Pre-Calculus Conic sections

Parabolas (standard form, focus, directrix, axis)

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

Match the equation to x squared equals 4 p y or y squared equals 4 p x, solve for p, and place the focus a distance p from the vertex.

Q: Which way does a parabola open?

The squared variable decides: x squared opens up or down; y squared opens left or right, with the sign of p giving the exact direction.

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Parabolas

Common Core Pre-Calculus • Standard G-GPE.2 • Conic Sections

Parabolas is a topic in Conic Sections in the Common Core State Standards. It is aligned to Standard G-GPE.2, which requires students to derive the equation of a parabola from its focus and directrix.

A parabola is the set of points equidistant from a focus and a directrix, with standard form \(x^2=4py\) or \(y^2=4px\).

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Theory

A parabola is the set of points equidistant from a fixed focus and a fixed line, the directrix. Its turning point is the vertex, halfway between them, and its axis of symmetry passes through both.

With the vertex at the origin, the standard forms are:

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

where \(|p|\) is the distance from the vertex to the focus (and to the directrix).

The squared variable tells the opening direction. \(x^2=\dots\) opens vertically; \(y^2=\dots\) opens horizontally.

Every point of a parabola is equidistant from the focus and the directrix.

Standard forms and the role of \(p\).

Standard forms and the focus/directrix:

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

\[y^2=4px:\ \text{focus }(p,0),\ \text{directrix }x=-p\]

for x squared equals 4 p y the focus is 0 comma p and directrix y equals negative p

Shifted vertex \((h,k)\): replace \(x,y\) with \(x-h,y-k\).

How to analyze a parabola

Identify the squared variable to get the opening direction.
Match to \(4p\) to find \(p\).
Locate the focus (distance \(p\) from the vertex) and the directrix (distance \(p\) on the other side).
Shift for a vertex \((h,k)\) using \(x-h,\ y-k\).

Example 1 — Focus and directrix

For \(x^2=8y\), find the focus and directrix.

Solution

Match \(x^2=4py\): \(4p=8\Rightarrow p=2\).

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

focus at 0 comma 2, directrix y equals negative 2

Example 2 — Opens sideways

Describe \(y^2=-12x\).

Solution

Match \(y^2=4px\): \(4p=-12\Rightarrow p=-3\); a negative \(p\) opens left.

\(\text{focus}\)	\(=\)	\((-3,0)\)
\(\text{directrix}\)	\(:\)	\(x=3\)

opens left, focus at negative 3 comma 0

Example 3 — Write the equation

Write the equation of a parabola with vertex \((0,0)\) and focus \((0,4)\).

Solution

Opens up with \(p=4\), so \(x^2=4py\).

\(x^2\)

\(=\)

\(4(4)y=16y\)

the equation is x squared equals 16 y

Example 4 — Vertex not at the origin

Give the vertex and focus of \((x-2)^2=8(y+1)\).

Solution

This is \(x^2=4py\) shifted to vertex \((2,-1)\) with \(p=2\).

\(\text{vertex}\)	\(=\)	\((2,-1)\)
\(\text{focus}\)	\(=\)	\((2,1)\)

vertex 2 comma negative 1, focus 2 comma 1

Common pitfalls

\(4p\), not \(p\). In \(x^2=8y\), \(4p=8\) so \(p=2\).

The squared variable sets the direction. \(x^2\) opens vertically, \(y^2\) horizontally.

Focus and directrix are equidistant from the vertex, on opposite sides.

Frequently asked questions

What is a parabola?

The set of all points equidistant from a fixed focus and a fixed line called the directrix.

How do you find the focus of a parabola?

Match the equation to \(x^2=4py\) or \(y^2=4px\), solve for \(p\), and place the focus a distance \(p\) from the vertex.

Which way does a parabola open?

The squared variable decides: \(x^2=\dots\) opens up or down; \(y^2=\dots\) opens left or right, with the sign of \(p\) giving the exact direction.

What is the directrix?

The fixed line such that every point of the parabola is the same distance from it as from the focus.

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Ellipses (standard form, foci, center at (h, k))

Parabolas (standard form, focus, directrix, axis)

Parabolas

📖 Theory

How to analyze a parabola

Common pitfalls

Frequently asked questions

Theory