**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.

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**

**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 the stuff given above, if you want to know more about "Families of quadratic functions", please click here

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

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**