Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Parabola"

m (→‎Graphing Parabolas)
m
 
(6 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
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]]).
+
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 ==
Line 11: Line 11:
 
==Graphing Parabolas==  
 
==Graphing Parabolas==  
 
[[Image:Parabola.png|thumb|right|150px|150px|The graph of <math>y=x^2</math>]]
 
[[Image:Parabola.png|thumb|right|150px|150px|The graph of <math>y=x^2</math>]]
−
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 />
+
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)
 +
<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></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==
 
==Problems==

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Parabola&oldid=198166"