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Difference between revisions of "Parabola"

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A '''parabola''' is a type of [[conic section]].  A parabola is a [[locus]] of points that are equidistant from a point (the [[vertex]]) and a line (the [[directrix]]).
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A '''parabola''' is a type of [[conic section]].  A parabola is a [[locus]] of points that are equidistant from a point (the [[focus]]) and a line (the [[directrix]]).  
  
 
== Parabola Equations ==
 
== Parabola Equations ==
  
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There are several "standard" ways to write the equation of a parabola. The first is polynomial form: <math>y = a</math><math>x^2</math><math>+bx+c</math> where a, b, and c are constants. The second is completed square form, or <math>y=a(x-k)^2+c</math> where a, k, and c are constants. The third way is the conic section form, or <math>y^2</math><math>=4px</math> or <math>x^2=4py</math> where the p is a constant, and is the distance from the focus to the directrix.
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There are several "standard" ways to write the [[equation]] of a parabola. The first is [[polynomial]] form: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manipulating the polynomial.  
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 +
The second is [[completing the square | completed square]] form, or <math>y=a(x-h)^2+k</math> where a, h, and k are constants and the [[vertex]] is (h,k). This is very useful for graphing the quadratic because the vertex and stretching factor are immediately before you.  
 +
 
 +
The third way is the conic section form, or <math>y^2</math><math>=4px</math> or <math>x^2=4py</math> where the p is a constant, and is the distance from the focus to the vertex.
 +
 
 +
==Graphing Parabolas==
 +
[[Image:Parabola.png|thumb|right|150px|150px|The graph of <math>y=x^2</math>]]
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Using the completed square form, <math>y - k = a(x - h)^2</math> or <math>x - h = a(y - k)^2</math>, the vertex of the graph is at the point <math>(h, k)</math>. The graph appears vertically if the <math>x</math> term is squared, and horizontal if the <math>y</math> term is squared. The graph will be oriented (opens up) upwards/right if <math>a</math> is positive, and will be downwards/left if <math>a</math> is negative.
 +
Here are the graphs of a few parabolas:
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<center> <math>3x^2-14x+8</math> (<math>a</math> is positive)
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<asy>
 +
import graph;
 +
size(300);
 +
Label f;
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f.p=fontsize(6);
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xaxis(-9,9,Ticks(f, 1.0));
 +
yaxis(-9,9,Ticks(f, 1.0));
 +
real f(real x)
 +
{
 +
return 3x^2-14x+8;
 +
}
 +
draw(graph(f,(7+2*sqrt(13))/3,(7-2*sqrt(13))/3),red+linewidth(1));
 +
</asy></center>
 +
<center><math>-x^2+2x+2</math> (<math>a</math> is negative)
 +
<asy>
 +
import graph;
 +
size(300);
 +
Label f;
 +
f.p=fontsize(6);
 +
xaxis(-8,8,Ticks(f, 2.0));
 +
yaxis(-8,8,Ticks(f, 2.0));
 +
real f(real x)
 +
{
 +
return -x^2+2x+2;
 +
}
 +
draw(graph(f,1-sqrt(11),1+sqrt(11)),green+linewidth(1));
 +
</asy></center>
 +
<br /><br /><br /><br /><br /><br /><br /><br /><br />
 +
 
 +
 
 +
==Video Description==
 +
https://youtu.be/Res-cddhRLw?si=a2XGd_hkEArrwOVG
 +
 
 +
~MathProblemSolvingSkills.com
 +
 
 +
 
 +
 
 +
==Problems==
 +
=== Introductory ===
 +
#A [[parabola]] with equation <math>y=x^2+bx+c</math> passes through the points (2,3) and (4,3).  What is <math>c</math>?<br><br><math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math><div style="text-align:right;">([[2006 AMC 10A Problems/Problem 8|2006 AMC 10A, Problem 8]])</div>
 +
 
 +
=== Intermediate ===
 +
Suppose that a parabola has vertex <math>\left(\dfrac{1}{4},-\frac{9}{8}\right)</math> and equation <math>y=ax^2+bx+c</math>, where <math>a>0</math> and <math>a+b+c</math> is an integer.  The minimum possible value of <math>a</math> can be written in the form <math>\dfrac{p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers.  Find <math>p+q</math>. <div style="text-align:right;">([[2011 AIME I Problems/Problem 6|2011 AIME I, Problem 6]])</div>
 +
 
 +
=== Olympiad ===
 +
Find the area of the largest triangle <math>ABC</math> [and prove this is the maximum] whose interior is entirely within the region bounded by <math>y=\sqrt{3}x-1</math> and <math>y=3x^2-12x+1</math>.
 +
 
 +
== See also==
 +
*[[Hyperbola]]
 +
*[[Circle]]
 +
*[[Ellipse]]

Latest revision as of 15:22, 13 September 2023

A parabola is a type of conic section. A parabola is a locus of points that are equidistant from a point (the focus) and a line (the directrix).

Parabola Equations

There are several "standard" ways to write the equation of a parabola. The first is polynomial form: $y = a{x}^2+b{x}+c$ where a, b, and c are constants. This is useful for manipulating the polynomial.

The second is completed square form, or $y=a(x-h)^2+k$ where a, h, and k are constants and the vertex is (h,k). This is very useful for graphing the quadratic because the vertex and stretching factor are immediately before you.

The third way is the conic section form, or $y^2$$=4px$ or $x^2=4py$ where the p is a constant, and is the distance from the focus to the vertex.

Graphing Parabolas

The graph of $y=x^2$

Using the completed square form, $y - k = a(x - h)^2$ or $x - h = a(y - k)^2$, the vertex of the graph is at the point $(h, k)$. The graph appears vertically if the $x$ term is squared, and horizontal if the $y$ term is squared. The graph will be oriented (opens up) upwards/right if $a$ is positive, and will be downwards/left if $a$ is negative. Here are the graphs of a few parabolas:

$3x^2-14x+8$ ($a$ is positive) [asy] import graph; size(300); Label f; f.p=fontsize(6); xaxis(-9,9,Ticks(f, 1.0)); yaxis(-9,9,Ticks(f, 1.0)); real f(real x) { return 3x^2-14x+8; } draw(graph(f,(7+2*sqrt(13))/3,(7-2*sqrt(13))/3),red+linewidth(1)); [/asy]
$-x^2+2x+2$ ($a$ is negative) [asy] import graph; size(300); Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 2.0)); yaxis(-8,8,Ticks(f, 2.0)); real f(real x) { return -x^2+2x+2; } draw(graph(f,1-sqrt(11),1+sqrt(11)),green+linewidth(1)); [/asy]











Video Description

https://youtu.be/Res-cddhRLw?si=a2XGd_hkEArrwOVG

~MathProblemSolvingSkills.com


Problems

Introductory

  1. A parabola with equation $y=x^2+bx+c$ passes through the points (2,3) and (4,3). What is $c$?

    $\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$

Intermediate

Suppose that a parabola has vertex $\left(\dfrac{1}{4},-\frac{9}{8}\right)$ and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written in the form $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Olympiad

Find the area of the largest triangle $ABC$ [and prove this is the maximum] whose interior is entirely within the region bounded by $y=\sqrt{3}x-1$ and $y=3x^2-12x+1$.

See also

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