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 = k

Family of parabolas : A group of parabolas which all share a common characteristic.

Family 1 :

If the values of a & 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

f(x)  =  -2(x² - 2x) + 1

f(x)  =  -2(x² - 2x(1) + 1² - 1²) + 1

f(x)  =  -2[(x - 1)² - 1²] + 1

f(x)  =  -2[(x - 1)² - 1] + 1

f(x)  =  -2(x - 1)² - 2 + 1

f(x)  =  -2(x - 1)² - 1

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

(i)  f(x)  =  -6x² + 12x - 7

f(x)  =  -6x² + 12x - 7

f(x)  =  -6(x² - 2x) - 7

f(x)  =  -6(x² - 2x(1) + 1² - 1²) - 7

f(x)  =  -6[(x - 1)² - 1²] - 7

f(x)  =  -6[(x - 1)² - 1] - 7

f(x)  =  -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

f(x)  =  10(x² - 2x) + 9

f(x)  =  10(x² - 2x(1) + 1² - 1²) + 9

f(x)  =  10[(x - 1)² - 1²] + 9

f(x)  =  10[(x - 1)² - 1] + 9

f(x)  =  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.

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

After having gone through the stuff given above, we hope that the students would have understood "Families of quadratic functions".

Apart from "Families of quadratic functions", if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6