Precalculus (2018 edition)
A parabola is the set of all points equidistant from a point (called the "focus") and a line (called the "directrix"). See this video to learn more about this.
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- How to prove that every point in the parabolic curve will have the same distance to the focus and directrix?(15 votes)
- Do all the parabolas have a directrix in the form
y = c?
Or there are some kind of parabolas that have the directrix in the form
y = mx + c?(8 votes)
- Sure, you could have something like x^2 - 2xy + y^2 - x * sqrt(2) - y * sqrt(2) = 0
Which is a parabola tilted 45 degrees, but it is also worth noting that it is not a function.(7 votes)
- What is the importance of having a directrix? Are there any applications of it in real life?(6 votes)
- if you had a paper with a focus and a directrix, and you folded the directrix onto the focus several times at different angles, you would end up with a parabola.(7 votes)
- Hi, I'm just a little stuck on this practice question:
The equation of a parabola is in the form y=kx^2. If the line 8x-y-4=0 is a tangent to the parabola, find the value of k.
I understand that the parabola has vertex 0 and is concave up (I gathered that from a quick sketch) but I'm not sure what to do beyond that...
Would really appreciate some help. Thank you!
P.S. Also, this is totally unrelated but how do you properly type indices and fractions in the comments? Thanks(3 votes)
- Unfortunately, KA currently does not support typesetting systems like LaTeX, so we are stuck with using Unicode characters.
You are correct in deducing that 𝑘 > 0 because the 𝑦-intercept of the tangential line is negative.
I presume you are familiar with the concept of derivatives. It forms the foundation of calculus. Essentially, it describes the slope or rate of change of a function. If you are not familiar with this concept, I would wait until you are solid in algebra, trigonometry, and precalculus before you worry about diving into calculus. There is a way to solve this without using calculus but I'll leave that up as a challenge for you. Feel free to comment if you want to see that solution. The calculus solution is as follows.
In our case, we have:
𝑓(𝑥) = 𝑘𝑥²
Differentiating this gives us:
𝑓'(𝑥) = 2𝑘𝑥
Thus, at a given point 𝑥, the slope of the tangent line to the curve is given by 2𝑘𝑥. The tangent line that we are given is:
𝑦 = 8𝑥 – 4
Which has a slope of 8. Thus, we must have 2𝑘𝑥 = 8 ⟹ 𝑘𝑥 = 4. Now consider the tangent line 𝑦 = 8𝑥 – 4. This line contains two well known points. The 𝑦-intercept of (0, -4) and the point of tangency (𝑥, 𝑘𝑥²). The slope of the line is known and is given by:
(𝑘𝑥² + 4)/𝑥 = 8
We simplify this as:
𝑘𝑥 + (4/𝑥) = 8
But recall that we have already deduced that 𝑘𝑥 = 4. Thus:
4 + (4/𝑥) = 8 ⟹ 𝑥 = 1
Hence we conclude 𝑘 = 4.
Comment if you have questions.(7 votes)
- so is the directrix always below the vertex (lowest point) of the parabola in an upward opening (positive) parabola and the directrix always above the vertex of a downward opening (negative) parabola?(3 votes)
- Correct. If the parabola opens to the right, the directrix is to the left of the vertex, and if the parabola opens to the left, then the directrix is to the right of the vertex. The focus is always inside, and the directrix is always outside.(5 votes)
- how do you find focus and directrix of a parabola(1 vote)
- The first instance is the best. If you have the parabola written out as an equation in the form y = 1/(2[b-k])(x-a)^2 + .5(b+k) then (a,b) is the focus and y = k is the directrix. This is for parabolas that open up or down, or vertical parabolas. For those that open left or right it is diffeent. You should get to the video that goes over these formula soon, or I could if you like.
If you only have a drawn graph you are in a tough spot. You should be able to get it into vertex form though. You do need to know how to get other forms into vertex form if you don't have that form I mentioned above. Literally EVERY other form will need to be put into vertex form.
You may be able to notice y = 1/(2[b-k])(x-a)^2 + .5(b+k) is kind of in vertex form. Vertex form is usually A(x-C)^2 + D. A is the vertical stretch, C is the horizontal shft and D is the vertical shift. There is no horizontal stretch/ shrink because you ca make any horizontal stretch/Shrink equal to some vertical one. Specifically it would look like A(B(x-c))+D where B is the horizonal shrink. But you can factor it out into AB^2(x-C)^2 + D so it's just easier to have one number acting as a vertical stretch. Also, with vertex form in the form A(x-C)^2 + D you should know the vertex is (C, D). The vertex is the only point A does not effect in stretching or shrinking it.
Anyway, if you compare A(x-C)^2 + D to 1/(2[b-k])(x-a)^2 + .5(b+k) you should see A = 1/(2[b-k]), C = a and D = .5(b+k) so it is a much more complicated vertex form. We can use this though.
Now, we know A = 1/(2[b-k]) and D = .5(b+k), so let's do a bit more work.
Starting with A = 1/(2[b-k]) you can use algebra to change it to 1/(2A) = b - k. So you will be able to find b - k. if we had one more equation with b and k we'd be able to get their exact values. Well, we do. D = .5(b + k) can eb written as 2D = b + k. So now, if you have a parabla in vertex form you can take A and D and find b and k, where b is the y value of the focus and k is the directrix of the form y = k. Of course the x value of the focus is just C. Again, only upward and downward opening parabolas.
If you'd like me to run through an example let me know.
There is also a kind of shortcut method. using a standard form you can rearrange your vertex form into (x-C)^2 = 4p(y-D) where 4p = 1/A or p = 1/[4A]. In this form you just need to memorize that the focus is (C, D + p) and the directrix is y = D - p. if you prefer not to use the p then you have (C, D + 1/[4A]) and y = D - 1/[4A]. Once again, this is for vertical parabola and the horizontal ones are just a bit different. You can us the same methods to find them though.
If there is anything you don't understand let me know and I can go more into it. Or in general if you'd like more of an explanation on something. This was kind of a complicated post. I hope it helped.
Thanks to muonsortsitout on reddit I've figured out how to explain that shhortcut method, with (x-C)^2 = 4p(y-D).
Starting back at 1/(2[b-k])(x-a)^2 + .5(b+k) you can rearrange things.
y = 1/(2[b-k])(x-a)^2 + .5(b+k)
(x - a)^2 = [2(y - (1/2)(b + k))](b - k)
If you set p = (b - k)/2 and q to (b + k)/2 you should see p is the distance in the y direction from the vertex to either the directrix or focus. Or at least the value. Specifically it is the distance to b. -p is the distance to the directrix. q though is the average of the two values, which is the value right between them which is the y coordinate of the vertex. Anyway this changes the equation.
(x - a)^2 = 4(y - q)p = 4p(y - q) in what I wrote before q was D. But yeah, that is how we this works.
the x coordinate of the focus is of course a
the y coordinate is q + p since q is the y coordinate of the vertex and p is the distance between the vertex and the focus
then the directrix is y = q - p since you are going in the other direction from the vertex.
Again, let me know if something here didn't make sense.
it might help to rarrange (x-a)^2 = 4p(y-q) into y = 1/(4p) (x-a)^2 + q just to sww how things correspond. We KNOW p is the distance from the vertex to either the(8 votes)
- How did the focus and directrix come to be? It seems a bit too random and out of the blue.(1 vote)
- The focus and directrix aren't just two random things. A cool property of this is that the distance from the focus to any point on the parabola will always be the same as the distance from that point on the parabola to the directrix.
Ever wonder how the headlights on your car work? It consists of a bulb at the focus of a parabolic mirror. When the light turns on, the light rays propagate spherically from the parabola's focus, when it comes into contact with mirror, the light will now be parallel to the ground. This is how your headlights concentrate the light to illuminate whats in front of your car. Hope this helps!(7 votes)
- Could a parabolic equation be written for a slanted directrix?(3 votes)
- Theoretically, I think you can write an equation for a rotated parabola for a rotated directrix. The math is messy, but doable with enough time and focus.
This video by Sen Zen explains how to rotate curves, parabolas being one of them; however, it does not talk about how to find the equation given a directrix after a parabola has been rotated:
https://www.youtube.com/watch?v=BPgq2AudoEo (copy and paste the entire thing, the formatting with the link is messed up, and clicking the link will not work)
I think it is possible, though. Given a directrix, you could figure out what angle it is being rotated by from the form y = "some number" or x = "some number". Imagine that the directrix started as straight up and down, or straight left to right, and then rotated by some angle. From there, you could construct a parabola using the y = "number" or x = "number" directrix, and then rotate it by the angle the directrix was rotated by.(2 votes)
- Is the vertex of a parabola always halfway between the focus and the directrix, or was that just an example Sal made?(2 votes)
- Yes; that is the definition of a parabola. All points on a parabola(including the vertex) are equidistant from the focus and directrix, which, if you think intuitively, makes sense, because the distances from the directrix and focus to the vertex are the same, and if you think of the distance from the focus to the directrix as 1, then the distances from both to the vertex are 1/2 of that distance.
Hope that helped!
- How can we prove that the equation ax^2 + bx + c creates a parabola? How can we prove that slicing a cone also creates a parabola?(2 votes)
- Slicing a cone is really just visual. If you look at all conic sections they are literally giving every possible way to slide a cone, or more specifically two cones positioned tip to tip, and describing what happens with each method.
As to prove that the standard form works the solution is to start with the most basic form of a parabola and work from there.
Now, before anything else you should eb familiar with graph transformations. Moving it left and right, or up and down, or manipulating how wide/ skinny it is. If you are familiar with graph transformations this should look familiar. A*f(B(x - C)) + D Where A is the vertical stretch, that is you multiply the y value of all coordinates by A except for where x=C. The point (C,D) is the only point that will not experience any stretches or shrinks. B is the horizontal shrink, so you divide all x values by B. c is the horizontal shift, so ALL x vales move C to the right, or if C is negative (so it would look like (x - -C) or (x + C) )you move to the left. D is the vertical shift so you move up D unless D is negative in which case you move down.
To write these transformations for x^2 s would look like A(B(x - C))^2 + D Although you could factor out the B to make it AB^2 * (x - C)^2 + D So AB^2 will give the effect of the horizontal and vertical stretch and shrink. This works because 8x^2, where x squared is getting stretched vertically by 8 is the same as 2(2x)^2 where x^2 is getting a vertical stretch by 2 and a horizontal shrink by 2. This may not be obvious, but if you use a point x = a you get a coordinate pair of (a, 8a^2) meanwhile if you do the same for 8x^2 (a, 8a^2). or if you'd rather use real numbers you could solve for 3. in 2(2x)^2 for x=3 you get (3, 72) meanwhile 8x62 gets you (3, 72). So hopefully you can understand 2(2x)^2 is the same as 8x^2.
NOW! I am going to say AB^2 is just A for ease. This leaves us with A(x - C)^2 + D. If you expand it you get Ax^2 - 2ACx + AC^2 + D. This leaves us witht he standard form where A = a, -2AC = b and AC^2 + D = c. So ax^2 + bx + c is just a rewritten version of x^2 with its transformations.
Does this make sense? If not you can agan think of it with real numbers. If you put actual numbers in for A, C and D then expand it you will get something in the form of ax^2 + bx + c where a, b and c are actual numbers.
If you are unaware there is also a way to change ax^2 + bx + c into A(x - C) + D, called changing from standard form to vertex form.
Let me know if you do need more explanation.(3 votes)
- What I want to do in this video is understand two words that you might hear associated with parabolas. And that is, that is a focus. That is a focus of a parabola, focus. And a directrix. Directrix. Directrix. Directrix, right over there. So what are these things? Well, a parabola can be defined as the set of all points, let me draw arbitrary axes right over here. So that's my y axis, this is my x axis. This is my x axis. And so a parabola can be defined as the set of all points that are equidistant to a point and a line. And that point is the focus of that parabola, and that line is the directrix of the parabola. So what am I talking about? So let's give ourselves a point. So let's say this point right over here, and we can even say that that is the point let's say that's the point, let's say the x coordinate is a, and the y coordinate is b right over here. So that is the point (a, b). And then let's give ourselves a line for the directrix. And actually I'm going to do this in a different color instead of just white, cos I did the coordinates in white. So I will do it in this magenta color. So that's (a,b) is the focus. And let's say y=c is the directrix. So this right over here is the line, this right over here is the line y is equal to c. So this on the y axis right over there, that is c, this is the line y=c. So a parabola, what does it mean to be the set of all points that are equidistant between a point and this line? Let's think about what those points might be. Well, this point right over here would be halfway between, between this point, between the focus and the directrix. And then as we move away from x=a, you're going to get points anywhere along this curve. Which is a parabola. And you might be saying, wait, I don't get this, I don't get why points along this curve are going to be equidistant. Well, let's just eyeball the distances. So this one, this distance, and obviously I'm drawing it by hand so it's not going to be completely precise, that distance needs to be equal to that distance. Well, that seems believable. Now this, if we take this point right over here of the parabola, this distance needs to be the same as that distance. Well, that seems believable. If you take this point on the parabola, this distance, this distance needs to be the same as this distance. So hopefully you get what I'm talking about when I say that the parabola is a set of all points that are equidistant to this focus and this directrix. So any point along this parabola, this point right over here, this point right over here, the distance to the focus, the distance to the focus, should be the same as the distance, as the distance to the directrix. Now, what you might realize is when you're taking the distance between a point and a point, the distance can, it'll be at, I guess you could say it could be at an angle. This one's straight up and down, this one is going from the top left to the bottom right. But when you take the distance from a point to a line, you essentially drop a perpendicular, you essentially go straight down. Or if the parabola was down here, you'd go straight up to find that distance. These are all, these are all right angles right over here. So that's all a focus and a directrix is. And every parabola is going to have a focus and a directrix, because every parabola is the set of all points that are equidistant to some focus and some directrix. So that's what they are. In future videos we'll try to think about, how do you relate these points, the focus and directrix, to the actual, to the actual equation, or the actual equation for a parabola.