12.3 The Parabola

Parabola Fundamentals

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). This geometric definition is what connects parabolas to the broader study of conic sections and gives rise to their distinctive U-shaped curve.

In analytic geometry, parabolas go beyond the simple $y = ax^2$ you've seen before. Here you'll work with the focus-directrix definition, derive equations from geometric properties, and use features like focal width to precisely describe a parabola's shape.

Graphing Parabolas

Every parabola is symmetrical about a line called the axis of symmetry, which passes through the vertex (the turning point of the curve).

• If the parabola opens upward, the vertex is a minimum. If it opens downward, the vertex is a maximum.
• The coefficient $a$ controls both direction and "width":
  • $a > 0$: opens upward
  • $a < 0$: opens downward
  • Larger $|a|$ makes the parabola narrower; smaller $|a|$ makes it wider

Vertex at the origin: $y = ax^2$, with axis of symmetry along the y-axis ($x = 0$).

Vertex at $(h, k)$: $y = a(x - h)^2 + k$, with axis of symmetry at $x = h$.

Equations in Standard Form

The vertex form (often called standard form in this context) is:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex and $a$ controls shape and direction.

Writing an equation in standard form from given information:

1. Identify the vertex $(h, k)$.
2. Determine $a$ using another known point on the parabola or information about the focus.
3. Substitute $h$, $k$, and $a$ into $y = a(x - h)^2 + k$.

Converting from general form $y = ax^2 + bx + c$ to vertex form:

1. Factor $a$ out of the first two terms: $y = a(x^2 + \frac{b}{a}x) + c$.
2. Complete the square inside the parentheses: take half of $\frac{b}{a}$, square it, then add and subtract that value inside.
3. Simplify to get $y = a(x - h)^2 + k$.

For example, convert $y = 2x^2 + 12x + 7$:

1. Factor out 2: $y = 2(x^2 + 6x) + 7$

2. Half of 6 is 3, and $3^2 = 9$. Add and subtract 9 inside: $y = 2(x^2 + 6x + 9 - 9) + 7$

3. Simplify: $y = 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11$

The vertex is $(-3, -11)$ and the parabola opens upward since $a = 2 > 0$.

Parabola Properties and Applications

Key Features of Parabolas

Vertex: The turning point, located at $(h, k)$.

Focus: A point on the axis of symmetry, inside the curve, at a distance of $\frac{1}{4|a|}$ from the vertex.

• For an upward-opening parabola ($a > 0$), the focus is above the vertex at $(h,\; k + \frac{1}{4a})$.
• For a downward-opening parabola ($a < 0$), the focus is below the vertex at $(h,\; k + \frac{1}{4a})$ (note $\frac{1}{4a}$ is negative here, so the focus shifts downward).

Directrix: A horizontal line on the opposite side of the vertex from the focus, also at a distance of $\frac{1}{4|a|}$ from the vertex.

• For $y = a(x-h)^2 + k$, the directrix is $y = k - \frac{1}{4a}$.

The defining property ties these together: every point on the parabola is exactly the same distance from the focus as it is from the directrix.

Latus rectum: The line segment that passes through the focus, runs perpendicular to the axis of symmetry, and has both endpoints on the parabola. Its length (the focal width) equals $\frac{1}{|a|}$. This measurement is useful because it tells you how wide the parabola is at the level of the focus, giving you a concrete reference for sketching.

Parabolas as Conic Sections

A parabola is one of the four conic sections, formed when a plane intersects a cone parallel to the cone's slant side.

• The eccentricity of a parabola is always exactly $e = 1$. (For comparison, ellipses have $e < 1$ and hyperbolas have $e > 1$.)
• Reflective property: Any ray traveling parallel to the axis of symmetry will bounce off the parabolic surface and pass through the focus. This is why satellite dishes and car headlights use parabolic shapes. A dish collects incoming signals at the focus, and a headlight placed at the focus sends light out in a parallel beam.

Applications of Parabolas

Parabolas appear frequently in real-world contexts:

• Projectile motion: An object launched near Earth's surface follows a parabolic path (ignoring air resistance).
• Reflectors and antennas: Satellite dishes, telescopes, and headlights use the reflective property described above.
• Architecture: Parabolic arches distribute weight efficiently, appearing in bridges and building design.

Solving applied problems with parabolas:

1. Identify what's given and what you need to find.
2. Set up a coordinate system (often placing the vertex at the origin or at a convenient point).
3. Write the appropriate parabolic equation using the given information.
4. Solve for the unknown quantity.

Example: A ball is thrown upward at 20 m/s from a height of 1.5 m. Its height after $t$ seconds is $h(t) = -4.9t^2 + 20t + 1.5$. Find the maximum height.

1. Since $a = -4.9 < 0$, the parabola opens downward, so the vertex gives the maximum height.
2. The time at the vertex is $t = \frac{-b}{2a} = \frac{-20}{2(-4.9)} \approx 2.04$ seconds.
3. Substitute back: $h(2.04) = -4.9(2.04)^2 + 20(2.04) + 1.5 \approx 21.9$ m.

The ball reaches a maximum height of approximately 21.9 m at about 2.04 seconds after being thrown.