Difference between revisions of "Parabola"

Revision as of 14:25, 21 February 2018

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 other parabolas:

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

$-x^2+2x+2$ $[asy] import graph; size(300); Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 2.0)); yaxis(-8,8,Ticks(f, 2.0)); real f(real x) { return -x^2+2x+2; } draw(graph(f,1-sqrt(11),1+sqrt(11)),green+linewidth(1)); [/asy]$

Problems

Introductory

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

$\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$
(2006 AMC 10A, Problem 8)

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 $p+q$ .

(2011 AIME I, Problem 6)

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 $y=3x^2-12x+1$ .

@@ Line 13: / Line 13: @@
 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 other parabolas:
-<math>3x^2-14x+8</math>
+<center> <math>3x^2-14x+8</math> (<math>a</math> is positive)
 <asy>
 import graph;
@@ Line 26: / Line 26: @@
 }
 draw(graph(f,(7+2*sqrt(13))/3,(7-2*sqrt(13))/3),red+linewidth(1));
-</asy>
+</asy></center>
 <math>-x^2+2x+2</math>
 <asy>

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