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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 [[focus]]) 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 ==
 +
{{Asy image|1=<asy>
 +
import graph;
 +
size(100); // Square canvas
 +
Label f;
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f.p = fontsize(6);
 +
 +
// Axes: from -8 to 8 with tick steps of 2
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xaxis(-8, 8, Ticks(f, 2.0));
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yaxis(-8, 8, Ticks(f, 2.0));
 +
 +
// Define the function
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real f(real x) { return x^2; }
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// Draw from x = -3 to x = 3 (main visible part of the parabola)
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draw(graph(f, -3, 3), blue+linewidth(1));
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</asy>|2=right|3=Graph of <math>y=x^2</math>}}
  
 
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.  
 
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 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.
 
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.
  
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==Graphing Parabolas==  
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== Graphing Parabolas ==  
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[[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.<br /><br /><br /><br /><br /><br /><br /><br /><br />
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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:
 +
<center> <math>3x^2-14x+8</math> (<math>a</math> is positive)
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<asy>
 +
import graph;
 +
size(300);
 +
Label f;
 +
f.p=fontsize(6);
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xaxis(-9,9,Ticks(f, 1.0));
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yaxis(-9,9,Ticks(f, 1.0));
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real f(real x)
 +
{
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return 3x^2-14x+8;
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}
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draw(graph(f,(7+2*sqrt(13))/3,(7-2*sqrt(13))/3),red+linewidth(1));
 +
</asy></center>
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<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)
 +
{
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return -x^2+2x+2;
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}
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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==
 
==Problems==
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=== Intermediate ===
 
=== Intermediate ===
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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>
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#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 ===
 
=== 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==
 
== See also==
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*[[Circle]]
 
*[[Circle]]
 
*[[Ellipse]]
 
*[[Ellipse]]
 +
 +
[[Category:Geometry]]

Latest revision as of 19:24, 22 June 2025

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

[asy] import graph; size(100); // Square canvas Label f; f.p = fontsize(6); // Axes: from -8 to 8 with tick steps of 2 xaxis(-8, 8, Ticks(f, 2.0)); yaxis(-8, 8, Ticks(f, 2.0)); // Define the function real f(real x) { return x^2; } // Draw from x = -3 to x = 3 (main visible part of the parabola) draw(graph(f, -3, 3), blue+linewidth(1)); [/asy]

Enlarge.png
Graph of $y=x^2$

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

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

  1. 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

  1. 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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