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.