Algebra 2 Quadratic functions (advanced)

Parabolas from focus, directrix, axis of symmetry

Q: What are the focus and directrix of \(x^2=4py\)?

Focus at 0, p and directrix y equals negative p.

Q: What is the axis of symmetry?

The line through the vertex and focus.

Q: How do you find \(p\)?

Set the coefficient equal to 4p and divide by 4.

20 practice questions 0 video lessons Theory + worked examples

Parabolas: Focus, Directrix, Axis

California Algebra 2 • Standard G-GPE.2 • Quadratic Functions

Parabolas: Focus, Directrix, Axis is a topic in Quadratic Functions in the California Common Core State Standards. It is aligned to Standard G-GPE.2, which requires students to derive the equation of a parabola given a focus and directrix.

A parabola is the set of points equidistant from a focus and a directrix; with vertex at the origin, \(x^2=4py\) has focus \((0,p)\).

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Practice questions

Every question with a fully worked solution.

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Theory

A parabola is the set of all points equidistant from a fixed focus and a fixed line, the directrix. With vertex at the origin:

\[x^2=4py\quad\Rightarrow\quad\text{focus }(0,p),\ \text{directrix }y=-p.\]

The axis of symmetry passes through the focus and vertex.

The vertex is halfway between the focus and the directrix.

Distances to the focus and directrix are equal.

Focus-directrix form.

Vertex at the origin:

\[x^2=4py\ (\text{up/down}),\qquad y^2=4px\ (\text{left/right})\]

x squared equals 4 p y opens up or down; y squared equals 4 p x opens sideways

\(p\) is the focus distance from the vertex.

How to use focus and directrix

Match the equation to \(x^2=4py\) or \(y^2=4px\).
Solve for \(p\).
Focus is \(p\) from the vertex; directrix is \(p\) on the other side.
The axis of symmetry runs through the vertex and focus.

Example 1 — Focus and directrix

Find the focus and directrix of \(y=\dfrac{1}{8}x^2\).

Solution

Write as \(x^2=4py\): \(4p=8\), so \(p=2\).

\(\text{focus}\)	\(=\)	\((0,2)\)
\(\text{directrix}\)	\(:\)	\(y=-2\)

focus 0 comma 2, directrix y equals negative 2

Example 2 — Write the equation

A parabola has focus \((0,3)\) and directrix \(y=-3\). Find its equation.

Solution

Here \(p=3\), so \(4p=12\).

\(x^2\)

\(=\)

\(12y\)

the equation is x squared equals 12 y

Example 3 — Opens sideways

Find the focus of \(y^2=4x\).

Solution

Compare with \(y^2=4px\): \(4p=4\), so \(p=1\).

\(\text{focus}\)	\(=\)	\((1,0)\)
\(\text{directrix}\)	\(:\)	\(x=-1\)

focus 1 comma 0, directrix x equals negative 1

Example 4 — Axis of symmetry

State the axis of symmetry of \(y=(x-2)^2+1\).

Solution

The axis passes through the vertex \((2,1)\).

\(\text{axis}\)

\(:\)

\(x=2\)

the axis of symmetry is x equals 2

Common pitfalls

\(4p\) is the whole coefficient — solve for \(p\) by dividing by 4.

\(x^2\) opens up/down; \(y^2\) opens left/right.

Directrix is opposite the focus, the same distance \(p\).

Frequently asked questions

What is a parabola, by definition?

The set of points equidistant from a focus and a directrix.

What are the focus and directrix of \(x^2=4py\)?

Focus \((0,p)\) and directrix \(y=-p\).

What is the axis of symmetry?

The line through the vertex and focus that splits the parabola in half.

How do you find \(p\)?

Set the coefficient equal to \(4p\) and divide by 4.

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Parabolas from focus, directrix, axis of symmetry

Parabolas: Focus, Directrix, Axis

📖 Theory

How to use focus and directrix

Common pitfalls

Frequently asked questions

Theory