Informative line

Graphical Transformations

Learn Graphical transformation function rules, practice how to graph & algebraic expressions functions with transformations, evaluate distinguishing between Cf(x) and F(x) and stretching and reflecting function.

Graphical Transformation

If we know the graph of a certain function say \(f(x)\) then we can sketch the graphs of many functions which are related to \('f'\) by making some appropriate changes in the graph of \('f'\).

The following tables show the function on the left and the transformation or change required to be done to the graph of \('f'\)  to obtain the graph of this function.

S.No. Function Transformation on \(f(x)\)
1 \(y=f(x)+c\;\;(c>0)\) Shift the graph 'c' units upwards
2 \(y=f(x)-c\;\;(c>0)\) Shift the graph 'c' units downwards
3 \(y=f(x-c)\;\;(c>0)\) Shift the graph 'c' units to right
4 \(y=f(x+c)\;\;(c>0)\) Shift the graph 'c' units to left.

 

e.g.

 

 

We have shifted the graph of \( f(x)\) by 1 units in the upward direction.

We have shifted the graph of\( f(x)\) by 1 units in the upward direction.

Illustration Questions

Which of the following is the graph of \(f(x) = \sqrt x + 2\)

A

Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 111]
Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 115]

B

Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 128]
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 132]

C

Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 145]
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 148]

D

Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 161]
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 165]

×

Consider \(f(x) = \sqrt x + 2 \space \rightarrow\) function whose graph we know.

image

So, for \(\sqrt {x}+2=f(x)\) graph we shift this graph vertically by 2 units

image

Which of the following is the graph of \(f(x) = \sqrt x + 2\)

Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 234]"> A
Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 237]
Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 244]
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 251]"> B
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 254]
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 259]
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 268]"> C
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 271]
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 278]
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 285]"> D
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 289]
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 296]

Option

Notice (8): Undefined variable: correctOption [APP/View/Elements/theorypage.ctp, line 306]
is Correct

Illustration Questions

Given below is the graph of \(f(x) = sin\,x\), which of the following option will represent the graph of \(g(x)=sin\,3x\) ?

A

B

C

D

×

To sketch the graph of \(f(c\,x)\) form the graph of \(f(x)\) we shrink the graph horizontally by \(c\) times. 

Given the graph of \(f(x)=sin\,x\) 

 

image

We divide each value on \(x\) axis by a factor of 3, to get the graph of \(sin\,3x\) as shown (effectively shrink the graph 3 times horizontally.)

image

Given below is the graph of \(f(x) = sin\,x\), which of the following option will represent the graph of \(g(x)=sin\,3x\) ?

image
A image
B image
C image
D image

Option A is Correct

Illustration Questions

Which of the following represent the graph of  \(f(x) = \sqrt {x-3}\) ?

A

B

C

D

×

Consider the graph of  \(f(x) = \sqrt x \) 

image

Now shift the graph horizontally by 3 units. towards right to get

 (dotted line)

image

Which of the following represent the graph of  \(f(x) = \sqrt {x-3}\) ?

A image
B image
C image
D image

Option A is Correct

Transformation of Trigonometric Functions

  • Suppose we know the graph of \(f(x)\) then graph of following function can be constructed using some appropriate transformation on \('f'\).
S.No. Function Transformation on \(f(x)\)
1 \(y=c\,f(x)\;\;(c>0)\) Stretch vertically by a factor of 'c'.
2 \(y=c\,f(x)\;\;(0<c<1)\) Shrink vertically by a factor of 'c'.
3 \(y=f(c\,x)\;\;(c>1)\) Shrink horizontally by a factor of 'c'.
4 \(y=f(c\,x)\;\;(0<c<1)\) Stretch horizontally by a factor of 'c'.
5 \(y=-f(x)\) Take reflection about \(x\)-axis.
6 \(y=f(-x)\) Take reflection about \(y\)-axis.
 

e.g.

Illustration Questions

Which of the following will represent the graph of \(f(x)=3\,cos\,x\)?

A

B

C

D

×

Consider the graph of \(g(x)=cos\,x\)

image

Now stretch vertically by a factor of 3. (Dotted line)

image

Which of the following will represent the graph of \(f(x)=3\,cos\,x\)?

A image
B image
C image
D image

Option A is Correct

Algebraic Expressions for Transformed Functions

  • If \(f(x) = P(x)\) then

                        \(f(Q(x)) = P(Q(x))\)

Illustration Questions

Let \(f(x)=2x^2+x-1\) be a function then the expression for \(f(x+3)\) will be

A \(2x^2+13x+20\)

B \(7x^2+8x+9\)

C \(-4x^2+8x+1\)

D \(20x^2+x-1\)

×

\(f(x+3)=2(x+3)^2+(x+3)-1\rightarrow\) Replace \(x\) by \(x+3\) in \(f(x)\)

\(=2(x^2+9+6x)+x+3-1\)

\(\)

\(=2x^2+13x+20\)

Let \(f(x)=2x^2+x-1\) be a function then the expression for \(f(x+3)\) will be

A

\(2x^2+13x+20\)

.

B

\(7x^2+8x+9\)

C

\(-4x^2+8x+1\)

D

\(20x^2+x-1\)

Option A is Correct

Distinguishing Between cf(x) and f(x)

(Vertical and horizontal stretching or shrinking)

  • To sketch  \(cf(x)\) , from \(f(x)\,\,for \,(c >1)\) we stretch vertically the graph of 'f' by a factor of c.
  • To sketch  \(f(cx)\)  from the graph of  \(f(x)\, for\, (c >1)\)  we shrink horizontally the graph  by a factor of 'c'

Illustration Questions

To sketch the graph of \(7\,f(x)\) from \(f(x)\) what are we suppose to do ?

A Stretch vertically the graph of \('f'\) by a factor of \(7\)

B Shrink vertically the graph of \('f'\) by a factor of \(7\)

C Stretch horizontally the graph of \('f'\) by a factor of \(7\)

D Shrink horizontally the graph of \('f'\) by a factor of \(7\)  

×

To sketch \(c\,f(x)\)  from \(f(x)\;(c>1)\) we stretch vertically the graph of \('f'\) by a factor of \('c'\)

 

To sketch the graph of \(7\,f(x)\) from \(f(x)\) what are we suppose to do ?

A

Stretch vertically the graph of \('f'\) by a factor of \(7\)

.

B

Shrink vertically the graph of \('f'\) by a factor of \(7\)

C

Stretch horizontally the graph of \('f'\) by a factor of \(7\)

D

Shrink horizontally the graph of \('f'\) by a factor of \(7\)

 

Option A is Correct

Practice Now