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. J

    Focus of parabola, checking my answer.

    I just had this question in an online test I took, and first I had 16ft, but for some unknown reason I chose to change it to 64ft right before I submitted the test, and of course I got it wrong. So I just want to confirm the answer is indeed 16ft.Homework Statement A parabolic antenna has a...
  2. L

    Shifting a parabola vs changing slope of a line

    Consider the graphs of two equations: [y = x] and [y = x^2] One is a line and the other is a parabola. If I include a 10x into each formula, to make: [y = 10x] and [y = 10x + x^2] then the affect it will have on the line is increase its slope. But the affect it will have on...
  3. V

    Area between a line and a parabola

    Homework Statement Find the area enclosed by the line y = x-1 and the parabola y^2 = 2x+6 Homework Equations I don't think there are any special formulas or anything The Attempt at a Solution Well, I graphed the two given equations as √2x+6 and x-1 and got intersects at...
  4. T

    Attempting to spin-cast parabola

    I am trying to make a mold of a deep parabola, by putting plaster of paris in a spinning flower pot. I've got another thread going as well. So far I've tried using and making all sorts of spinning bases, however inconsistencies in the rotation of the pot have to be almost invisible for the...
  5. A

    Finding the equation of a parabola with only 2 x-intercepts and the vertex

    1. Write an equation of the quadratic function f whose graph has x-intercepts 3 and 7 and f(5) = 8. 2. I really don't know where to go from here :frown: 3. obviously, is the x-int. are 3 and 7 that means they are points (3,0) and (7,0). Also, f(5) = 8 becomes the point (5,8) which i have found...
  6. T

    Find a point on the parabola with given point and distance

    Homework Statement Find a point on the parabola y = x^2 whose distance from (18,0) is 4*sqrt(17)Homework Equations I'm guessing I need to use the distance formula, but I'm not entirely sure. D = sqrt( (x2 - x1)^2 + (y2 - y1)^2) The Attempt at a Solution Well I really don't think I'm doing...
  7. Vorde

    Determining distance from Focus to Point on Parabola

    Hi, As part of my portfolio for my end of semester math classes, I am trying to develop the equation for a parabola at an arbitrary rotation. I have seen it done based off of substituting X and Y variables with trig terms, and I am confident I could prove it that way, but I am curious if I...
  8. G

    Parameterize part of a Parabola

    Homework Statement Find a parametric equation for a part of a parabola. Given: y=-2x2 initial point: (-2,-8) terminal point: (1,-2) Homework Equations x(t)=a+t(c-a) y(t)=b+t(d-b) The Attempt at a Solution x(t)=-2+t(1-(-2)) =3t-2 y(t)=-8+t(-2-(-8)) =6t-8...
  9. A

    How Do You Correctly Derive the Equation of a Parabola with a Horizontal Axis?

    Hi! I am trying to find the equation of a parabola with vertex as (h,k) and axis parallel to the x-axis. However, I am not able to derive the correct result. (1) I shift the origin to the point (h,k). (2) Now the equation of the parabola in the new system becomes y^{2} = 4ax. (3) Now, we know...
  10. F

    Infinite line intersecting parabola only once

    I think I know the answer to this problem however I am not 100% sure. I must describe the conditions in which an infinite continuous line (ax+by=c) will only cross a parabola once. I believe the only answer is a straight, vertical line because while the slope of the parabola may approach...
  11. M

    Derivation of Effective Mass from a Parabola

    There is some fundamental about effective mass that I am misunderstanding about effective masses. I understand the relation E\left(k\right) = E_0 + \frac{1}{2m^*}k^2 But I'm not sure when it's appropriate to fit this to a parabola. I would have thought the fitting is only done when...
  12. K

    Finding the Centroid of a Parabola: Calculating Coordinates for y = 10m

    The equation y = x2 describes a parabola. Find the x and y coordinates of the centroid of the area bounded by this curve, and the line y = 10m
  13. O

    Finding the slope of line tangent to a parabola

    Homework Statement Find the equations of both lines through the point (2,-3) that are tangent to the parabola y=(x^2)+x Homework Equations The Attempt at a Solution Took the derivative and got a slope of 5 and the slope of the normal line being -1/5, but the answer was marked...
  14. S

    Finding parametric equation of the line tangent to the parabola

    Homework Statement Find a parametric equation of the line that satisfies the condition: The line that is tangent to the parabola y=x^2 at the point (-2,4) The Attempt at a Solution My answer came out to <x,y> = <-2,4> + t<1,2>
  15. rude man

    Feynman exercises bead parabola accelerometer

    In the subject accelerometer (see link) the answer given is a = kgx. I get a = 2kgx. I'm wondering if anyone cares to discuss this problem. The answerer used an energy argument that I'm not sure I followed, whereas I used a simpler force-balance method. F = force on bead by parabola and is...
  16. S

    Making the equation of a parabola

    Homework Statement So i have a parabola graphed, i need to express y in terms of X. The vertex is 2,10 and the roots are 0,0 and 4,0. Homework Equations ax^2+bx+c=y quadratic The Attempt at a Solution I thought of working the quadratic in a reverse process, but it doesn't...
  17. L

    How Do You Fit 3D Coordinate Data to a Parabola?

    Homework Statement I'm trying to go about fitting some (x,y,z) coordinate data that I got out of a simulation to a parabola, but I'm not entirely sure how. Is there a general equation for a parabola that instead of having usual (x,y) coordinates has (x,y,z) coordinates? Homework Equations...
  18. N

    What is the solution to this parabola word problem?

    Hi all. I'm a nurse in the middle of a nightmare math class. It is second half of college algebra but it may as well be Greek. The professor is a PhD in engineering and is teaching way over my head- and others from what I hear. I have never failed a class in my life but this may be a first...
  19. V

    Convention in parametric equation of parabola.

    general equation of parabola is y^2=4*a*x. it's parametric equation is ((a*t^2),(2*a*t)) [as in my book] but i think there can be other kind of parametric equations also like( ((t^2)/4*a),t) it defines a parabola easily. is using ((a*t^2),(2*a*t)) as parametric equation of parabola is convention...
  20. S

    Projectile motion parabola shape

    why the projectile motion's graph distance against inclination angle is parabola shape? ...thank you
  21. M

    Tricky Oblique Parabola Question?

    Please help with this! I cannot seem to figure it out. A parabola has a focus (1,1) and a vertex (4,5). Find the equation of the directrix, the endpoints of the latus rectum, and the equation in standard form. I have plotted the points but I have no idea how to get the equation because...
  22. T

    Find the equation of a parabola with a given focus and vertex.

    Homework Statement Find the equation of a parabola with a vertex of (1,-3) and focus of (1,-1)? Homework Equations The Attempt at a Solution
  23. 0

    Partition for the equivalence relation of a parabola

    Homework Statement Let f: R -> R, x -> x^2 What does the partition for the equivalence relation of this function look like? Homework Equations The Attempt at a Solution Uh...I have no idea. Sorry, the book only has examples of like integers from modulo n, if anybody could...
  24. L

    Suspension Bridge Parabola Problem

    Please help me. I can't figure out how to solve this problem. The cable of a suspension bridge hangs in the form of a parabola when the load is evenly distributed horizontally. The distance between the two towers is 150m, the points of support of the cable on the towers are 22m above the...
  25. T

    Epsilon-Delta limits, *not* for a straight line. Parabola example

    So the general tactic for straight lines: f(x) = 2x Show by epsilon-delta definition of limit, as lim x->2, f(x) tends to 4. let, ε>0, and choose 0<δ<ε/2 (|x - 2|<δ)→ |f(x) - 4| = |2x - 4| = 2|x - 2| < 2δ < ε No problem, but what about for a parabola? g(x) = ax2 for some a in R. Show as lim x...
  26. AlexChandler

    Bead Parabola Accelerometer

    Homework Statement An accelerometer is made of a piece of wire with a bead on it that can slide on the wire with no friction. The wire is formed as a parabola y = kx2, as shown in the drawing. The bead rests at the lowest point of the parabola when it is at rest. When accelerated parallel to...
  27. Z

    Reaction force on a semi-cubical parabola

    Homework Statement A particle of mass m moves without friction along a semi-cubical parabolic curve given by y2=4ax3 with a constant speed v. The reaction force of the curve on the particle when it is at the point (x = 1, y = 2) is given approximately by The Attempt at a Solution...
  28. W

    Radio telescope parabola question

    If a particular radio telescope is 100ft in diameter and has a cross section modeled by the equation x^2=167y, how deep is the parabolic dish? What is the location of the focus? can someone show me some steps to solving this? I have (167/4,0) as the focus for the second part but I am not...
  29. M

    Parabola Midpoint: Find Equation of Locus

    Homework Statement A variable tangent at P to the parabola y2=4ax meets the y-axis at Q. Find the equation of the locus of the midpoint of PQ. As shown below, my final answer is 2y2=9ax but the answer provided is y2=9ax. Can anyone correct me? Homework Equations The Attempt at a Solution
  30. B

    Linear algebra with with unknown coefficientt for parabola

    Homework Statement Find the numbers a0, a1 and a2 so that the parabola with the equation: y = a0 + a1 x + a2 x2 Goes through the points: (2, −3), (9, 4), and (t, 4) For any value of t, specify the amount of solutions Homework Equations y = a0 + a1 x + a2 x2 The Attempt...
  31. S

    Finding parabola equation and only given a graph

    Homework Statement The math problem I am supposed to do is, find an equation for the given graph. Homework Equations There are no equations given, but here's an image for it. The Attempt at a Solution I understand there is an equation like y=ax^2+bx+c but I'm not sure how I can...
  32. S

    Area between a parabola and line?

    Homework Statement Find the area of the region enclosed by the curves: 2y=sqrt(4x) y=5 and 2y+3x=7 Homework Equations A = integral from a to b of f(x)-g(x) dx The Attempt at a Solution Tried to integrate this with respect to y. Found the intersection points to be y=2...
  33. U

    Proof that this system is tangent to a parabola

    Homework Statement Well I would like to prove that any equation that follows the pattern y=rx-r^(-1) is tangent to some sideways parabola (I know this to be true). Problem is that I need help in finding the parabola in question and actually proving my conjecture. I do know, after graphing...
  34. Char. Limit

    Line Integral along a Parabola

    Homework Statement a. Find a parametric equation to describe a parabola from the point (1,1) to the point (2,4). b. Evaluate the line integral \int_C x ds along the parabolic segment in part a. Homework Equations \int_C x ds = \int_{t1}^{t2} x(t) |r'(t)| dt The Attempt at a Solution Well...
  35. D

    Is It a Parabola or an Ellipse?

    Parabola or an ellipse?? Homework Statement I have a curved structure which looks like this: http://imageupload.org/pt-1412920046522.html I haven't been told specifically what it is. I assumed it to be ½ellipse. It has a height of 36m. So, I also found out the equation y=...
  36. R

    Find the equation of the parabola

    Find the equation of the parabola (in standard form) whose vertex is at the origin, has a vertical axis, and passes through the point (-2,5). Thanks y = ax^2 sub in the point (-2, 5) and solve for a 5 = a(-2)^2 5 = a(4) a = 5/4 Final equation is y = (5/4)x^2. Is that right?
  37. J

    Finding the Vertex of a Parabola: A Quick Guide

    Homework Statement Find the vertex of the parabola y = (a-b)(a+b) Homework Equations x = -b/2a The Attempt at a Solution This question was extra credit on my Pre-Calc test today. I got the answer and it took almost a page to do it. But I'm very anxious and I just can't wait until i get my...
  38. N

    Parabola definition parboiling my frontal lobe

    Parabola definition parboiling my frontal lobe! Homework Statement The general equation of second degree is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. the values of the coefficients in the general eqn. for a parabola shown above could be: A B C D E F +ve 0 0 0 +ve 0 -ve 0 0 0 +ve 0 0 0...
  39. C

    Half-circle Parabolas: A Misconception?

    Greetings, Is a half-circle a parabola? I am guessing yes because it is a conic section, the very bottom of the cone? If yes, why aren't antenna dishes and telescope mirrors half-spheres? Wouldn't that be simpler? Thanks
  40. T

    Finding Common Ratio of Geometric Progression for 3 Points on a Parabola

    Homework Statement (p,a) , (q,b) and (r,c) are the coordinates of three points on the parabola y^2=3x. If the x-coordinate for these three points form a geometric progression whereas the corresponding y-coordinate form an arithmetic progression, find the common ratio of the geometric...
  41. srfriggen

    Interesting Parabola: Learn About t Slope and dx/dt Properties

    On a test in multivariable calculus I came across, what I thought, was an interesting parabola. Figured I'd ask the forum because my professor is pretty unavailable. In parametric form it is, x=t+4 , y = (1/2)t^2 + 2 The reason I found it interesting is because t is always equal to...
  42. Z

    Proving Circle-Parabola Intersection: a>b>1 | x2 + y2 = 1, y = ax2 - b

    Homework Statement For a > 0, prove that the circle x2 + y2 =1 and the parabola y=ax2 - b intersect at four distinct points, provided a>b>1. 2. The attempt at a solution This is the solution given in my book. Since a>0, by figure -b<-1 i.e. b>1 also when y=0 x2=b/a (from the equation...
  43. B

    Finding the equation for the parabola

    Homework Statement Find the parabola of the form y=ax^2+bx, whose tangent at given point P has equation y. y=5x-8 P=(2,2) Homework Equations I guess the equations involved would be the equation for the parabola: y=ax^2+bx, and the y-y1=m(x-x1) for the slope. The Attempt at a...
  44. T

    How Do Parabolas Relate to Gaussian Elimination?

    Got it
  45. M

    Value of when a line touches a parabola at one point.

    The Question asks: Find the value of m > 0 for which the line y = mx touches the parabola y = (x - 1)2 + 1 at just one point. So far what I've done... I know that if the line touches the parabola at one point, it is tangential. So I put together the 2 equations in order to find the...
  46. A

    3d parabola (projectile motion) to hit a target on an arbitrary plane

    Hi everyone, I'm a software engineer (sorry in advance if I made/will make some math mistake) and I'm trying to calculate the projectile motion from one point to hit a specific target in a 3d space so that the parabola will lie on an arbitrary plane. Is several days that I'm trying to...
  47. R

    Find the extremities of latus rectum of the parabola

    Find the extremities of latus rectum of the parabola y=(x^2)-2x+3. Please someone post its solution. Ans. is (1/2,9/4) (3/2,9/4). i just need full solution. I tried a lot but didn't get this correct answer.
  48. I

    Equation of a Parabola Passing Through (2,-2√2) Opening to the Right

    Homework Statement Write the equation of a parabola passing through the point (2,-2√2) and opening to the right. Homework Equations Parabola with a horizontal axis and vertex (0,0) y^2 = 4cx The Attempt at a Solution Since the parabola opens right I will use the equation y^2 =...
  49. H

    How to define a parabola in 3d coordinate system.

    Currently I am using a graphying application called "Autograph" and modeling a building with a dome shaped roof on top. I need to define parabolic shapes in 3d system. But i can't do it ( my math knowledge is pretty elementary) What would be the basic parabolic function in 3d that i can base...
  50. T

    Rotate Parabola: Eliminate xy-Term

    Homework Statement Rotate the axis to eliminate the xy-term. 3x^2-2\sqrt{3}xy+y^2+2x+2\sqrt{3}y=0Homework Equations \cot2\theta=\frac{A-C}{B} x=x'\cos\theta-y'\sin\theta y=x'\sin\theta+y'\cos\theta The Attempt at a Solution Find the Angle of Rotation...
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