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]]). | + | 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 10: | Line 10: | ||
==Graphing Parabolas== | ==Graphing Parabolas== | ||
− | [[Image:Parabola.png|thumb|right| | + | [[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. | 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== | ||
=== Introductory === | === Introductory === | ||
− | #A [[parabola]] with equation <math> | + | #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 === | === 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 === | === 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== | ||
Line 24: | Line 65: | ||
*[[Circle]] | *[[Circle]] | ||
*[[Ellipse]] | *[[Ellipse]] | ||
− | |||
− |
Latest revision as of 16:22, 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).
Contents
Parabola Equations
There are several "standard" ways to write the equation of a parabola. The first is polynomial form: where a, b, and c are constants. This is useful for manipulating the polynomial.
The second is completed square form, or 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 or
where the p is a constant, and is the distance from the focus to the vertex.
Graphing Parabolas
Using the completed square form, or
, the vertex of the graph is at the point
. The graph appears vertically if the
term is squared, and horizontal if the
term is squared. The graph will be oriented (opens up) upwards/right if
is positive, and will be downwards/left if
is negative.
Here are the graphs of a few parabolas:
![$3x^2-14x+8$](http://latex.artofproblemsolving.com/d/d/a/dda1c136b9dc5d5fb5a7dbf5b6ae246bc68b3c1e.png)
![$a$](http://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png)
![[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]](http://latex.artofproblemsolving.com/e/f/0/ef05cddb62e9f4a3185f556405ec61a88c1c91b7.png)
![$-x^2+2x+2$](http://latex.artofproblemsolving.com/c/e/a/ceaad52176efe9df90583ed9f5c7d51661fe5ead.png)
![$a$](http://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png)
![[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]](http://latex.artofproblemsolving.com/c/0/d/c0d9ba587243d9b1155bf2e547df7685ebabd990.png)
Video Description
https://youtu.be/Res-cddhRLw?si=a2XGd_hkEArrwOVG
~MathProblemSolvingSkills.com
Problems
Introductory
- A parabola with equation
passes through the points (2,3) and (4,3). What is
?
Intermediate
Suppose that a parabola has vertex and equation
, where
and
is an integer. The minimum possible value of
can be written in the form
where
and
are relatively prime positive integers. Find
.
Olympiad
Find the area of the largest triangle [and prove this is the maximum] whose interior is entirely within the region bounded by
and
.