The asymptote of a function y = f(x) or an implicit function f(x,y) = 0 is a straight line such that the distance between the curve of the function y=f(x) or the implicit function(x, y)=0 and the straight line tends to zero when the points on the curve approach to infinity. The number of asymptote of a curve is one or more than one. There are three types of asymptotes (i) Horizontal asymptote (ii) Vertical asymptote and (iii ) Oblique asymptote.

(i) Horizontal asymptote: If the equation of an asymptote is parallel to x-axis, then it is called horizontal asymptote. As for example, a function y=f(x)=1/x has horizontal asymptote y=0. See above the graph of the y=f(x)=1/x.
(ii) Vertical asymptote: If the equation of an asymptote is parallel to y-axis, then it is called vertical asymptote. As for example, a function y=f(x)=1/x has vertical asymptote x=0. See above the graph of the y=f(x)=1/x.

(iii)Oblique asymptote: If the equation of an asymptote is neither parallel to x-axis nor parallel to y-axis , then it is called oblique asymptote. As for example, a function y=f(x)=x+1/x has oblique asymptote y=x. See above the graph of the y=f(x)=1/x.

A linear function is a mathematical function whose graph is a straiht line. A linear function has one independent variable and one dependent variable which are both power one. Let us consider f(x)=ax+b is a linear function which is always represent a straight line and has a slope=a and y-intercept=b units.

A polynomial function of degree n is defined by P(x)=a_nxⁿ+a_n-1x^n-1+a_n-2x^n-2+... ... ..+a₂x^x+a₁x¹+a₀ where a_n ≠0 and n is positive integer. The most common types of polynomial functions are
1. Constant Polynomial Function: P(x) = a = ax⁰
2. Linear Polynomial Function or linear function: P(x) = ax + b
3. Quadratic Polynomial Function or Qudratic function: P(x) = ax²+bx+c
4. Cubic Polynomial Function or cubic function: P(x)=ax³+bx²+cx+d
5. Quartic Polynomial Function: P(x)=ax⁴+bx³+cx²+dx+e, etc.

Polynomial equation

Polynomial equation of degree n is defined by P(x)=0 i.e. a_nxⁿ+a_n-1x^n-1+a_n-2x^n-2+... ... ..+a₂x^x+a₁x¹+a0 =0 where a_n ≠0 and n is positive integer. The most common types of polynomial equation are
1. Linear Polynomial equation or linear equation of one variable: ax + b=0
2. Quadratic Polynomial equation or Qudratic equation of one variable: ax²+bx+c=0
3. Cubic Polynomial equation or cubic equation of one variable: ax³+bx²+cx+d=0
4. Quartic Polynomial equation of one variable: ax⁴+bx³+cx²+dx+e=0, etc.

Let us consider a function f(x). If f''(x)>0 on a interval, then the function is concave up on that interval and if f''(x)'<'0 on a interval, then the function is concave down on that interval. As for example, f(x)=x³-8x is a polynomial function. Now we will find the concavity of the function. We get f''(x)=6x. If f''(x)>0 or 6x>0 or x>0, then the function f(x) is concave up on the interval (0, ∞). Again If f''(x)<0 or 6x<0 or x<0, then the function f(x) is concave down on the interval (-∞, 0). You can also solve the inequality by the graphical method with graphing of inequality. The graph of the function f(x)and its concavity is given bellows-