Understanding Parabolas in Conic Sections

Chelsea

12/8/2025

Pre-Calculus

Introduction to Conics: Parabolas

Subjects

Pre-Calculus

Ellipses & Hyperbola

Hyperbolas Not Centered at the Origin

132

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Dec 8, 2025

•

4 pages

Understanding Parabolas in Conic Sections

Chelsea

@zuncho

Conics are fascinating geometric shapes formed when a plane intersects... Show more

1 / 4

Conics
Introduction to Conics: Parabolas
- A conic section (or simply conic) is the intersection of a plane and a
double-napped cone
- In th

Introduction to Parabolas

A parabola is a set of points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex sits halfway between the focus and directrix, while the axis runs through both the focus and vertex.

Several key elements determine a parabola's appearance. The axis affects its symmetry, the focus determines its size, the directrix influences its direction, and the vertex establishes its location. When you see a parabola in standard form, you can immediately tell which way it opens and where it's positioned.

The general form of a parabola comes from the conic section equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0. For parabolas specifically, B = 0, and either A = 0 or C = 0 (but not both). This gives us two possible forms:

If C = 0: Ax² + Dx + Ey + F = 0 $opens up/down$
If A = 0: Cy² + Dx + Ey + F = 0 $opens left/right$

Try This! Next time you see a satellite dish or the path of a ball tossed in the air, you're looking at a parabola in real life! The shape's special reflective properties make it perfect for focusing signals or following physical laws.

Standard Equations of Parabolas

The standard form of a parabola with vertex at (h,k) comes in two main types, depending on which way it opens:

$x-h$ ² = 4p $y-k$ for vertical parabolas
$y-k$ ² = 4p $x-h$ for horizontal parabolas

The value of p is crucial - it represents the directed distance from the vertex to the focus. When p is positive, the parabola opens upward or to the right. When p is negative, it opens downward or to the left.

For vertical parabolas, the axis of symmetry is vertical $x=h$ , and the directrix is horizontal $y=k-p$ . For horizontal parabolas, the axis is horizontal $y=k$ , and the directrix is vertical $x=h-p$ . These relationships help you quickly visualize the parabola's orientation.

Let's look at an example: To convert x²-12x-2y+20=0 into standard form, we complete the square:

x²-12x = 2y-20
$x-6$ ² = 2y+16
$x-6$ ² = 2 $y+8$ This gives us vertex (6,-8) with p=½, meaning it opens upward.

Remember: The variable that's squared tells you the orientation of the parabola. If x is squared, the parabola opens up/down; if y is squared, it opens left/right.

Graphing Parabolas

Graphing parabolas becomes simple with a step-by-step approach. First, convert the equation to standard form. Then plot the vertex (h,k) and determine the value of p. The variable that's squared tells you the orientation - horizontal or vertical axis.

When the directrix is horizontal, the parabola opens up (p>0) or down (p<0). When the directrix is vertical, it opens right (p>0) or left (p<0). After plotting the directrix and focus, draw the parabola opening away from the directrix.

For example, with x²-12x-2y+20=0, we found the standard form $x-6$ ²=2 $y+8$ . This gives us vertex V(6,-8) and p=½. Since x is squared and p is positive, the parabola opens upward.

Another example: 4y+16x=44-y². Converting to standard form gives $y+2$ ²=-16 $x-3$ , with vertex at (3,-2) and p=-4. Since y is squared and p is negative, the parabola opens to the left with focus at (-1,-2) and directrix at x=7.

Quick Tip: When graphing, always start with the vertex as your anchor point. Everything else - the direction, focus, and directrix - can be determined from there and the value of p.

More Parabola Examples

Let's work through another graphing example: x²-6x+8y+41=0. Converting to standard form:

x²-6x = -8y-41
$x-3$ ² = -8y-32
$x-3$ ² = -8 $y+4$

From this, we identify h=3, k=-4, and p=-2. Since x is squared and p is negative, this parabola opens downward with vertex at (3,-4), focus at (3,-6), and directrix at y=-2.

We can also write equations when given specific information. For example, with a vertex at (-3,5) and focus at (-3,1), we know:

The focus is 4 units below the vertex, so p=-4
Since the focus and vertex share the same x-coordinate, the axis is vertical
Thus, $x+3$ ² = -16 $y-5$

Similarly, with vertex (4,-1) and directrix x=2:

The directrix is 2 units to the left of the vertex, so p=2
This is a horizontal parabola opening right
The equation is $y+1$ ² = 8 $x-4$

Master Skill: You can create any parabola if you know its vertex and either its focus or directrix. This is powerful for modeling real-world situations like designing reflectors or predicting projectile motion.

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Subjects

Pre-Calculus

Ellipses & Hyperbola

Hyperbolas Not Centered at the Origin

Pre-Calculus

•

132

•

Dec 8, 2025

•

4 pages

Understanding Parabolas in Conic Sections

Chelsea

@zuncho

Conics are fascinating geometric shapes formed when a plane intersects with a double-napped cone. This intersection creates four basic shapes: circles, ellipses, parabolas, and hyperbolas. In this summary, we'll focus on parabolas - those U-shaped curves you've seen in everything... Show more

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Introduction to Parabolas

If C = 0: Ax² + Dx + Ey + F = 0 $opens up/down$
If A = 0: Cy² + Dx + Ey + F = 0 $opens left/right$

Try This! Next time you see a satellite dish or the path of a ball tossed in the air, you're looking at a parabola in real life! The shape's special reflective properties make it perfect for focusing signals or following physical laws.

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By signing up you accept Terms of Service and Privacy Policy

Standard Equations of Parabolas

The standard form of a parabola with vertex at (h,k) comes in two main types, depending on which way it opens:

$x-h$ ² = 4p $y-k$ for vertical parabolas
$y-k$ ² = 4p $x-h$ for horizontal parabolas

Let's look at an example: To convert x²-12x-2y+20=0 into standard form, we complete the square:

x²-12x = 2y-20
$x-6$ ² = 2y+16
$x-6$ ² = 2 $y+8$ This gives us vertex (6,-8) with p=½, meaning it opens upward.

Remember: The variable that's squared tells you the orientation of the parabola. If x is squared, the parabola opens up/down; if y is squared, it opens left/right.

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Graphing Parabolas

For example, with x²-12x-2y+20=0, we found the standard form $x-6$ ²=2 $y+8$ . This gives us vertex V(6,-8) and p=½. Since x is squared and p is positive, the parabola opens upward.

Quick Tip: When graphing, always start with the vertex as your anchor point. Everything else - the direction, focus, and directrix - can be determined from there and the value of p.

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More Parabola Examples

Let's work through another graphing example: x²-6x+8y+41=0. Converting to standard form:

x²-6x = -8y-41
$x-3$ ² = -8y-32
$x-3$ ² = -8 $y+4$

From this, we identify h=3, k=-4, and p=-2. Since x is squared and p is negative, this parabola opens downward with vertex at (3,-4), focus at (3,-6), and directrix at y=-2.

We can also write equations when given specific information. For example, with a vertex at (-3,5) and focus at (-3,1), we know:

The focus is 4 units below the vertex, so p=-4
Since the focus and vertex share the same x-coordinate, the axis is vertical
Thus, $x+3$ ² = -16 $y-5$

Similarly, with vertex (4,-1) and directrix x=2:

The directrix is 2 units to the left of the vertex, so p=2
This is a horizontal parabola opening right
The equation is $y+1$ ² = 8 $x-4$

Master Skill: You can create any parabola if you know its vertex and either its focus or directrix. This is powerful for modeling real-world situations like designing reflectors or predicting projectile motion.