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Polynomial And Quadratic Functions

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.

Illustration Questions

The expression for quadratic function whose graphs is as shown will be

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

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

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

D \(2x^2\)

×

Assume    \(f(x) = ax^2 + bx + c\)

\(f(1) = 4\)

\(\Rightarrow 4 = a + b + c\)       ......(1)

\(f(0) = -1\)

\(\Rightarrow -1=c\)       ......(2)

\(f(-2) = 1\)

\(\Rightarrow 1=4 a-2b + c\)       ......(3)

Put   \(c = –1\)  in equation (1) & (2)

\(\Rightarrow a+b=5\) and \(4a-2b=2\)

\(\Rightarrow 2a-b=1\)

\(\therefore\;\) on adding \(3a=6\)

\(\Rightarrow a=2, \; b=3,\; c=-1\)

\(\therefore\,f(x)=2{x^2}+3x-1\)

The expression for quadratic function whose graphs is as shown will be

image
A

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

.

B

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

C

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

D

\(2x^2\)

Option B is Correct

Polynomial Functions

  • 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.

Illustration Questions

The degree of the polynomial     \(f(x) = 5x^4 + 7x^3 + 8x^2 + 6x - 3\) is

A –2

B 7

C 8

D 4

×

Degree of a polynomial is the highest power of \(x\) occurring in the expression. In this case it is 4. So degree is 4.

The degree of the polynomial     \(f(x) = 5x^4 + 7x^3 + 8x^2 + 6x - 3\) is

A

–2

.

B

7

C

8

D

4

Option D is Correct

Power 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.

Illustration Questions

Graph of \(x^4, x^6 \space and \space x^8\) are given in the following figure, which one is that of \(f(x) = x^8\)

A 1

B 2

C 3

D none

×

Graph of \(f(x) = x^8\) will be the steepest among the three.

If x > 1 → \(x^8 > x^6 > x^4\) → so steepest & highest after x > 1

If 0 < x < 1 → \(x^8 < x^6 < x^4\) → so will be the lowest in this interval.

Correct answer is 3 (Option C).

Graph of \(x^4, x^6 \space and \space x^8\) are given in the following figure, which one is that of \(f(x) = x^8\)

image
A

1

.

B

2

C

3

D

none

Option C is Correct

Graphs of Quadratic Expression

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 \(a <0\)

Illustration Questions

Which of the following can be graph of \(y=2ax^2+bx+c\) where \(b\) and \(c\) are real number?

A

B

C

D

×

\(y=2ax^2+bx+c\) will be an upward opening parabola.

\(\therefore\;\;\) Option (D) is correct.

Note that 'b' and 'c' have no role to play in the opening of parabola.

Which of the following can be graph of \(y=2ax^2+bx+c\) where \(b\) and \(c\) are real number?

A image
B image
C image
D image

Option D is Correct

Identifying the Quadratic Function from its Graph

  • 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.

Illustration Questions

The expression for the quadratic whose graph is as shown is

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

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

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

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

×

Let \(f(x)=ax^2+bx+c\)

\(f(0)=0\;\Rightarrow c=0\)

\(f(1)=5\;\Rightarrow a+b=5\)

\(f(-2)=2\;\Rightarrow 4a-2b=2\)

Solving we get \(a=2,\;b=3\)

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

The expression for the quadratic whose graph is as shown is

image
A

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

.

B

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

C

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

D

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

Option A is Correct

Graphs of Odd and Even Powers of \(x\)

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

Illustration Questions

Which of the following graph does not represent correctly the function indicated against it?

A

B

C

D

×

Graph of odd & even power of \(x\) are as indicated

image image

\(\therefore\) 'B' is incorrect.

image

Which of the following graph does not represent correctly the function indicated against it?

A image
B image
C image
D image

Option B is Correct

Vertex of Parabola and Greatest Value of the Quadratic Function

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 \(\left(\dfrac{-b}{2a}, \dfrac{-D}{4a}\right)\) .

Illustration Questions

Find the co- ordinates of vertex of the parabola  \(y= 2x^2+3x-1\) .

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

×

The vertex of the parabola  \(y= ax^2 +bx+c\)  is \(\left(\dfrac{-b}{2a},\dfrac{-D}{4a} \right)\)

 

In this  case the given quadratic is  \(y= 2x^2+3x-1\) 

\(\therefore \)  vertex is \(\left(\dfrac{-3}{2×2}, \dfrac{-(9-4× -1×2)}{4×2}\right)\) 

i.e \(\left(\dfrac{-3}{4}, \dfrac{-17}{8}\right)\)

Find the co- ordinates of vertex of the parabola  \(y= 2x^2+3x-1\) .

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

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

Option B is Correct

Illustration Questions

Find the greatest value of the quadratic function  \(f(x) = -4x^2-3x+7\) .

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

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

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

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

×

If \(a<0\)  ,the greatest  value of  \(f(x) = ax^2+bx+c\)   is given by \(\dfrac{-D}{4a}\)  where \(D= b^2-4ac\)

 

\(\therefore\)The  greatest value of \(f(x)= -4x^2-3x+7\) is 

\(\Rightarrow- \left(\dfrac{9-4× 7×-4}{4×-4}\right) = \left(\dfrac{-121}{-16}\right) = \dfrac{121}{16}\)

Find the greatest value of the quadratic function  \(f(x) = -4x^2-3x+7\) .

A

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

.

B

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

C

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

D

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

Option B is Correct

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