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.

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

A

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$pos2 = false$header = ''include - APP/View/Elements/theorypage.ctp, line 115
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108

B

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

                                        <div class="col-sm-6">

                                            <p><span id="ring-brd" class="col-xs-2">B</span> <span><?php echo strip_tags($bucketData[$i]['Bucket']['QuestionOption'][1]['QuestionOption']['description']); ?></span>
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View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 132]Code Context

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$pos2 = false$header = ''include - APP/View/Elements/theorypage.ctp, line 132
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108

C

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$pos2 = false$header = ''include - APP/View/Elements/theorypage.ctp, line 145
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 148]Code Context

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$pos2 = false$header = ''include - APP/View/Elements/theorypage.ctp, line 148
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108

D

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

                                        <div class="col-sm-6">

                                            <p><span id="ring-brd" class="col-xs-2">D</span> <span><?php echo strip_tags($bucketData[$i]['Bucket']['QuestionOption'][3]['QuestionOption']['description']); ?></span>
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Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
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×

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

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

### 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]Code Context                                            <span class="col-xs-1 span-design">A</span>

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View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
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View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108
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]Code Context                                            <span class="col-xs-1 span-design">B</span>

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View::_evaluate() - CORE/Cake/View/View.php, line 935
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include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
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[main] - APP/webroot/index.php, line 108
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]Code Context                                            <span class="col-xs-1 span-design">C</span>

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View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
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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]Code Context

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'
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$header = ''$m = (int) 2include - APP/View/Elements/theorypage.ctp, line 296
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 108

Option

Notice (8): Undefined variable: correctOption [APP/View/Elements/theorypage.ctp, line 306]Code Context                                </div>

                                <div class="col-xs-12 col-sm-8 col-sm-offset-2 col-md-4 col-md-offset-4 hence-answer">

                                     <p class=""> Option <span><?php echo $correctOption; ?></span> is Correct</p> $viewFile = '/home/bo6uxs8yrzio/public_html/app/View/Elements/theorypage.ctp'
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is Correct

#### 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 $$f(x)=sin\,x$$

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.)

### 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

Option A is Correct

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

A

B

C

D

×

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

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

(dotted line)

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

A
B
C
D

Option A is Correct

# Algebraic Expressions for Transformed Functions

• If $$f(x) = P(x)$$ then

$$f(Q(x)) = P(Q(x))$$

#### 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

# 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.

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

A

B

C

D

×

Consider the graph of $$g(x)=cos\,x$$

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

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

A
B
C
D

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'

#### 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