Lesson 12.3: The Parabola

New Concept

A parabola is a U-shaped curve defined by a focus and directrix. We'll use its standard equations, like $x^2=4py$, to graph its key features—vertex, focus, and axis of symmetry—and solve real-world problems involving parabolic reflectors.

What’s next

This is just the beginning. Next, you'll work through interactive examples and practice cards to master graphing parabolas and writing their equations.

Property

A parabola is the set of all points $(x, y)$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix. The axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus. The line segment that passes through the focus and is parallel to the directrix is called the latus rectum.

Examples

• A point $P$ on a parabola is 7 units away from its focus. Therefore, the distance from point $P$ to the parabola's directrix must also be exactly 7 units.

• If a parabola has its focus at $(0, 3)$ and its directrix at $y = -3$, its vertex must be at the midpoint, which is $(0, 0)$.

Property

For a parabola with its vertex at the origin $(0, 0)$, the standard form depends on its axis of symmetry.
If the axis of symmetry is the x-axis, the equation is $y^2 = 4px$. The focus is $(p, 0)$, the directrix is $x = -p$, and the endpoints of the latus rectum are $(p, \pm 2p)$.
If the axis of symmetry is the y-axis, the equation is $x^2 = 4py$. The focus is $(0, p)$, the directrix is $y = -p$, and the endpoints of the latus rectum are $(\pm 2p, p)$.
The sign of $p$ determines the direction the parabola opens.

Examples

• For the equation $y^2 = 20x$, we see that $4p = 20$, so $p=5$. Since $y$ is squared and $p$ is positive, the parabola opens right with a focus at $(5, 0)$.

• For the equation $x^2 = -12y$, we have $4p = -12$, so $p=-3$. Since $x$ is squared and $p$ is negative, the parabola opens down with a focus at $(0, -3)$.

Property

To write the equation for a parabola given its focus and directrix:

1. Determine the axis of symmetry. If the focus is $(p, 0)$, the axis is the x-axis, and the form is $y^2 = 4px$. If the focus is $(0, p)$, the axis is the y-axis, and the form is $x^2 = 4py$.
2. Identify the value of $p$ from the focus coordinates.
3. Substitute the value of $p$ into the correct standard form equation.

Examples

• Given a focus at $(0, 5)$ and directrix at $y=-5$, we know $p=5$ and the parabola opens up. The equation is $x^2 = 4(5)y$, or $x^2 = 20y$.

• For a parabola with focus $(-3, 0)$ and directrix $x=3$, we know $p=-3$ and it opens left. The equation is $y^2 = 4(-3)x$, or $y^2 = -12x$.

Property

If a parabola is translated so its vertex is at $(h, k)$, we replace $x$ with $(x - h)$ and $y$ with $(y - k)$.
For an axis of symmetry parallel to the x-axis ($y=k$): $(y - k)^2 = 4p(x - h)$. The focus is $(h + p, k)$ and the directrix is $x = h - p$.
For an axis of symmetry parallel to the y-axis ($x=h$): $(x - h)^2 = 4p(y - k)$. The focus is $(h, k + p)$ and the directrix is $y = k - p$.

Examples

• The equation $(x - 3)^2 = 8(y + 2)$ describes a parabola with its vertex at $(3, -2)$. Since $4p=8$, $p=2$, and it opens up.

• For the parabola $(y - 1)^2 = -16(x - 5)$, the vertex is $(5, 1)$. Here, $4p=-16$, so $p=-4$. It opens to the left, and its focus is at $(5-4, 1) = (1, 1)$.

Property

To graph a parabola from a general form equation like $x^2 - 8x - 28y - 208 = 0$, first rewrite it in standard form. Isolate the terms containing the squared variable. Complete the square for these terms. Factor the other side of the equation to match the form $(x - h)^2 = 4p(y - k)$ or $(y - k)^2 = 4p(x - h)$. From standard form, identify the vertex, $p$, focus, and directrix to sketch the graph.

Examples

• To convert $y^2 + 2y + 12x + 25 = 0$, isolate y-terms: $y^2 + 2y = -12x - 25$. Complete the square: $(y+1)^2 = -12x - 24$. Factor to get $(y+1)^2 = -12(x+2)$.

• Starting with $x^2 + 10x - 8y + 41 = 0$, we write $x^2 + 10x = 8y - 41$. Completing the square gives $(x+5)^2 = 8y - 16$, which factors to $(x+5)^2 = 8(y-2)$.

Property

When rays of light parallel to the parabola’s axis of symmetry are directed toward any surface of the mirror, the light is reflected directly to the focus. This unique reflecting property is why parabolic mirrors can focus energy, like light or sound, to a single point.

Examples

• A satellite dish is a parabolic reflector. It collects incoming satellite signals that are parallel to its axis and concentrates them onto the receiver, which is placed at the focus for the strongest signal.

• The reflector in a car's headlight is a parabola with the bulb at its focus. This design causes the light to shoot out in a strong, straight beam.