10.3 The Parabola

Parabola Equations and Graphing

A parabola is a conic section defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This geometric definition drives everything about how parabolas behave, from their equations to their reflective properties used in satellite dishes and headlights.

Graphing parabolas with various vertices

The vertex form of a parabola equation is $y = a(x - h)^2 + k$, and each part tells you something specific about the graph:

• $(h, k)$ is the vertex, the turning point of the parabola
  • If the vertex is at the origin, then $h = 0$ and $k = 0$, giving you $y = ax^2$
  • A vertex at $(3, -2)$ means $h = 3$ and $k = -2$, so the equation includes $(x - 3)^2 + (-2)$
• $a$ controls both the direction and the width
  • $a > 0$: opens upward (U-shape)
  • $a < 0$: opens downward (inverted U-shape)
  • Larger $|a|$ means a narrower parabola; smaller $|a|$ means a wider one

To graph a parabola from vertex form:

1. Plot the vertex $(h, k)$
2. Use the sign of $a$ to determine whether it opens up or down
3. Find additional points by plugging in $x$-values on either side of the vertex
4. Use symmetry across the axis of symmetry ($x = h$) to mirror points

For example, with $y = 2(x - 1)^2 + 3$, the vertex is $(1, 3)$ and the parabola opens upward. Plugging in $x = 2$ gives $y = 2(1)^2 + 3 = 5$, so $(2, 5)$ is on the graph, and by symmetry so is $(0, 5)$.

Equations of parabolas in standard form

The standard form is $y = ax^2 + bx + c$, where $a \neq 0$. This form doesn't show the vertex directly, but you can extract all the key features:

• Vertex x-coordinate: $x = -\frac{b}{2a}$
• Vertex y-coordinate: Substitute that $x$-value back into the equation
• Axis of symmetry: The vertical line $x = -\frac{b}{2a}$, which divides the parabola into two mirror-image halves
• y-intercept: Set $x = 0$ and solve. The y-intercept is always $(0, c)$
• x-intercepts (roots/zeros): Set $y = 0$ and solve the quadratic $ax^2 + bx + c = 0$. These exist only when the discriminant $b^2 - 4ac \geq 0$

For example, given $y = 2x^2 - 4x + 1$:

• Vertex x-coordinate: $x = -\frac{-4}{2(2)} = 1$
• Vertex y-coordinate: $y = 2(1)^2 - 4(1) + 1 = -1$, so the vertex is $(1, -1)$
• y-intercept: $(0, 1)$
• x-intercepts: Solve $2x^2 - 4x + 1 = 0$ using the quadratic formula

Real-world applications of parabolic models

Many physical situations follow parabolic paths:

• Projectile motion: A ball thrown in the air traces a parabola. The equation $y = -16t^2 + v_0t + h_0$ models height over time (in feet), where $v_0$ is initial velocity and $h_0$ is initial height. The maximum height occurs at the vertex, found at $t = \frac{v_0}{32}$.
• Suspension bridge cables: The main cables of bridges like the Golden Gate Bridge hang in a parabolic shape under uniform load.
• Parabolic reflectors: Satellite dishes and car headlights use the reflective property of parabolas: signals hitting a parabolic surface all reflect to the focus.

To solve these problems:

1. Identify what the variables represent (time, distance, height, etc.)
2. Match the given information to parts of a parabolic equation
3. Write the equation and use it to find the quantity you need (maximum height, time of landing, etc.)

Parabola Properties and Relationships

Parabola equations vs geometric properties

This section ties together how the algebra and the geometry connect. Much of this reinforces what's above, but the relationships are worth seeing side by side:

The x-intercepts (also called roots or zeros) are found by setting $y = 0$ in either form. The discriminant $b^2 - 4ac$ tells you how many x-intercepts exist: two if positive, one if zero (the vertex touches the x-axis), and none if negative.

Conic sections and parabola properties

In the context of analytic geometry, a parabola is one of the four conic sections (along with circles, ellipses, and hyperbolas), formed when a plane intersects a cone parallel to one of its slant sides.

The conic definition introduces several important terms:

• Focus: A fixed point inside the parabola. For $y = ax^2$, the focus is at $(0, \frac{1}{4a})$
• Directrix: A fixed line outside the parabola. For $y = ax^2$, the directrix is $y = -\frac{1}{4a}$
• Focal length ($p$): The distance from the vertex to the focus, equal to $\frac{1}{4|a|}$. The vertex sits exactly halfway between the focus and the directrix.
• Latus rectum: A line segment through the focus, perpendicular to the axis of symmetry, with length $|4p| = \frac{1}{|a|}$. This segment gives you a quick sense of the parabola's width at the focus.
• Eccentricity: For a parabola, this is always exactly $1$. Ellipses have eccentricity less than 1, and hyperbolas have eccentricity greater than 1, so this value is what distinguishes a parabola from the other conics.

The conic form of a vertical parabola with vertex at the origin is $(x - h)^2 = 4p(y - k)$, where $p > 0$ means it opens upward and $p < 0$ means it opens downward. For horizontal parabolas, the form is $(y - k)^2 = 4p(x - h)$, opening right when $p > 0$ and left when $p < 0$. Being comfortable converting between this form and vertex form is essential for analytic geometry problems.