Graphs of Some Algebraic Functions
Polynomial Functions
A function f: R → R defined as f(x) = a₀ + a₁x + a₂x² + a₃x³ + ... + aₙ₋₁xⁿ⁻¹ + aₙxⁿ where a₀, a₁, ..., aₙ are all real numbers and n (called the degree of polynomial) is a non-negative integer is called a polynomial function. For example, f(x) = 2x³ - x² + x is a polynomial function of degree 3 having leading coefficient 2.
The highest exponent of the variable involved in a polynomial is called its degree. The coefficient of highest degree term in a polynomial is called the leading coefficient.
(i) Zero Polynomial Function
A polynomial function of no degree is called a zero polynomial function. This function is of the form y = f(x) = 0. The graph of y = 0 is a straight line that lies on the x-axis.
(ii) Constant Function
A polynomial function of degree zero is called a constant function. This function is of the form, y = f(x) = a = ax⁰, where a is any non-zero constant. For example, y = 3, y = -1/2 etc. The graph of y = 1 is a straight line parallel to x-axis.
Do You Know? A vertical line intersects the graph of a function only at one point. If it intersects the graph at more than a point, then that graph will not be the graph of a function. This is called vertical line test.
(iii) Linear Function
A polynomial function of degree one is called a linear function. This function is of the form, y = f(x) = ax + b, where a,b ∈ R and a ≠ 0. For example, y = 2x + 3, y = 2x etc. The graph of a linear equation is a straight line.