Pre Calculus 8.1 - Conic Sections and a New Look at Parabolas

Summary

In this engaging session with John Swanson, we dive into Chapter 8 of Pre-Calculus, unraveling the fascinating world of conic sections, with a special focus on parabolas. The lecture explores the different conic sections—circles, ellipses, parabolas, and hyperbolas—and how they arise from slicing a cone at various angles. It also elucidates the properties and equations of parabolas, showing how they differ from mere quadratic functions and introducing concepts like the vertex, focus, directrix, and focal width.

Highlights

• John Swanson introduces the concept of conic sections, exploring their unique properties. 📚
• The slicing of a cone results in different shapes like circles, ellipses, parabolas, and hyperbolas. 🌀
• Parabolas are treated as not just quadratic functions, highlighting their deeper geometric properties. 📐
• Practical applications of parabolas in technology, such as satellite dishes and parabolic heaters, are discussed. 🔥
• The process of graphing and transforming equations of parabolas is elaborated. 📉

Key Takeaways

• Conic sections are derived from slicing a cone at various angles. ✂️
• Conic sections include circles, ellipses, parabolas, and hyperbolas. 🌐
• Parabolas can open horizontally or vertically and have applications in real-world technologies, like satellite dishes and flashlights. 🛰️
• The vertex, focus, and directrix are key components in defining a parabola. 💡
• Graphing parabolas requires understanding of their direction and equation transformations. 📊

Overview

John Swanson kicks off Chapter 8 by introducing conic sections, emphasizing their departure from traditional function-based studies in earlier years. He walks through the origins of each section by exploring how slicing a cone in different ways yields distinct geometrical shapes: circles, ellipses, parabolas, and hyperbolas.

Digging deeper into parabolas, Swanson explains their unique characteristics that extend beyond being simple quadratic functions. He covers essential elements such as the vertex, focus, and directrix, how they are defined mathematically, and their real-world applications, from satellite receivers to flashlights and heating systems.

Lastly, Swanson delves into the technical aspects of graphing parabolas, discussing how to formulate and transform their equations for practical use in graphing calculators. He highlights the nuances involved in determining their orientations and the importance of accuracy when calculating focal points and widths.

Chapters

• 00:00 - 01:30: Introduction to Conic Sections Chapter eight, titled 'Introduction to Conic Sections,' offers a transition from the traditional study of functions to exploring the broader scope of conic sections. This chapter promises an engaging diversion from previous mathematical concepts, providing a fresh perspective on geometry not confined to function-based analysis.
• 01:30 - 03:00: Parabolas and Learning Targets This chapter introduces the concept of conic sections and revisits the topic of parabolas. Although quadratic functions have been studied extensively, parabolas encompass more than these functions alone. The chapter aims to expand the understanding of parabolas beyond their representation in quadratic functions.
• 03:00 - 05:00: Understanding Conic Sections This chapter focuses on the study of parabolas, which are a specific type of conic section. The key learning objectives include identifying critical components of parabolas such as the vertex, focus, directrix, and focal width. Additionally, the chapter covers the skills needed to graph parabolas and provide equations of parabolas under certain conditions. It also addresses proving whether an equation represents a parabola and solving real-world application problems involving parabolas.
• 05:00 - 07:30: Non-Degenerate and Degenerate Conic Sections This chapter discusses conic sections which originate from cutting a cone. If a cone, placed upright, is cut horizontally, it results in a circle. However, if the cone is cut at an angle, different shapes like ellipses, hyperbolas, and parabolas can be obtained. The chapter explores both non-degenerate conics, such as circles and ellipses, and degenerate conics, like points, lines, and pairs of lines that occur through different sections of the cone.
• 07:30 - 10:30: Graphing Conic Sections with Calculators The chapter 'Graphing Conic Sections with Calculators' discusses the geometric figures obtained from slicing a cone. It explains that slicing the cone parallel to its side results in an ellipse, while a vertical cut produces a parabola. The chapter also highlights how these shapes can be visualized and analyzed using calculators.
• 10:30 - 14:00: Exploring Parabolas and Hyperbolas In this chapter, the discussion focuses on the differences between parabolas and hyperbolas. It is clarified that what is often mistaken for a part of a parabola is actually a hyperbola. Additionally, a brief insight into the geometric concept of a cone is provided, explaining that a cone is not just a simple shape but can be visualized as a line being pivoted, which forms what we understand as one half of a cone.
• 14:00 - 17:00: Applications of Parabolas in Real-World The chapter discusses the concept of a cone, illustrating how it is essentially two cones connected at a single point, forming a hyperbola. It explains that when you pivot around this point, you can visualize the hyperbola as two separate pieces. These connections and visualizations play a crucial role in understanding the application of parabolas in real-world scenarios.
• 17:00 - 21:00: Parabolas Equations and Properties This chapter discusses the geometric properties of parabolas. It begins by examining how different types of conic sections (like circles and ellipses) are formed by slicing a cone in various ways. The focus is on explaining why a parabola is formed when the slice is parallel to the edge of the cone. The chapter highlights the unique behavior of parabolas compared to other conic sections, noting that parabolas do not intersect the opposite side of the cone like hyperbolas do. This distinct property is emphasized to underline the uniqueness of parabolas within conic sections.
• 21:00 - 24:00: Standard Form Equations and Examples The chapter introduces various conic sections including circles, ellipses, parabolas, and hyperbolas, categorizing them as non-degenerate conic sections. It slightly touches upon the concept of degenerate conic sections, which is a term not necessary to delve deeply into.
• 24:00 - 31:00: Solving Parabola Equations and Graphing The chapter "Solving Parabola Equations and Graphing" focuses on understanding the different outcomes derived from slicing a cone, also known as conic sections. It begins with the explanation of how cutting a cone horizontally results in a singular point, not a conic section. When the slice is perfectly along a particular line, it results in a complete line. Further slicing vertically results in other shapes of conic sections, leading up to the parabola equations and their graphical representations.
• 31:00 - 36:00: Conclusion and Next Steps This chapter discusses the concept of geometric degenerate cases, drawing an analogy to algebraic expressions. When certain conditions are met, such as when triangles or linear sections align at a single point, the normal properties of these shapes or equations no longer apply, making them degenerate cases. This is compared to altering the quadratic equation y = ax^2 + bx + c by setting a = 0, which transforms the equation from a quadratic to a linear equation, illustrating a similar type of transition. The chapter emphasizes understanding these degenerate cases as crucial in mathematical analysis and interpretation.

Pre Calculus 8.1 - Conic Sections and a New Look at Parabolas Transcription

• 00:00 - 00:30 welcome back to chapter eight uh chapter eight is on conic sections which is uh i think kind of fun uh it's different than what we've been doing so far um not only this year but in the past number of years you've worked with functions conic sections are not functions so we get to get into something that's just a little bit bigger than what we've been dealing with before in this section
• 00:30 - 01:00 we're going to see what conic sections are and then we're going to have a new look at some parabolas now you might think that wait no you know everything to know about parabolas i mean we've been working on them since algebra one that's not exactly true we have been working with quadratic functions whose graph is a parabola but parabolas are bigger than that so we'll see what
• 01:00 - 01:30 those are our learning targets today we'll find the vertex focus directrix and focal width of parabolas i will graph parabolas we'll find equations of parabola satisfying given conditions and we'll prove that the graph of an equation is a parabola and then we'll solve application problems involving parabolas so lots of parabolic activity so before we get into parabolas what are
• 01:30 - 02:00 conic sections well conic is from cone so these all involve cones and specifically cutting a cone if we cut a cone that's just like sitting on a desk if we cut it horizontally we get a circle if we cut it at an angle like that we get
• 02:00 - 02:30 this oval which is an ellipse if we cut it parallel to the side so like that like this cut is parallel to the the side of the cone we get that shape which is in fact a parabola and then if we cut it vertically so straight up and down we're going to get this thing that looks kind of like a
• 02:30 - 03:00 parabola but it's not this is actually called a hyperbola in fact this isn't all of a hyperbola because this isn't actually all of a cone cones aren't just the shape in fact this is half of a cone a cone is defined as taking a line so we have a line and we're going to pivot it
• 03:00 - 03:30 around a point so if we pivot that around the point we get those two so a cone is actually what we think of as two cones connected at that point so that hyperbola that we saw earlier not only is it right down here but it's also up here so it makes two different pieces
• 03:30 - 04:00 now that's the only one that does that because if we cut it sideway if we cut it horizontally to get our circle that's not going to go into the other part of the cone if we cut it at an angle that's not going to go into the other part of the cone if we cut it parallel to get the parabola that's not going to go in the other part of the cone because it's going to continue to be parallel down here so hyperbola is the only one that goes
• 04:00 - 04:30 into both of them like that so our conic sections that we'll look at we have circles ellipses parabolas and hyperbolas and these are called non-degenerate conic sections i know it's a really big word it's not one that you really need to know too much of but it's non-degenerate conic sections a degenerate conic section goes through
• 04:30 - 05:00 that point so if we cut it like perpendicular or so like horizontal it's going to be just a point that's not really a conic section it's a point if we cut it perfectly along that line then we have a line and then if we cut it going straight up and down we have
• 05:00 - 05:30 like two triangle deals so each of these don't go through that point if they do it's degenerate and that's something else it's kind of like having our quadratic the y equals ax squared plus bx plus c where a equals zero all right that's no longer a quadratic it's a line so same kind of idea i'm seeing all of the cuts at one place we have
• 05:30 - 06:00 this one will make a circle that one will make an ellipse this one which is parallel to this side we'll make a parabola and that one will make a hyperbola so we have we have those i'll try to even have it in class so you guys can kind of see it and play with it a little bit um one of the things we'll be doing with these this isn't just with
• 06:00 - 06:30 the parabolas that we're going to be looking at today but it'll be with each of them is graphing them on a calculator now in our calculators you can't just type in a conic section and graph them because it graphs functions it also graphs polar coordinates and parametric equations which you can make this as polar we're not getting there right now um we're going to write it as a function
• 06:30 - 07:00 um in desmos you can actually just type it in as a as a conic section it graphs it for you really quite nicely but looking at it on a calculator what do we have to do we have to get it to be y equals something so if we start out with this guy right here 5y squared equals x we need to get it to be y equals so divide by 5 square root now when we take
• 07:00 - 07:30 the square root we need a plus or minus so we have plus or minus root x over 5. now which is it plus or minus it's both that's the key it's both you need to graph both the positive and the negative so if we were to do this we'd go in we'd clear out what we need
• 07:30 - 08:00 to we can put this one comma negative one there and that actually does plus or minus for us or we could use y one and y two and have it do um like one of them be positive one of them be negative it doesn't really change anything other than just gives us in one equation there's the first one and there's the
• 08:00 - 08:30 negative part and that makes a parabola that's sideways we've never seen sideways parabolas before our parabolas have always looked like y equals x squared this is x equals y squared it makes it sideways now if you do look at this that very closely where it's coming together there's a little gap there right on that x-axis now there's not actually a gap in the function that's
• 08:30 - 09:00 a um an issue with how the calculator calculates it because it goes in the next value over would be non-existent because we'd end up with the square root of a negative number and so it graphs a series of points and connects the dots and it won't get all the way there here the gap is pretty small when we're graphing circles ellipses things like that the graph can get a little bit bigger
• 09:00 - 09:30 so as i said that's a parabola the vertex is at 0 0 we'll see what the vertex is but it's we know what the vertex of a parabola is it's where it turns around and this is opening to the right so now they can open to the right or left in addition to up or down in fact they can even be rotated to open in any direction at any angle those would be oblique parabolas we're not looking at those today so you don't have to worry about those
• 09:30 - 10:00 quite yet but know that they do exist let's try this one we have x squared equals y squared plus nine we have to solve for y so subtract nine square root gives us plus or minus and remember the square root of x squared minus nine is not x minus 3 it is just the square root of x squared minus 9. so graphing it
• 10:00 - 10:30 we go back we can keep these this one common negative one notice those are with the the curved brackets those are the second parentheses and then just type in x squared minus nine graph it this is a weird looking shape here but
• 10:30 - 11:00 it's still coming there's the negative half of it and so notice this one has two it looks kind of like a parabola but there's two of them that's a hyperbola and it isn't just like two parabolas there are fairly big differences um but they do kind of look like it with this you can notice that gap at the x-axis in both places remember that's not actually a gap it does connect
• 11:00 - 11:30 there so this is hyperbola the center here is at 0 0 that's right in the center of it it opens horizontally because it's both left and right we have some vertices there's two of them in a hyperbola and they'll be exactly where you think they are as far as like what we know about vertices for parabolas they're at the turning points in this case they're at negative 3 and 0
• 11:30 - 12:00 and positive 3 0. so what is a parabola we know all there is to know about parabolas right it's it's a quadratic no no it's not um parabolas and in fact all conic sections are defined um as a distance a like a circle is the all the points that are equidistant from that center point a parabola is all points are equidistant
• 12:00 - 12:30 from a fixed point called a focus and a line called a directrix so we have a focus right over here we have this directrix over here and every point on this parabola is going to be the same distance so those two distances m and n are the same that's for
• 12:30 - 13:00 every single point on the parabola um notice the distance to m or this m this is going to be a 90 degree so as the point moves over here that line would be over there it's always perpendicular to the directrix so but m will equal n for every single point because of this we can use the distance formula to
• 13:00 - 13:30 come up with an equation of a parabola and we'll actually do an example where we see that it's not the only way of doing it but it can be nice you can actually derive the formula for a parabola like the equation from that as well um so before we get too far into the algebra what are the uses of parabolas and more specifically paraboloids a paraboloid is like a three-dimensional
• 13:30 - 14:00 parabola if we took the parabola and just rotated it so we had like a dish or a bowl that's a paraboloid and being that we live in a three-dimensional world we're going to use the three-dimensional ones more than the two-dimensional ones so what are some uses of these this is a parabolic heater and what it does is it has a heating
• 14:00 - 14:30 element right here that's at the focus what happens with parabolas is that anything from the focus is going to radiate out to the surface of the parabola or paraboloid and it's going to reflect straight away so the cool thing about this heater is if you put your hand just aside like you actually you feel
• 14:30 - 15:00 like a wave of heat it's kind of like a fan where if you step like half a foot to the side you don't feel it then you can step into it it's like that but with heat it's crazy um a parabolic microphones this is the same thing but backwards in a parabolic microphone the dish this dish is not the microphone it's just a reflective surface the microphone is right here and it sits at the focus pointed
• 15:00 - 15:30 back into the dish and all the sound waves come in and get reflected to the microphone which takes a big area of sound waves and concentrates it satellite dishes same exact idea the receiver is this thing right here that's sitting at the focus we have all these rays coming in from
• 15:30 - 16:00 space hitting the dish and then they all bounce straight to the receiver allowing us to get high definition signals from space because we're collecting an area of it a flashlight same thing this bulb should be at about the focus all the light bounces and then goes straight out the flashlight
• 16:00 - 16:30 which is why we get a beam on flashlights that look kind of like this where you can kind of rotate it to adjust the beam on how how tight you want it how much of a flood light you want it what that does is it pulls the light in like the light bulb in or pushes it out so it's moving it away from where the focus is and so all of a sudden it's not all going out in a straight line it's it's going out at angles which
• 16:30 - 17:00 gives you a wider a wider view because the the light the angle of the light is wider so you can see more but it's more diffused so parabolas what are they we have some horizontal parabolas and we'll have vertical parabolas the vertical parabolas are the ones that we already know a bit about because those are actually functions vertical parabolas are the only conic
• 17:00 - 17:30 sections the non-degenerate ones that are functions so a horizontal parabola is going to look kind of like this now this one opens to the right it could be flipped over and open to the left vertical ones open up or could be reflected opening down they have a focus the focus is that point that's inside the parabola
• 17:30 - 18:00 the closer the focus is to the vertex the wider the parabola is we have a directrix the directrix is that line and then the distance from the directrix to the vertex and from the vertex to the focus oh so the vertex is exactly where we
• 18:00 - 18:30 think it would be and then anyway so the distance from the focus to the vertex and the vertex to the directrix is the same because the distance from any point on the parabola to the focus and that point to the directrix is the same the vertex is going to be that shortest
• 18:30 - 19:00 distance and that we call p p that distance is called the focal length we have something called the focal width as well
• 19:00 - 19:30 the focal width is the line it's the cord because it goes from part of the parabola to another part of the parabola and it's going to be parallel to the directrix and it goes through the focus the focal width is actually 4p now p could be positive or negative
• 19:30 - 20:00 though in these cases they're positive if it opened to the left or it opened down p would be negative because p is the directional like the directed length from between the vertex and the focus so the focal width is actually just a distance so it's going to be the absolute value of 4p it'll always be positive so let's look at some equations
• 20:00 - 20:30 so for the horizontal one this is centered at the origin we have y squared equals 4px now i've seen some places that make it like x equals something with y squared um our text doesn't it puts the squared by itself so we have y squared equals 4px if we want to move the vertex
• 20:30 - 21:00 it's the whole h and k thing we've seen this with parabolas before the x turns into x minus h the y turns into y minus k um and you might be thinking that wait a second way back when when we looked at parabolas we had y equals a times x minus h squared plus k right that's the vertex form where a oh
• 21:00 - 21:30 a is for p that's kind of nice um this plus k like it was always kind of weird because it was always x minus h like every translation was x minus x minus x minus and we just added k why is it plus k it's because we already solved for y this would be a y minus k equals a times x minus h squared
• 21:30 - 22:00 so it's actually subtraction for both of them we just added the k over to the other side so here being that it has to be squared we keep it over there anyway the focus so the focus is going to be p away from the vertex in a horizontal direction so we're just going to add p to the x value
• 22:00 - 22:30 and if the vertex is at 0 0 then it's just p comma 0. if it's at h k it's h plus p comma k we just add it to the x value if p is negative it would end up going backwards because if it was negative 1 then we're going to be 1 to the left of h so we don't even need to worry about a separate equation for that we just need to know if p is positive or negative
• 22:30 - 23:00 the directrix is just the other direction it's going to be h minus p and being that's the line it'll be x equals h minus p so if the vertex is at 0 0 it's just x equals negative p if the vertex is at h k it's going to be x equals h minus p and then the axis of symmetry the axis of symmetry is going to be
• 23:00 - 23:30 um that same line in a horizontal one the axis of symmetry will be horizontal and it goes through the y part of the vertex so it'll be just y equals the y part of the vertex so if the vertex is at 0 0 it's y equals 0. it's the x axis if the vertex is at h comma k it's y equals k how about for the vertical ones same thing except
• 23:30 - 24:00 backwards so we have x squared equals 4py again sometimes it's y equals one four one over four p times x squared where one over four p would be that a value or we can change where the um change where the vertex is h k notice again it's x minus h and y minus k those don't switch the h is always associated with x
• 24:00 - 24:30 the y is always associated with k the the um what is squared however changes um the focus we're going to go up or down now and so it'll be the k plus zero or k plus p so if it's at zero zero it'll be zero comma p if it's h k we'll have h comma k plus p
• 24:30 - 25:00 the directrix now is a horizontal line so it's y equals um and then it's just whatever the vertex is y equals negative p because we're going backwards so y equals negative p or y equals k minus p and then the axis of symmetry is vertical line so it'll be x equals the x part of the vertex so either 0 or x equals h
• 25:00 - 25:30 so those are a lot of the equations then again p is always the focal length and the focal width is always the absolute value of 4p that doesn't change for any of these so what are we going to do with the parabolas we can use the distance formula to write the equation of a parabola with a given focus and directrix now again you don't need to use the distance formula all the time but it can sometimes it's
• 25:30 - 26:00 important especially if you have a focus in directrix so if we have focus of zero comma negative five and a directrix of y equals five we need some points to find the distance one of the points is going to be zero comma negative five the point on the line will be just x y well on the parabola will be x comma y and then we need a
• 26:00 - 26:30 point on the directrix because we're going to have the distance between x y and 0 negative five has to equal the distance from x y to the directrix well that's a point on a line if you remember it'll always be perpendicular to the line so it will share the same x value or the same y value being that this is y equals 5 it will share the same x value since we're at x
• 26:30 - 27:00 comma y this point will be at x comma 5. we can plug this into a distance formula so we have the distance between the focus and our xy point so x minus zero squared plus y minus negative five squared will equal the distance between x y and x comma five so we have x minus x squared
• 27:00 - 27:30 plus y minus five squared we can square both sides because if the stuff underneath the square roots if we have a square root equals a square root that means the stuff underneath the square roots has to be equal so x squared plus y plus 5 squared will equal 0 plus y minus 5 squared multiply this stuff out so we have x squared plus y squared plus 10y plus 25 equals y
• 27:30 - 28:00 squared minus 10y plus 25 because remember y plus 5 squared does not equal y squared plus 5 squared it's y plus 5 times y plus 5. don't be that guy uh we can simplify first of all the y squareds cancel in a parabola only one of our variables is squared it's either x squared or y squared it's not both um 25s also cancel because we have plus 25 in my n plus 25
• 28:00 - 28:30 so we can combine our like terms so by subtracting 10y to get x squared equals negative 20y squared and this would be our equation here we have a vertex of 0 0 the focal length that p-value remember the negative 20 equals 4p so p equals negative 5 which makes sense if
• 28:30 - 29:00 the vertex is 0 0 and the focus is at 0 negative 5 we had to go 5 over and we could have actually seen the vertex was 0 0 because it's halfway in between negative 5 and positive 5 as well so we have x squared equals negative 20y which could be y equals negative 1 20th x squared that's like the kind that we are used to seeing this is vertical it's opening down
• 29:00 - 29:30 which that negative p-value tells us so but this one is how the answer generally looks we can write an equation standard form of from a graph so things to notice vertex is it zero zero and here it gives us the directrix
• 29:30 - 30:00 which means that that distance is two notice it's counting by twos so p will be two which means the focus if we needed it would be right there but since that's two p equals two and the vertex is 0 0 being that our equation and it's going horizontal which means it'll be x squared equals
• 30:00 - 30:30 4p y if p is 2 that means we'll have x squared equals 8y or starting out there plug it in yes y squared because y equals x squared is vertical that's a bummer i'm going to erase
• 30:30 - 31:00 we didn't see that y squared equals 4p x which means y squared will equal 8x yes that's what i meant to say from the beginning um so we have y squared equals 8x so it did it right there um so
• 31:00 - 31:30 you know this brings up a great point make sure you're checking yourself you wouldn't want to do something horrible like switch your x and your y and square the wrong thing i don't know anyone that would ever do that so um yep what else are we going to do with parabolas write an equation standard form if we have a vertex and a focus
• 31:30 - 32:00 so the focus notice it's 8 to the left which means p equals negative 8 which we can plug in being that it's to the left we have the y squared or like that of y squared equals 4p x plug in the negative 8 for p
• 32:00 - 32:30 and you get y equals negative 32 y squared equals negative 32 x so a lot of times just finding that p value is all you really need to do and making sure you know which way it's going and the vertex too i suppose so find the vertex value of p axis symmetry focus and directrix then graph
• 32:30 - 33:00 so we have y plus 1 equals 1 16 x minus 2 squared which could be x minus 2 squared times 16 or eq x minus 2 squared equals 16 times y plus 1. we could multiply that 16 over but this one notice it's x equals y equals x squared it's y equals x squared which is a function type parabola it's going to open vertically it's positive so it'll be opening
• 33:00 - 33:30 up so what do we need to find vertex h comma k is 2 comma negative 1. remember it's y minus k and x minus h so we have to change the sign of both of them so we have vertex is two comma negative one the value of p is four because 4 times 4 equals 16 right it's 4p equals 16. so p equals 4.
• 33:30 - 34:00 the axis of symmetry x equals 2. since it's a vertical parabola the axis of symmetry is vertical and it goes through the the x part of the vertex so x equals 2. the focus we're going to have to add p to the y value so instead it's negative one plus four so it'll be a three so we have two comma three the directrix we subtract the p
• 34:00 - 34:30 value from the that y value of the vertex so negative one minus 4 is negative 5 so it'll be y equals negative 5. notice the axis and symmetry and the directrix are perpendicular ones x equals the others y equals then graph it so we have our vertex is it two comma negative one
• 34:30 - 35:00 we have our focus is it two comma three the directrix is at y equals negative five and the parabola will go like that um you can always graph an extra point the distance since the focal width is 4p so this distance is 4p notice that this distance is
• 35:00 - 35:30 2p 2p so that's a good way of checking without actually having to find other points you go from the focus you already know p that's how you found the focus just go each direction to p and graph a point and that can help guide that parabola so this was an introduction to conic sections and at new look at parabolas we will continue looking at circles and
• 35:30 - 36:00 ellipses and then hyperbolas in the next couple um i will see you in class but until then keep working problems keep asking questions and as always happy mathing