Parabola Definition and 363 Threads

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.
Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.
The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.

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  1. C

    How Do You Find the Y-Coordinate of a Parabola's Vertex?

    Homework Statement Sorry for such an elementary question, but I'm struggling thru Int Alg. Today's nightmare involves a downward parabola. An arch supporting a bridge has the equation y=-0.0022x^2 + 1.578x + 0. Position of left side parabolic arch is (0,0). Using x = -b/2a, I calculated the...
  2. C

    Can Geometry and Algebra Explain a Triangle's Romance with a Parabola?

    Homework Statement Here's the problem: refer to question no. 6 http://img1.uploadscreenshot.com/images/orig/5/12314183569-orig.jpg" Homework Equations I am not sure :S The Attempt at a Solution This, honestly, is a question which I have no clue how to go about... Thanks...
  3. A

    Finding a function for the parabola

    Note: I don't need any answer, all I want to know is whether this question is possible. Homework Statement "What is the function of the parabola which has the points (1, 1) (2, 2) and (3, 3)?" I just asked my teacher to get the question, It's not stated in my textbook. Homework...
  4. r-soy

    Quetion to find the equation of a parabola i want check

    Hi all Use the definition of a parabola and the distance formula to find the equation of a parabola with a ) directix x = -4 and focus (2,2 ) B ) directix x = 2 and focus (6,-4 ) my answer :
  5. r-soy

    Use the definition of a parabola and the distance

    Hi all Use the definition of a parabola and the distance formula to find the equation of a parabola with a ) directix x = -4 and focus (2,2 ) B ) directix x = 2 and focus (6,-4 ) How i solve like this queation please hle me the steps to solve that thanks
  6. r-soy

    Find the equaion of the parabola

    Find the equaion of the parabola having its vertex at the origin , its axis as indicated : A ) x-axis : (-4, - 20 ) b) y - axis : (30,-15 ) C) y - axis : (-root2,3) my answer A ) we have (-4, - 20 ) that mean the focus ( - 20 , 0 ) now we do comper 4ay...
  7. T

    Solve Absolute Parabola: Sketch |x^2 - 2x - 3| Graph

    Homework Statement Sketch the graph of |x^2 - 2x - 3| Homework Equations None The Attempt at a Solution |x^2 - 2x - 3| = { x^2 - 2x + 3 x^2 - 2x - 3 >= 0 x >= 3 OR x >= -1 {-x^2 + 2x + 3 x^2 - 2x - 3 < 0 x < 3 OR x < -1 if x < -1 then y = -x^2 + 2x + 3 if -1...
  8. R

    Archimedes' rule that area of a parabola is 4/3 times an inscribed triangle

    Homework Statement Let ABC be a piece of a parabola. The point B is chosen such that the tangent to the parabola at B is parallel to the line AC. Archimedes proved the Area of parabola inside ABC is 4/3 times the triangle ABC. Prove this using calculus Homework Equations The integral...
  9. J

    Parabola for 18 aluminum mirror

    Parabola for 18" aluminum mirror Help; need to configure inputs into parabola formula to machine an 18" dia., 1.5"h aluminum disc. for use as primary objective mirror in telescope.
  10. L

    Find equation of parabola with focus and directrix

    Homework Statement Find the equation of the parabola with focus (0, -4) and directrix y = -2 i barely understand this thing. i mean how do i find the vertex. the book says D1 = focus to any point on the graph, and D2= the point on the graph to the directrix. and D1 = D2. how do i graph if...
  11. T

    Solve Parabola Equation: (10,0) ((13,27) (16,0)

    Homework Statement What is the equation of this parabola: (10,0) ((13,27) (16,0) Homework Equations IDK what this is please help The Attempt at a Solution 0=a(10)^2+b(10)+c 0=100a+10b+c -100a-10b=c I'm stuck please help
  12. L

    Mass of chain in shape of parabola

    Homework Statement A chain in the shape of y = x^{2} between x = -1 and x = 1, has density of |x|. Find M, and CM. Homework Equations The Attempt at a Solution \int^{1}_{-1}|x|dx = \int^{0}_{-1}-xdx + \int^{1}_{0}xdx = 1 I got this far and realized that I did nothing with...
  13. L

    Find Equation of Parabola Given Zero and Point

    stupid question, but if i know a zero of a parabola and a point it goes through, how do i find it's equation?
  14. A

    Checking whether a curve is a parabola or not

    Hi everyone, I'm generating some hypar surfaces for a structure through parametric design techniques. In order to assess their structural abilities, I am finding their cross section curves and would like to check whether they are parabolas or not. So, in 3d space, (x,y,z) I have a curve - I...
  15. M

    Parabola General to standard form

    Homework Statement write the standard form of the equation x^2-4x+8y+12=0 find the focus Homework Equations The Attempt at a Solution i have it down to (x-2)^2=-8(y-8) i'm not sure if you have to multiply the -8 on the right side of the equations by 4 since the focus is 1/4a
  16. M

    Velocity vector along a parabola

    Hello, I'm new here and wasn't sure if this should be put into the homework section. It's not a homework question but the nature of the problem is homework-ish in nature I suppose. Anyway I'm trying to understand why a velocity vector along a parabola would have the same initial velocity as...
  17. G

    Apostol's Archimedes area proof for a parabola

    Homework Statement My question is on the last bit of Apostol's proof, in his book Calculus Vol I 2nd ed, where he shows that the area under the parabola = b^{3}/3 where b is the base of the rectangle enclosing the parabola. The bit I am confused about is where his contradiction...
  18. C

    Find the equation of the tangent of the parabola y^2= 4px

    hello everyone, :confused:i'm having can't seem to solve these two questions... 1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact? 2) Find the equation of the tangent to the parabola Y^2= 10x, at the...
  19. D

    Solving for Acceleration on a Parabolic Path | Physics Tutorial

    Hi everybody we know that if we have an object sliding on a frictionless ramp, the acceleration force will be constant, and it equals to a=g * sin(theta) where theta is the ramp angle w.r.t. the ground so the path of motion in this problem can be written mathematically as a...
  20. J

    Small Oscillations on a Parabola problem.

    Homework Statement Find the frequency of oscillations of a particle (mass m) which is free to move along the parabola y= -ax^2 + 2ax - a, and is attached to an ideal spring whose other end is fixed at (1,l) A force F is required to extend the spring to length l. a can be any real number...
  21. J

    What is the equation of a parabola?

    Homework Statement i am given the equation of a parabola to be y=2x^2-2x+3 and asked to sketch the parabola Homework Equations y=1/2(l-p)(x-k)^2+(l+p)/2 (l+p)/2=h vertex is at (k,h) equation of the directrix is y=p distance[(k,h) to y=p]=distance[(x,y) to (k,h)] The Attempt...
  22. Mentallic

    Focus of a Parabola Q6ciii: Australian HSC Mathematics Extension 2 Exam

    Homework Statement This problem is from the Australian HSC mathematics extension 2 exam. Q6ciii) It states: Find the focus, S, of the parabola y^2=r^2+c^2-2cx where r and c are constants. The Attempt at a Solution I couldn't figure out how to convert this into the parabola focus...
  23. M

    Parabola, tangebt line and Normal Line intersect

    Homework Statement Where does the normal line to the parabola y= x - x^2 at the point (1,0) intersect the parabola a second time? Illustrate with a sketch Homework Equations y= x - x^2 The Attempt at a Solution y' = 1-2x slope of tangent = -1 (after substituting value of x in...
  24. S

    Point on a parabola where the tangent intersects a point

    1. A car is traveling on a highway shaped like a parabola with its vertex at the origin. The car begins at a point 100m west and 100m north of the origin and is traveling easterly. There is a statue 100m east and 50m north of the origin. At what point on the highway will the car`s headlights...
  25. L

    Proof of a focus point on parabola and tangent line equal angles

    Homework Statement Prove that a vertical line and a line going from a point on a parabola to the focus of the parabola form equal angles with the tangent line of the point on the parabola. Homework Equations Focus = 1/4a (maybe relevant) The Attempt at a Solution I know how to...
  26. Y

    What Is the General Equation of a Parabola with Directrix x=p and Focus (h, k)?

    For a parabola whose Directrix is given by the equation x=p and whose Focus is (h,k). Is this by any chance the correct general form of the parabola? x=1/2(h-p) [y^2 - 2yk + h^2+k^2-p^2]
  27. S

    Rotating a Parabola: 30o Anti-Clockwise About Origin

    The parabola y = x2 is rotated anti-clockwise about the origin through the angle 30o. What is the new equation of the new curve (relative to the standard basis)?
  28. Z

    Parabola (Horizontal) - finding vertex, focus and equation of line

    Homework Statement Use completing the square method to rewrite the equation of the parabola y^2 – 4y – 44 = 16x in the form (y-y0)^2 = 4A(x-x0) Hence find: a) the coordinates of the vertex b) the coordinates of the focus c) the equation of the line that passes through the focus...
  29. S

    ANOTHER parabola related question ._.

    Homework Statement a cannon launches a projectile. H, is the height in metres that the cannon ball is above the horizon by t seconds. h= -0.75t^2 + 16t + 3 Homework Equations What is the maximum height reached by the cannonball? The Attempt at a Solution Ok so i tried completing...
  30. D

    Intersection of a parabola with another curve

    Homework Statement For a any parabola with the equation y=kx^{2} I'm trying to find a curve that intersect every point of the parabola at right angles. Homework Equations For a perpendicular intersection the slope is -\frac{1}{m}The Attempt at a Solution I took the derivative and then took...
  31. S

    Is a Paper Strip Contracted on a Table a Parabola?

    When you take a paper strip, put it on the flat surface (table), and contract it so it stands up - is it parabola or something else?
  32. J

    Distance from vertex to focus in a parabola

    Homework Statement I am a bit rusty on parabolae. I am doing a question on projectiles and have found the coordinates of the vertex of a parabola as: (\frac{(v_0)^2\sin\alpha\cos\alpha}{g},\frac{(v_0)^2(\sin^2\alpha)}{2g}) The question now requires you to show that the distance...
  33. W

    Finding the function of a Parabola

    1. The graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all number less than or equal to 4. Determine an expression for the function. 2. none 3. I do not know how to use the intercepts and the range and manipulate it into a function...
  34. S

    Draw parabola using these materials

    You have a flat wooden surface, some rope, some nails and a pencil - how do you draw parabola using just that?
  35. D

    What Is the General Equation of a Parabola and Its Derivation?

    I read a Wikipedia article titled "Parabola" that listed an equation of the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 How is this derived? or where did it come from?
  36. S

    Circle inscribed in a parabola

    Homework Statement A circle with radius 1 inscribed in the parabola y=x^2. Find the center of the circle.Homework Equations equation of the parabola: y=x^2 circle: (x-h)^2 + (y-k)^2 = 1 The Attempt at a Solution After ghosting the forums and reading through every post with this exact same...
  37. M

    Determining the equation of an ellipse from its intersection with a parabola

    Homework Statement The vertex of the parabola y^2=2px is the center of an ellipse. The focus of the parabola is an end of one of the principle axes of the ellipse, and the parabola and ellipse intersect at right angles. Find the equation of the ellipse. Homework Equations...
  38. R

    Exploring Limits at (0,0) on a Parabola

    Homework Statement http://img40.imageshack.us/img40/39/20688555.gif a) Show that if C is any straight line through (0,0) then \lim_{(x,y) \to (0,0)} along C exists and equals 1. b) Show that the limit as (x,y) -> 0 doesn't exist. Homework Equations The Attempt at a...
  39. D

    Where is the vertex of the parabola f(x) = x^2 - 5x + 6?

    See, there is an equation which is f(x) = x^2 - 5x + 6. On the points 2 and 3 of the x axis, they reach 0. So where is the vertex? I'm guessing its -.25, but just to make sure i had to ask. Well, our class seemed to have a lot of it and our teacher, Ms. Knudsvig, didn't know.
  40. C

    Find the equation of a parabola

    Find the equation of a parabola with the following characteristics: range Y <= 8 x-coordinate of the turning point is -4 y-intercept = -6 I have tried to substitute all the information into y = a(x-p)^2 + q which gives me y = a(x+4)^2 + q and substituted the y-intercept into the...
  41. U

    Transformation that takes all point on parabola onto unit circle

    Homework Statement Show that the transformation _ __ _ __ | 0 -2 1 || x | | -2 2 0 || y | |_2 -2 1 _||_ 1_| takes all points on parabola y2=x onto the unit circle x2+y2=1 Homework Equations The Attempt at a Solution I can't find out what to do I just...
  42. B

    Parametrizing and Line integrals (of a line, parabola, curve.)

    Homework Statement In each part, evaluate the integral \int(3x+2y)dx+(2x-y)dy (A) The line segment from (0,0) to (1,1). (b) The parabolic arc y=x^2 (c) The curve y=sin(pi(x)/2) from (0,0) to (1,1) (D) The curve x=y^3 from (0,0) to (1,1). Homework Equations \int...
  43. K

    Shortest Distance on a Parabola: Point to Point Calculation

    2. Find the shortest distance from the point (1,4) to a point on the parabola y^2 = 2x Not really sure what to do here next. I'd imagine i might have to fuse implicit differentiation? But not really sure.
  44. S

    What Is the Equation of a Parabola for a Bridge Needing Specific Dimensions?

    Homework Statement 1. It is a word problem I need done by tomorrow morning. so please help: A new bridge is being constructed. The space between the support needs to be 1050 feet; the height at the center of the arch needs to be 350 feet. An empty tanker needs a 280 foot clearance to pass...
  45. P

    What Part of a Ball Hitting a Parabola Hits the Focal Point?

    1" ball hitting Parabola If a 1" ball hits the inside of a parabola, what part of the ball hits the focal point? Is it the edge that strikes the inside of the parabola or the center of the ball? Thank You
  46. J

    Standard form of the equation of parabola

    Homework Statement Write the equation in standard form for the parabola with an axis of symmetry y=0 and focus(-5,0). Homework Equations I think ax^2+bx+c. Also maybe x = -b/2a. But I don't know how to apply these. The Attempt at a Solution I know that focus and vertex lie on...
  47. turin

    Transforming parabola to straight line

    To the moderator: This isn't a HW question, but it probably sounds like one, so I appologize. Please move this to the HW forum if need be. I have an integration domain inside three intersecting curves. Two of the curves are straightlines and the third is a parabola. These three boundaries...
  48. P

    Tangents of a parabola perpendicular to a line

    A Tangent to the parabola y = 3x^2 - 7x + 5 is perpendicular to x + 5y - 10 = 0 Determine the equation of the tangent i really don't know where to start!
  49. E

    Projectile Motion parabola effect

    Homework Statement I'm still kind of confused with projectile motion. I know when you throw something, you throw it at an angle and will have a negative parabola effect. An exception to when you throw something up at a 90° angle where it will go up vertically and come down vertically, with no...
  50. I

    Vertical distance between two parabola

    Homework Statement Write a function for d(x), the vertical distance between the two curves, and find the minimum value of d(x). Homework Equations The equation for parabola one is y = x^2 + 6, for parabola two, y = -(x-2)^2 + 6 The Attempt at a Solution The answer in the back...
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