An expression in x is said to have nth degree, if the highest degree of the terms of the expression in x is n. Example: 3x^5+5x^3+5x+7, is an expression of 5th degree as the degree of the highest term , 3x^5 , with exponent (or degree) ,5 is 5.
A quadratic expression of the variable x is of the form ax^2+bx+c, where a, b,c are real numbers. It is also an expression of the second degree in x.
Therefore, y(x) < or = >ax^2+bx+c is a quadratic relation between y and x , where y is depending on x. When the equality holds it is an equation, a particular form of relation.
A relation between x and y which is not of second degree is not a quadratic relation. Example:
y < or = or >ax^3+bx^2+cx+d is of higher than second degree . It is not a quadratic relation. It is a cubic relation
y < or = or > ax+b is not a quadratic relation , because ax+b is not of second degree. It is an expression of dgree 1. Moreover it represents a straight line. It is called a linear relation.
Similarly x(y) < or = or > ky^2+ly+ m is a quadratic relation btween y and x, where x is depending upon y.
The graph the function y=ax^2+bx+c, is a parabola, symmetrical about the line x= -b/2a which is parallel to y axis and its vertex at (x,y)= ( -b/(2a), (-b^2+4ac)/4a.
The general expression of quadratic relation between x and y is form:
ax^2+bxy+cy^2+dx+ey+f < or = or > 0 is a form of equality/ inequality, which is in second degree in both x and y. Note that the highest degree terms here : ax^2 of degree 2 in x, bxy of degre 2 for x and y together and cy^2 of degree 2 in y.
Hope this helps.
An equation containing x and y is quadratic if...
There is at least one x^2. (with no y in the term)
If there is more than one x^2 term, the sum of these terms is not zero.
There is no term with a degree higher than 2 (i.e. x^3, x^4, etc)