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.
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.
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include - APP/View/Elements/theorypage.ctp, line 111
View::_evaluate() - CORE/Cake/View/View.php, line 935
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View::_renderElement() - CORE/Cake/View/View.php, line 1195
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B Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 128]
<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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'
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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 420
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 132]Code Context
<?php
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'created' => '0', 'updated' => '0' ), 'Topic' => array( 'id' => '126', 'chapter_id' => '23', 'parent_id' => '206', 'name' => 'Graphical Transformations', 'is_paid' => '0', 'price' => '0.00', 'pre_request' => '{"1":{"text":"Please Try this Goal Definition and Value of Function","id":"120"}}', 'position' => '1', 'status' => '1', 'publish' => '1', 'is_deleted' => '0' ), 'Goal' => array( 'id' => '2', 'name' => 'Goal 2', 'position' => '2', 'status' => '1', 'created' => '0', 'updated' => '0' ), 'Theory' => array( (int) 0 => array( [maximum depth reached] ) ) ) ) $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' $subjectdataHeader = array( (int) 0 => array( 'Subject' => array( 'id' => '6', 'name' => 'Calculus 1', 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935 View::_render() - CORE/Cake/View/View.php, line 897 View::render() - CORE/Cake/View/View.php, line 466 Controller::render() - CORE/Cake/Controller/Controller.php, line 952 Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192 Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160 [main] - APP/webroot/index.php, line 420
C Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 145]
<div class="col-md-6">
<div class="col-sm-6">
<p><span id="ring-brd" class="col-xs-2">C</span> <span><?php echo strip_tags($bucketData[$i]['Bucket']['QuestionOption'][2]['QuestionOption']['description']); ?></span>
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include - APP/View/Elements/theorypage.ctp, line 145
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::_renderElement() - CORE/Cake/View/View.php, line 1195
View::element() - CORE/Cake/View/View.php, line 412
include - APP/View/Students/subject.ctp, line 10
View::_evaluate() - CORE/Cake/View/View.php, line 935
View::_render() - CORE/Cake/View/View.php, line 897
View::render() - CORE/Cake/View/View.php, line 466
Controller::render() - CORE/Cake/Controller/Controller.php, line 952
Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192
Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160
[main] - APP/webroot/index.php, line 420
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 148]Code Context
<?php
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935 View::_render() - CORE/Cake/View/View.php, line 897 View::render() - CORE/Cake/View/View.php, line 466 Controller::render() - CORE/Cake/Controller/Controller.php, line 952 Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192 Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160 [main] - APP/webroot/index.php, line 420
D Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 161]
<div class="col-sm-6">
<p><span id="ring-brd" class="col-xs-2">D</span> <span><?php echo strip_tags($bucketData[$i]['Bucket']['QuestionOption'][3]['QuestionOption']['description']); ?></span>
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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 420
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 165]Code Context
<?php
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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 420
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
</div>
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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $header = '' $m = (int) 2
include - APP/View/Elements/theorypage.ctp, line 234 View::_evaluate() - CORE/Cake/View/View.php, line 935 View::_render() - CORE/Cake/View/View.php, line 897 View::_renderElement() - CORE/Cake/View/View.php, line 1195 View::element() - CORE/Cake/View/View.php, line 412 include - APP/View/Students/subject.ctp, line 10 View::_evaluate() - CORE/Cake/View/View.php, line 935 View::_render() - CORE/Cake/View/View.php, line 897 View::render() - CORE/Cake/View/View.php, line 466 Controller::render() - CORE/Cake/Controller/Controller.php, line 952 Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192 Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160 [main] - APP/webroot/index.php, line 420
Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 237]Code Context<span class="col-xs-1 span-design">A</span>
<?php
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Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 244]Code Context?>
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?>
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<div class="col-md-5 col-xs-12 options-in-solSteps <?php if($bucketData[$i]['Bucket']['QuestionOption'][1]['QuestionOption']['is_correct']) { echo 'correct-Answer'; $correctOption="B"; } ?>">
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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $header = '' $m = (int) 2
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 420
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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Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 259]Code Context<?php
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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $header = '' $m = (int) 2include - 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() - 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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $header = '' $m = (int) 2
include - APP/View/Elements/theorypage.ctp, line 268 View::_evaluate() - CORE/Cake/View/View.php, line 935 View::_render() - CORE/Cake/View/View.php, line 897 View::_renderElement() - CORE/Cake/View/View.php, line 1195 View::element() - CORE/Cake/View/View.php, line 412 include - APP/View/Students/subject.ctp, line 10 View::_evaluate() - CORE/Cake/View/View.php, line 935 View::_render() - CORE/Cake/View/View.php, line 897 View::render() - CORE/Cake/View/View.php, line 466 Controller::render() - CORE/Cake/Controller/Controller.php, line 952 Dispatcher::_invoke() - CORE/Cake/Routing/Dispatcher.php, line 192 Dispatcher::dispatch() - CORE/Cake/Routing/Dispatcher.php, line 160 [main] - APP/webroot/index.php, line 420
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 271]Code Context<span class="col-xs-1 span-design">C</span>
<?php
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Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 278]Code Context<?php
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?>
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<div class="col-md-5 col-xs-12 options-in-solSteps <?php if($bucketData[$i]['Bucket']['QuestionOption'][3]['QuestionOption']['is_correct']){ echo 'correct-Answer'; $correctOption="D"; }?>">
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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $header = '' $m = (int) 2
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 420
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 289]Code Context
<?php
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'created' => '0', 'updated' => '0' ), 'Topic' => array( 'id' => '126', 'chapter_id' => '23', 'parent_id' => '206', 'name' => 'Graphical Transformations', 'is_paid' => '0', 'price' => '0.00', 'pre_request' => '{"1":{"text":"Please Try this Goal Definition and Value of Function","id":"120"}}', 'position' => '1', 'status' => '1', 'publish' => '1', 'is_deleted' => '0' ), 'Goal' => array( 'id' => '2', 'name' => 'Goal 2', 'position' => '2', 'status' => '1', 'created' => '0', 'updated' => '0' ), 'Theory' => array( (int) 0 => array( [maximum depth reached] ) ) ) ) $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' $subjectdataHeader = array( (int) 0 => array( 'Subject' => array( 'id' => '6', 'name' => 'Calculus 1', 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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $header = '' $m = (int) 2include - 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 420
Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 296]Code Context<?php
}
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reached] ), (int) 5 => array( [maximum depth reached] ) ) ), (int) 5 => array( 'Subject' => array( 'id' => '10', 'name' => 'Grade 7 Maths Algebra', 'short_display_name' => 'G-7 Algebra', 'position' => '6', 'status' => '1', 'is_deleted' => '0' ), 'Chapter' => array( (int) 0 => array( [maximum depth reached] ), (int) 1 => array( [maximum depth reached] ), (int) 2 => array( [maximum depth reached] ) ) ) ) $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 $pos2 = false $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 420
Option Notice (8): Undefined variable: correctOption [APP/View/Elements/theorypage.ctp, line 306]
is Correct </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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include - APP/View/Elements/theorypage.ctp, line 306
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 420
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.)
Option A is Correct
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)
Option A is Correct
\(f(Q(x)) = P(Q(x))\)
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 \(f(x)\)
\(=2(x^2+9+6x)+x+3-1\)
\(\)
\(=2x^2+13x+20\)
\(2x^2+13x+20\)
.\(7x^2+8x+9\)
\(-4x^2+8x+1\)
\(20x^2+x-1\)
Option A is Correct
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.
A
B
C
D
Consider the graph of \(g(x)=cos\,x\)
Now stretch vertically by a factor of 3. (Dotted line)
Option A is Correct
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 \('c'\).
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