How To Draw Graph Of Quadratic Polynomials

Polynomials of Class 10

A quadratic polynomial graph is a parabola. The roots of the quadratic figure are points where the parabola cuts the x axis i.e. points where the quadratic polynomial value is zero. There are certain steps you need to follow while making graph for quadratic polynomials:

Write the given quadratic polynomial f(x)= ax2 + bx+ c as Y = ax2 + bx + c

Calculate the zeros of the polynomial, if exist, by putting y = 0 i.e., ax2 + bx + c

Calculate the points where the curve meets y-axis by putting x = 0.

Calculate D = b2 – 4ac

if D > 0, graph cuts x-axis at two points.

D = 0, graph touches x-axis at one point.

D < 0, graph is far away from x-axis.

Find Quadratic Polynomials which is the turning point of curve.

Make a table of selecting values of x and corresponding values of y, two to three values on left and two to three values on right of turning point

Draw a smooth curve through these points by free hand. The graph so obtained is called a parabola.

Q1. Draw the graph of the polynomial f(x) = x2 – 2x – 8

Sol. Let y = x2 - 2x - 8.

The following table gives the values of y or f(x) for various values of x.

x

-4

-3

-2

-1

0

1

2

3

4

5

6

y = x2 – 2x - 8

16

7

0

-5

-8

-9

-8

-5

0

7

16

Let us plot the points (-4, 16), (-3, 7), (-2, 0), (-1, -5), (0, - 8), (1, - 9), (2, - 8), (3, - 5), (4, 0), (5, 7) and (6, 16) on a graphs paper and draw a smooth free hand curve passing through these points. The curve thus obtained represents the graphs of the polynomial f(x) = x2 - 2x - 8. This is called a parabola.

The lowest point P, called a minimum point, is the vertex of the parabola. Vertical line passing through P is called the axis of the parabola. Parabola is symmetric about the axis. So, it is also called the line of symmetry.

How To Draw Graph Of Quadratic Polynomials

Observations:

For the graphs of the polynomial f(x) = x2 - 2x - 8, following observations can be drawn:

  • The coefficient of x2 in f(x) = x2 - 2x - 8 is 1 (a positive real number) and so the parabola opens upwards.
  • D = b2 - 4ac = 4 + 32 = 36 > 0. So, the parabola cuts X-axis at two distinct points.
  • On comparing the polynomial x2 - 2x - 8 with ax2 + bx + c, we get a = 1, b = –2 and c = –8.

The vertex of the parabola has coordinates (1, -9) i.e. How To Draw Graph Of Quadratic Polynomials, where D ≡ b2 - 4ac.

  • The polynomial f(x) = x2 – 2x – 8 = (x – 4) (x + 2) is factorizable into two distinct linear factors (x – 4) and (x + 2). So, the parabola cuts X-axis at two distinct points (4, 0) and (–2, 0). The x-coordinates of these points are zeros of f(x).

For additional information related to the subject you can check the Maths Formula section.

Graph of a Cubic Polynomial

Graphs of a cubic polynomial does not have a fixed standard shape. Cubic polynomial graphs will always cross X-axis at least once and at most thrice.

Q1. Draw the graphs of the polynomial f(x) = x3 - 4x.

Sol. Let y = f(x) or, y = x2 – 4x.

The values of y for variable value of x are listed in the following table :

x

–3

–2

–1

0

1

2

3

y = x3 – 4x

–15

0

3

0

–3

0

15

Thus, the curve y = x3 – 4x passes through the points (–3, –15), (–2, 0), (–1, 3), (0 ,0), (1, –3), (2, 0), (3, 15), (4, 48) etc.

Plotting these points on a graph paper and drawing a free hand smooth curve through these points, we obtain the graph of the given polynomial as shown figure.

How To Draw Graph Of Quadratic Polynomials

Observations:

For the graphs of the polynomial f(x) = x3 – 4x, following observations are as follows:

  1. The polynomial f(x) = x3 - 4x = x(x2 – 4) = x(x – 2) (x + 2) is factorizable into three distinct linear factors. The curve y = f(x) also cuts X-axis at three distinct points.
  2. We have, f(x) = x (x – 2) (x + 2). Therefore 0, 2 and -2 are three zeros of f(x). The curve y = f(x) cuts X-axis at three points O (0, 0), P(2, 0) and Q (-2, 0).

Relationship Between Zeros and Coefficient of a Quadratic Polynomial

Let α and β be the zeros of a quadratic polynomial f(x) = ax2 + bx + c. By facto r theorem ( x - α) and (x - β) are the factors of f(x).

∴ f(x) = k ( x - α) (x - β) are the factors of f(x)

⇒ ax2 + bx + c = k{x2 - (α + β)x + αβ}

⇒ ax2 + bx + c = kx2 - k(α + β)x + kαβ

Comparing the coefficients of x2, x and constant terms on both sides, we get a = k, b = –k (α+ β) and kαβ

How To Draw Graph Of Quadratic Polynomials

How To Draw Graph Of Quadratic Polynomials

Hence,

Sum of the zeros How To Draw Graph Of Quadratic Polynomials

Product of the zeros How To Draw Graph Of Quadratic Polynomials

REMARKS:

If α and β are the zeros of a quadratic polynomial f(x). The , the polynomial f(x) is given by

f(x) = k{x2 - (α + β)x + αβ}

or f(x) = k{x2 - (Sum of the zeros) x + Product of the zeros}

Q1. Find a quadratic polynomial whose zeros are 5 +√2 and 5 - √2.

Sol. Given α = 5 +√2, β = 5 - √2

∴ f(x) = k{x2 - (α + β)x + αβ}

Here, How To Draw Graph Of Quadratic Polynomials

and How To Draw Graph Of Quadratic Polynomials

= 25 – 2 = 23

∴ f(x) = k {x2 – 10x + 23}, where, k is any non-zero real number.

Relationship Between Zeros and Coefficient of a Cubic Polynomial

Let α,β,γ be the zeros of a cubic polynomial f(x) = ax3 + bx2 + cx + d, a ≠ 0 Then, by factor theorem,(x - α), (x - β) and (x - γ) are factors of f(x). Also, f(x) being a cubic polynomial cannot have more than three linear factors.

∴ f(x) = k (x - α)(x - β) (x - γ)

⇒ ax3 + bx2 + cx + d = k (x - α)(x - β) (x - γ)

⇒ ax3 + bx2 + cx + d = k{x3 How To Draw Graph Of Quadratic Polynomials}

⇒ ax3 + bx2 + cx + d = k x3 - k How To Draw Graph Of Quadratic Polynomials

Comparing the coefficients of x3, x2, x and constant terms on both sides, we get

a = k, b = –k How To Draw Graph Of Quadratic Polynomials

How To Draw Graph Of Quadratic Polynomials

⇒ Sum of the zeros How To Draw Graph Of Quadratic Polynomials

⇒ Sum of the products of the zeros taken two at a time How To Draw Graph Of Quadratic Polynomials

⇒ Product of the zeros How To Draw Graph Of Quadratic Polynomials

Remarks :

Cubic polynomial having α,β and γ as its zeros is given by

∴ f(x) = k (x - α)(x - β) (x - γ)

 

f(x) = k {x3 - How To Draw Graph Of Quadratic Polynomials} where k is any non-zero real number.

Q1. Form a cubic polynomial with zeros α = 3, β = 2, γ = –1 .

Sol. α = 3, β = 2, γ = –1

Required polynomial = k (x - α)(x - β) (x - γ)

= k[(x - 3) (x - 2) (x + 1)]

= k[(x2 - 5x + 6)(x + 1)]

= k[x3 - 5x2 + 6x + x2 - 5x + 6]

= k[x3- 4x2 + x + 6]

where k is a non-zero constant.

Students can also access the class 10 Math Notes from here.

Also Check

  1. Quadratic Equation
  2. Methods Of Solving Quadratic Equation
  3. CBSE Worksheet of Quadratic Equation
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