Quadratic Relations & Conic Sections: Parabolas

[envelope]

The text book I borrowed from my school reads "you already know that the graph of y = ax^2 is a parabola whose vertex (0,0) lies on its axis of symmetry x=0." On the contrary, I know not of what it reads. I am unaware of what y symbolizes, as well as what a & x^2 symbolizes. The axis of symmetry as defined by the text book is "the line perpendicular to the parabola's directrix & passing through its focus. In particular, the axis of symmetry is the vertical line through the vertex of the graph of a quadratic function." So, from this, I've deduced that the axis of symmetry is the line perpendicular to a fixed line associated with the parabola, which passes through the focus (which aids in decsribing the conic section). & not only do parabolas have a vertical axis of symmetry, opening up or down, but a horizontal axis of symmetry, opening left or right. It then goes onto list & show each case:

 

x^2 = 4py, p > 0

x^2 = 4py, p < 0

y^2 = 4px, p > 0

y^2 = 4px, p < 0

 

The top two being parabolas with a vertical axis of symmetry, opening up & down. The bottow two being parabolas with a horizontal axis of symmetry, opening left or right. What I don't understand is what the equation is stating. In particular, what x^2 is symbolizing, what 4py is symbolizing, & p > 0 is symbolizing, & so on & so forth throughout each case. Another thing I am not understanding are the standard equations of a parabola with its vertex as its origin. The equations being:

 

x^2 = 4py, its focus (0,p), its directrix y = -p, and its axis of symmetry being vertical (x = 0)

y^2 = 4px, its focus (p,0), its directrix x = -p, and its axis of symmetry being horizontal ( y = 0)

 

I am guessing that in order for me to under the standard equations, I should first understand what each individual component means. Pertaining to graphing an equation of a parabola, I am having trouble grasping a few things taken from the text book. First off, it says identify the focus and directrix of the parabola given by x = -1/6y^2. In order to get rid of the fraction, I understand you multiply by -6 on each side. So:

 

(-6)x = -1/6y^2(-6) giving you -6x = y^2.

 

Because the variable is y^2 the axis of symmetry is horizontal. Now, the equation above, the text book says, will aid in finding the focus and directrix. It then says that "since 4p = -6, you know p = -3/2. The focus is (p,0) = (-3/2, 0) and the directrix is x = p = 3/2." It then goes onto saying that since p < 0, only negative x-values should be chosen to create a table of values. & then the table of values goes onto showing:

 

x = -1, -2, -3, -4, -6

y = +-2.45, +-3.46, +-4.24, +-4.90, +-5.48

 

What I am not understanding is how the y values are arrived at. As you can probably already tell, I'm as lost as they get. If you haven't already labeled me mathematically deficient, I would greatly appreciate any help. In-depth explanations would be ideal.

[Cap'n Refsmmat]

The text book I borrowed from my school reads "you already know that the graph of y = ax^2 is a parabola whose vertex (0,0) lies on its axis of symmetry x=0." On the contrary, I know not of what it reads. I am unaware of what y symbolizes, as well as what a & x^2 symbolizes. The axis of symmetry as defined by the text book is "the line perpendicular to the parabola's directrix & passing through its focus. In particular, the axis of symmetry is the vertical line through the vertex of the graph of a quadratic function." So, from this, I've deduced that the axis of symmetry is the line perpendicular to a fixed line associated with the parabola, which passes through the focus (which aids in decsribing the conic section). & not only do parabolas have a vertical axis of symmetry, opening up or down, but a horizontal axis of symmetry, opening left or right.

Right, you're going in the right direction. Let me explain further.

 

If you are familiar with the Cartesian plane (x and y coordinate system), x is associated with the horizontal position of a point, and y with the vertical position. Thus, (1, 2) is a point 1 unit to the right and 2 units above the origin (center of the graph).

 

When you have an equation such as [math]y = ax^2[/math], it is describing the relationship between those points. That means that for each point, the y value (vertical position) is the square of the horizontal position (x), multiplied by a (some arbitrary value; when you're given a real equation, a will be a number).

 

The axis of symmetry of a parabola like this one:

would be a line going vertical directly through its center. It's called a line of symmetry because if you were to divide the parabola down this line, each side would be symmetrical - totally identical, only flipped. Assuming that the center of that parabola is located where x = 0 on the plane (the center of the graph), the axis of symmetry is the graph of the line [math]x = 0[/math].

 

I'm afraid I don't know what some of the other equations you have are showing, but I can continue here:

 

I am guessing that in order for me to under the standard equations, I should first understand what each individual component means. Pertaining to graphing an equation of a parabola, I am having trouble grasping a few things taken from the text book. First off, it says identify the focus and directrix of the parabola given by x = -1/6y^2. In order to get rid of the fraction, I understand you multiply by -6 on each side. So:

 

(-6)x = -1/6y^2(-6) giving you -6x = y^2.

 

Because the variable is y^2 the axis of symmetry is horizontal.

You can shortcut that: if the equation is x = something, it will be horizontal; if the equation is y = something, it will be vertical.

 

Here's a good link which links to other pages that describe everything:

http://en.wikipedia.org/wiki/Parabola

If you don't know what something is, type it in to the search box on the left.

 

Don't worry about us labeling you mathematically deficient; it is apparent that you care enough about learning to come to us for help, and you are very good at explaining what you need help with. Unfortunately, I don't know some of the terms myself; I'm guessing I just learned parabolas differently.

 

Good luck.

[JackMuChabas]

Problems with conical indices given a quantuum figure.