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 \(f(x) = \sqrt x + 2\)

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include - APP/View/Elements/theorypage.ctp, line 111
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: 0 [APP/View/Elements/theorypage.ctp, line 115]Code Context                                    

                                            <?php 

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$description = 'Learn Graphical transformation function rules, practice how to graph & algebraic expressions functions with transformations, evaluate distinguishing between Cf(x) and F(x) and stretching and reflecting function.'
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$url = 'https://www.actucation.com/'
$i = (int) 0
$j = (int) 3
$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
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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

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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$description = '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.'
$SubtopicName = 'graphical transformations'
$subject = 'Calculus 1'
$subjectUrl = 'calculus-1'
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$url = 'https://www.actucation.com/'
$i = (int) 0
$j = (int) 3
$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
$pos = false
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include - APP/View/Elements/theorypage.ctp, line 128
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                                    

                                            <?php 

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'
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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

Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 145]Code Context                                    <div class="col-md-6">

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

                                            <p><span id="ring-brd" class="col-xs-2">C</span> <span><?php echo strip_tags($bucketData[$i]['Bucket']['QuestionOption'][2]['QuestionOption']['description']); ?></span>
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$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
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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                                    

                                            <?php 

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$description = 'Learn Graphical transformation function rules, practice how to graph & algebraic expressions functions with transformations, evaluate distinguishing between Cf(x) and F(x) and stretching and reflecting function.'
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$subjectUrl = 'calculus-1'
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$url = 'https://www.actucation.com/'
$i = (int) 0
$j = (int) 3
$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
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$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

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>
$viewFile = '/home/bo6uxs8yrzio/public_html/app/View/Elements/theorypage.ctp'
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$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
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include - APP/View/Elements/theorypage.ctp, line 161
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: 3 [APP/View/Elements/theorypage.ctp, line 165]Code Context                                    

                                            <?php 

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include - APP/View/Elements/theorypage.ctp, line 165
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

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]

Code Context

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include - APP/View/Elements/theorypage.ctp, line 234
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

"> A

Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 237]Code Context                                            <span class="col-xs-1 span-design">A</span>

                                            <?php 

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include - APP/View/Elements/theorypage.ctp, line 237
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: 0 [APP/View/Elements/theorypage.ctp, line 244]Code Context                                            ?>

                                            <?php 

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$description = 'Learn Graphical transformation function rules, practice how to graph & algebraic expressions functions with transformations, evaluate distinguishing between Cf(x) and F(x) and stretching and reflecting function.'
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$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
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include - APP/View/Elements/theorypage.ctp, line 244
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 251]

Code Context

                                            ?>

                                        </div>

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include - APP/View/Elements/theorypage.ctp, line 251
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 254]Code Context                                            <span class="col-xs-1 span-design">B</span>

                                            <?php 

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include - APP/View/Elements/theorypage.ctp, line 254
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 259]Code Context                                            <?php 

                                            }

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$description = 'Learn Graphical transformation function rules, practice how to graph & algebraic expressions functions with transformations, evaluate distinguishing between Cf(x) and F(x) and stretching and reflecting function.'
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$url = 'https://www.actucation.com/'
$i = (int) 0
$j = (int) 3
$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
$pos = false
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$header = ''
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include - APP/View/Elements/theorypage.ctp, line 259
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 268]

Code Context

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include - APP/View/Elements/theorypage.ctp, line 268
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

Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 271]Code Context                                            <span class="col-xs-1 span-design">C</span>

                                            <?php 

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$description = 'Learn Graphical transformation function rules, practice how to graph & algebraic expressions functions with transformations, evaluate distinguishing between Cf(x) and F(x) and stretching and reflecting function.'
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$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
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include - APP/View/Elements/theorypage.ctp, line 271
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 278]Code Context                                            <?php 

                                            }

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$str = '<p>We have shifted the graph of<span class="math-tex">\( f(x)\)</span> by 1 units in the upward direction.</p>
'
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include - APP/View/Elements/theorypage.ctp, line 278
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: 3 [APP/View/Elements/theorypage.ctp, line 285]

Code Context

                                            ?>

                                        </div>

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include - APP/View/Elements/theorypage.ctp, line 285
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 289]Code Context                                            

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include - APP/View/Elements/theorypage.ctp, line 289
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: 3 [APP/View/Elements/theorypage.ctp, line 296]Code Context                                            <?php 

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'
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include - 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>
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is Correct

Illustration Questions

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

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 \(g(x)=sin\,3x\) ?

Option A is Correct

Illustration Questions

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

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 \(f(x) = \sqrt {x-3}\) ?

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

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

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

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

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

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.

Illustration Questions

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

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 \(f(x)=3\,cos\,x\)?

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 ?

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

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

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

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

Option A is Correct