the parabola for Example 3
Figure 4. \( x = \dfrac{(y-2)^2}{4} + 10 \).

Standard Forms at a Glance

Axis Equation Vertex Focus Directrix
Vertical \( y = \dfrac{(x-h)^2}{4p} + k \) \( (h,k) \) \( (h, k+p) \) \( y = k - p \)
Horizontal \( x = \dfrac{(y-k)^2}{4p} + h \) \( (h,k) \) \( (h+p, k) \) \( x = h - p \)

Interactive Parabola Grapher

Explore \( y = \dfrac{1}{4p}(x - h)^2 + k \) dynamically. Change \( p, h, k \) and click Plot Equation to see the vertex, focus, and directrix shift. The grapher below plots the vertical parabola only.

Hover over the graph to read coordinates.

Interactive Activities

  1. With the defaults \( p=1, h=2, k=3 \), compare the plotted vertex and focus with your manual calculations.
  2. Set \( p=4, h=1, k=-4 \) and predict the directrix equation before you click Plot Equation.
  3. Exercise: rewrite \( x^2 - 4x - 4y = 0 \) in standard form, identify \( p, h, k \), then graph it with the app to check your work.

Frequently Asked Questions

What is the definition of a parabola?

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

What is the standard equation of a vertical parabola?

For a vertical axis with vertex \( (h,k) \): \( y = \dfrac{(x-h)^2}{4p} + k \). Focus: \( (h, k+p) \). Directrix: \( y = k - p \).

What is the standard equation of a horizontal parabola?

For a horizontal axis with vertex \( (h,k) \): \( x = \dfrac{(y-k)^2}{4p} + h \). Focus: \( (h+p, k) \). Directrix: \( x = h - p \).

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

Write the equation in standard form to read off \( p \), \( h \), and \( k \). For a vertical axis: focus \( (h, k+p) \), directrix \( y = k-p \). For a horizontal axis: focus \( (h+p, k) \), directrix \( x = h-p \).

More Resources & Related Topics

Last updated: October 17, 2026