Learn concept of graphing polynomial, quadratic and power function. Practice polynomial of degree 2 are quadratic functions & equation examples, and find the vertex of parabola an equation.

A \(5x^2 + 8x + 1\)

B \(2x^2 + 3x - 1\)

C \(x^2 + 9x + 3\)

D \(2x^2\)

- A function P of the form

\(P(n) = a_n x^n + a_{n-1} x^{n-1}+ a_{n-2}x^{n-2} + .......... a_1x + a_0 \)

is called a polynomial function of degree n

(where n is non-negative integer) and \(a_n \neq 0\).

\(\star\) The domain of all polynomial function is \(R = (-\infty, \infty)\).

\(\star\) Linear and quadratic functions are special cases of polynomial functions with degree 1 and degree 2.

\(\star\) A polynomial of degree 3 is of the form

\(f(x) = ax^3 + bx^2 + cx + d\) → It is called a cubic function.

\(\star\) Graph of some special cubic functions.

- A function of the form

\(f(x) = x^a\) is called power function.

Here 'a' is any constant value. Depending on what value of 'a', we have different category of function.

- If a = n where n is a positive integer.

\(f(x) = x^n\) (polynomial with one term)

Note that graphs of \(x^2, x^4, x^6, ......, x^{2n}\) one of similar shape whereas those of \(x^3, x^5, x^7, ......\) are similar. As n increases they become steeper.

A quadratic expression given by

\(f(x)=ax^2+bx+C\)

where \(a, b, c\) are real and \(a\neq0\) has always a parabolic shape of its graph.

- The parabola opens upwards or downwards accordingly as 'a' is positive or negative.
- If \(x=ay^2+by+c\), then the graph of y v/s x will be a rightward or leftwards opening parabola depending on where \(a >0\) or \(a <0\)

- A function of the form \(f(x) = ax^2 + bx + c \) is called a quadratic function. Its graph is always of parabolic shape. The parabola opens upward or downward depending on whether a > 0 or a < 0.

A \(f(x)=2x^2-3x\)

B \(f(x)=4x^2+x\)

C \(f(x)=-3x^2+x\)

D \(f(x)=5x^2+9x\)

Graph of odd power of x are symmetric about origin and those of even power are symmetric about y-axis.

Consider the quadratic function \(f(x) = ax^2 +bx+c\)

\(= a\left(x^2+\dfrac{b}{a}x +\dfrac{c}{a}\right) = a \left(\left(x+\dfrac{b}{2a}\right)^2-\dfrac{b^2}{4a^2}+\dfrac{c}{a}\right)\)

\(= a\left(\left(x+\dfrac{b}{2a}\right)^2-\dfrac{D}{4a^2}\right)\) where \(D= b^2 -4ac.\)

- Now if \( a>0 \to\) The graph is an upward opening parabola.
- If \( a<0 \to\) The graph is an downward opening parabola.
- If \(a>0,\) the expression \(f(x)\) has least value when \(x+\dfrac{b}{2a} =0\Rightarrow x=\dfrac{-b}{2a}\) and corresponding value of \(f(x)\) is \(\dfrac{-D}{4a}\) . There is no greatest value as the graph goes to \(\infty\) .

If \(a<0\) the expression \(f(x)\) has greatest value when \(x=\dfrac{-b}{2a}\) and corresponding value of \(f(x)\) is \(\dfrac{-D}{4a}\) . There is no least value as the graph goes to \(-\infty\) .

In both the cases the vertex of the parabola \(f(x) = ax^2 +bx+c\) is the point with the co- ordinates \(\left(\dfrac{-b}{2a}, \dfrac{-D}{4a}\right)\) .

A \(\left(\dfrac{-3}{4},\dfrac{15}{8}\right)\)

B \(\left(\dfrac{-3}{4},\dfrac{-17}{8}\right)\)

C \(\left(\dfrac{5}{2},\dfrac{-17}{8}\right)\)

D \(\left(\dfrac{2}{3},1\right)\)

A \(\dfrac{54}{5}\)

B \(\dfrac{121}{16}\)

C \(\dfrac{132}{5}\)

D \(\dfrac{42}{33}\)