Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Advanced Algebra II: Conceptual Explanations » Finding a Parabolic Function for any Three Points

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Bookshare

    This collection is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech Initiative

    Comments:

    "DAISY and BRF versions of this collection are available."

    Click the "Bookshare" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This is the "concepts" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Busbee's Math Materials display tagshide tags

    This collection is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy Busbee

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Finding a Parabolic Function for any Three Points

Module by: Kenny M. Felder. E-mail the author

Summary: This module introduces the concept of modeling data with parabolic functions in Algebra.

Finding a Parabolic Function for any Three Points

Any two points are joined by a line. Any three points are joined by a vertical parabola.

Let’s start once again with the exceptions. Once again, if any two of the points are vertically aligned, then no function can join them. However, there is no an additional exception—if all three points lie on a line, then no parabola joins them. For instance, no parabola contains the three points (1,3), (2,5), and (5,11). In real life, of course, if we wanted to model those three points, we would be perfectly happy to use the line y = 2 x + 1 y=2x+1 instead of a parabola.

However, if three points are not vertically aligned and do not lie on a line, it is always possible to find a vertical parabola that joins them. The process is very similar to the process we used for a line, except that the starting equation is different.

Example 1: Finding a Vertical Parabola to Fit Three Points

Find a vertical parabola containing the points (-2,5), (–1,6), and (3,–10).

The problem. As with our example earlier, this problem could easily come from an attempt to find a function to model real-world data.

y = a x 2 + b x + c y=a x 2 +bx+c

This is the equation for any vertical parabola. Our job is to find a a, b b, and c c. Note that this starting point is the same for any problem with three points, just as any problem with two points starts out y = m x + b y=mx+b.

5 = a ( -2 ) 2 + b ( –2 ) + c 6 = a ( –1 ) 2 + b ( –1 ) + c –10 = a ( 3 ) 2 + b ( 3 ) + c 5 = a ( -2 ) 2 + b ( –2 ) + c 6 = a ( –1 ) 2 + b ( –1 ) + c –10 = a ( 3 ) 2 + b ( 3 ) + c

Each point represents an (x,y)(x,y) pair that must create a true equation in our function. Hence, we can plug each point in for xx and yy to find three equations that must be true. We can now solve for our 3 unknowns.

Rewrite the above three equations in a more standard form:

4 a 2 b + c = 5 a b + c = 6 9 a + 3 b + c = –10 4 a 2 b + c = 5 a b + c = 6 9 a + 3 b + c = –10

Uh-oh. Now what? In the linear example, we used elimination or substitution to solve for the two variables. How do we solve three? Oh, yeah. Matrices! Rewrite the above three equations as [ A ] [ X ] = [ B ] [A][X]=[B], where [ X ] = [ a b c ] [X]=[ a b c ] is what we want.

[ A ] = 421111931[A]=421111931 size 12{ left [ matrix { 4 {} # - 2 {} # 1 {} ## 1 {} # - 1 {} # 1 {} ## 9 {} # 3 {} # 1{} } right ]} {} [ B ] =5610[B]=5610 size 12{ left [ matrix { 5 {} ## 6 {} ## - "10" } right ]} {}

[ A ] –1 [ B ] = 125 [ A ] –1 [B]=125 size 12{ left [ matrix { - 1 {} ## - 2 {} ## 5 } right ]} {}

From the calculator, of course. Remember what this means! It means that a = –1 a=–1, b = –2 b=–2, and c = 5 c=5. We can now plug these into our original equation, y = a x 2 + b x + c y=a x 2 +bx+c.

y = x 2 2 x + 5 y= x 2 2x+5

So this is the equation we were looking for.

Did it work? Remember that we were looking for a parabola that contained the three points(–2,5), (–1,6), and (3,–10). If this parabola contains those three points, then our job is done. Let’s try the first point.

5 = ? - ( - 2 ) 2 - 2 ( - 2 ) + 5 5 = ? - ( - 2 ) 2 -2(-2)+5
(1)

5 = - 4 + 4 + 5 5=-4+4+5 graphics3.png

So the parabola does contain the point (-2,5). You can confirm for yourself that it also contains the other two points.

Finally, remember what this means! If we had measured some real-world phenomenon and found the three points (–2,5), (–1,6), and (3,–10), we would now suspect that the function y = x 2 2 x + 5 y= x 2 2x+5 might serve as a model for this phenomenon.

This model predicts that if we make a measurement at x = - 3 x=-3 we will find that y = 2 y=2. If we made such a measurement and it matched the prediction, we would gain greater confidence in our model. On the other hand, if the measurement was far off the prediction, we would have to rethink our model.

A surprising application: “secret sharing”

Bank vaults are commonly secured by a method called “secret sharing.” The goal of a secret sharing system runs something like this: if any three employees enter their secret codes at the same time, the vault will open. But any two employees together cannot open the vault.

Secret sharing is implemented as follows.

  • Choose a parabolic function—that is, choose the numbers aa, b b, and c c in the equation y = a x 2 + b x + c y=a x 2 +bx+c. This function is chosen at random, and is not programmed into the vault or given to any employee.
  • The actual number that will open the vault is the y-intercept of the parabola: that is, the y-value of the parabola when x = 0 x=0. This number is not given to any employee.
  • Each employee’s secret code is one point on the parabola.

When three employees enter their secret codes at the same time, the vault computer uses the three points to compute aa, b b, and c c for the parabola. As we have seen, this computation can be done quickly and easy using inverse matrices and matrix multiplication, both of which are easy algorithms to program into a computer. Once the computer has those three numbers, it computes the y-value when x = 0 x=0, and uses this number to open the vault.

Any three employees—that is, any three points—are enough to uniquely specify the parabola. But if you only have two points, you are no closer to the answer than when you started: the secret y value could still be, literally, any number at all.

Note also that the system is easily extendable. That is, if you want to say that four employees are required to open the vault, you just move up to a third-order polynomial, y = a x 3 + b x 2 + c + d y=a x 3 +b x 2 +c+d. The resulting equations—four equations with four unknowns—are just as easy, with matrices, as three were.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks