Why do linear and quadratic equations form straight line and parabola? I've been trying to figure this out. I searched on the internet but couldn't find the answers either. Why exactly does a quadratic equation form a parabolic shape? and why does a linear equation form a straight line?
It might help you to go back to the original Euclidean geometry.  For example, one theorem in 'The Elements' is: A straight line is the locus of all points equidistant from two (distinct) given points" ('locus of points' just means 'the shape all of the points fall upon and/or trace out').  If you transpose to a Cartesian coordinate system, and accept the obvious translation of the Pythagorean Theorem into an equation for distance between points as a function of coordinates, then you can translate this theorem into an equation between two formulae defining the distance of each 'locus' point from each of the given points.  That equation can be simplified and rearranged into the usual linear expression.

Similarly, for Euclid a parabola is -defined- to be 'the locus of all points equidistant from a given line ('directrix') and a given point ('focus') which is not on that line'.  Restrict the case to a horizontal line [greatly simplifying the equation for the distance from a point to that line] and a point above the line, then translate the definition into an equation of two distance formulae, and you get a form of the quadratic equation.
This is an interesting question.  However, it is currently not answerable.  I will need to put additional constraints on it.

First, from the comments above, you want to know why it is that the linear equation, when plotted, produces a straight line, and why the parabola, when plotted, produces...a parabola.  However, it's not enough to just say "when plotted".  We need to specify coordinate axes. 

Let's begin by working in Euclidean space, where the x and y axes are perpendicular to each other, the origin is at (0,0), and the scales on the x and y axes are the same and the "ticks" on the axes are evenly spaced.  Under these restrictions, then yes, we get straight lines and other things.  Let's go a bit deeper.

We'll begin by stating a fact without proof:

Any differentiable function in the above space that has a constant derivative is representable by a linear equation of the form y = mx+b.

Now, we will think about something: what is the derivative?  Well, it's the rate of change of y with respect to x (in this case, yes it's more complex in other cases, but we don't care at this point).  So, what is a constant derivative?  It would mean that our function's y values change by a constant amount for some given change in the function's x values.  So far, however, this isn't very satisfying - why would this produce the straight line when graphed?  Is there some other way to describe the derivative that might help out?

In fact, one great way to describe it is as the direction of the line that is tangent to the function at a point.  This is just another way to say "slope", but calling it a "direction" allows us to define the derivative geometrically as follows:

Let's say that the derivative at a point $x_0$ is $f'(x_0) = C$.  What this means is this:  If you begin at the point $x_0$ and, initially, you are facing in the same direction as the positive x axis, and you turn clockwise by an angle of $\tan^-1(C)$, then you will be facing in the same direction as the tangent line at that point.  If you now never change your direction and keep walking forever, then a few things happen:

1) you walk straight, never turning, and therefore you trace out a line
2) You are walking along the tangent line at that point.

So the question for why a linear equation traces out a line is that its tangent line (and hence the direction you have to walk) never changes, no matter where you start.

Now, for the parabola, unfortunately the best I can tell you is that it is curved, and that's because the direction of the tangent line DOES change.  However, I can't tell you why it specifically traces out that particular shape.  I'm sure someone else can, but it's not as easy as saying "because the tangent changes", since many other functions have changing tangents as well