Informative line

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.

#### 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

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.

#### 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.

#### 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 $$f(x) = x^8$$

A

1

.

B

2

C

3

D

none

Option C is Correct

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

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 $$a <0$$

#### 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
B
C
D

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.

#### 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

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.

#### 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

$$\therefore$$ 'B' is incorrect.

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

A
B
C
D

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

#### 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

#### 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