# Analytic Geometry

## The Parabola


### Learning Objectives

1. Graph vertical parabolas. (IA 11.2.1)
2. Graph horizontal parabolas. (IA 11.2.2)

### Objective 1: Graph vertical parabolas. (IA 11.2.1)

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here.

### Practice Makes Perfect


Graph vertical parabolas.


### Objective 2: Graph horizontal parabolas. (IA 11.2.2)


Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.


### Practice Makes Perfect

Katherine Johnson is the pioneering NASA mathematician who was integral to the successful and safe flight and return of many human missions as well as satellites. Prior to the work featured in the movie Hidden Figures, she had already made major contributions to the space program. She provided trajectory analysis for the Mercury mission, in which Alan Shepard became the first American to reach space, and she and engineer Ted Sopinski authored a monumental paper regarding placing an object in a precise orbital position and having it return safely to Earth. Many of the orbits she determined were made up of parabolas, and her ability to combine different types of math enabled an unprecedented level of precision.  Johnson said, "You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off."
Johnson's work on parabolic orbits and other complex mathematics resulted in successful orbits, Moon landings, and the development of the Space Shuttle program. Applications of parabolas are also critical to other areas of science. Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.

### Graphing Parabolas with Vertices at the Origin

In The Ellipse, we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola. See .
Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points 
       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.
In Quadratic Functions, we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See . Notice that 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. The endpoints of the latus rectum lie on the curve. By definition, the distance 
  from the focus to any point 
  on the parabola is equal to the distance from 
     to the directrix.
To work with parabolas in the coordinate plane, we consider two cases: those with a vertex at the origin and those with a vertex at a point other than the origin. We begin with the former.
Let 
     be a point on the parabola with vertex 
     focus 
     and directrix 
 
as shown in . The distance 
  from point 
     to point 
 
on the directrix is the difference of the y-values: 
     The distance from the focus 
  to the point 
     is also equal to 
     and can be expressed using the distance formula.
Set the two expressions for 
     equal to each other and solve for 
     to derive the equation of the parabola. We do this because the distance from 
     to 
     equals the distance from 
     to 
 


We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.
The equations of parabolas with vertex 
     are 
     when the x-axis is the axis of symmetry and 
     when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. See . When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.
A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the latus rectum, these lines intersect on the axis of symmetry, as shown in .

###  Writing Equations of Parabolas in Standard Form

In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

### Graphing Parabolas with Vertices Not at the Origin

Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated 
       units horizontally and 
       units vertically, the vertex will be 
       This translation results in the standard form of the equation we saw previously with 
       replaced by 
       and 
       replaced by 
      
    
To graph parabolas with a vertex 
       other than the origin, we use the standard form 
       for parabolas that have an axis of symmetry parallel to the x-axis, and 
       for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.

### Solving Applied Problems Involving Parabolas 

As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting 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. See . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.
Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.

### Key Equations


### Key Concepts

1. A parabola is the set of all points 
         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.
2. The standard form of a parabola with vertex 
         and the x-axis as its axis of symmetry can be used to graph the parabola. If 
         the parabola opens right. If 
         the parabola opens left. See .
3. The standard form of a parabola with vertex 
         and the y-axis as its axis of symmetry can be used to graph the parabola. If 
         the parabola opens up. If 
         the parabola opens down. See .
4. When given the focus and directrix of a parabola, we can write its equation in standard form. See .
5. The standard form of a parabola with vertex 
         and axis of symmetry parallel to the x-axis can be used to graph the parabola. If 
         the parabola opens right. If 
         the parabola opens left. See .
6. The standard form of a parabola with vertex 
         and axis of symmetry parallel to the y-axis can be used to graph the parabola. If 
         the parabola opens up. If 
         the parabola opens down. See .
7. Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See .

### Section Exercises


### Verbal


###  Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.
For the following exercises, rewrite the given equation in standard form, and then determine the vertex 
         focus 
         and directrix 
  of the parabola.
      

### Graphical

For the following exercises, graph the parabola, labeling the focus and the directrix.
For the following exercises, find the equation of the parabola given information about its graph.
For the following exercises, determine the equation for the parabola from its graph.

###  Extensions

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.

###  Real-World Applications
