FAMILIES OF QUADRATIC  FUNCTIONS

A group of quadratic functions which all share a common characteristic is called family of quadratic functions.

To know more about different families of quadratic functions, we have to know the different forms in which quadratic functions can be expressed. 

Let us come to know the different forms of quadratic functions. 

Different Forms of Quadratic Functions

Quadratic functions can be expressed in the following three different algebraic forms. 

Standard form : f(x) = ax² + bx + c

Factored form : f(x) = a(x - r)(x - s)

Vertex form : f(x) = a(x - h)² + k

In vertex form, vertex of the parabola is (h, k) and the axis of symmetry is x = h.

Family of parabolas :

A group of parabolas which all share a common characteristic. 

Families of Quadratic Functions

Family 1 :

If the values of a and b are varied in a quadratic function expressed in standard form, f(x)  =  ax² + bx + c, a family of parabolas with the same y–intercept is created.

Common characteristic  :  

Same y - intercept

Family 2 :

If the value of a is varied in a quadratic function expressed in factored form, f(x)  =  a(x - r)(x - s), a family of parabolas with the same x–intercepts and axis of symmetry is created.

Common characteristic  :

 Same x - intercepts and Axis of symmetry

Family 3 :

If the value of a is varied in a quadratic function expressed in vertex form, f(x)  =  a(x - h)² + k, a family of parabolas with the same vertex and axis of symmetry is created.

Common characteristic  :

 Same vertex and Axis of symmetry

Determining the Family of Quadratic Functions

Example :

Write each function in vertex form and check whether they all belong to the same family.

(i) f(x) = -2x² + 4x + 1

(ii) f(x) = -6x² + 12x - 7

(iii) f(x) = 10x² - 20x + 9

Solution :

Let us write the all the given three quadratic functions in vertex form

(i) f(x) = -2x² + 4x + 1

f(x) = -2x² + 4x + 1

= -2(x² - 2x) + 1

= -2(x² - 2x(1) + 1² - 1²) + 1

= -2[(x - 1)² - 1²] + 1

= -2[(x - 1)² - 1] + 1

= -2(x - 1)² + 2 + 1

f(x) = -2(x - 1)² + 3

Vertex (1, 3) and Axis of symmetry is x = 1

(ii)  f(x)  =  -6x² + 12x - 7

f(x) = -6x² + 12x - 7

= -6(x² - 2x) - 7

= -6(x² - 2x(1) + 1² - 1²) - 7

= -6[(x - 1)² - 1²] - 7

= -6[(x - 1)² - 1] - 7

= -6(x - 1)² + 6 - 7

f(x) = -6(x - 1)² - 1

Vertex (1, -1) and Axis of symmetry is x = 1

(iii)  f(x) = 10x² - 20x + 9 

f(x) = 10x² - 20x + 9

= 10(x² - 2x) + 9

= 10(x² - 2x(1) + 1² - 1²) + 9

= 10[(x - 1)² - 1²] + 9

= 10[(x - 1)² - 1] + 9

= 10(x - 1)² - 10 + 9

f(x) = 10(x - 1)² - 1

Vertex (1, -1) and Axis of symmetry is x = 1

When we write all the given three quadratic functions in the form, we get same vertex and same axis of symmetry for all the three parabolas. 

So, all the given three quadratic functions are belonging to the same family. 

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