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292 lines
14 KiB
HTML
292 lines
14 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/sets/function-transformations.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:00:37 GMT -->
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<title>Function Transformations</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Function Transformations</h1>
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<table border="0">
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<tr>
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<td><img src="../geometry/images/congruent-flip.svg" alt="move and flip" /></td>
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<td>Just like <a href="../geometry/transformations.html">Transformations in Geometry</a>, we can move and resize the graphs of functions</td>
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</tr>
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</table>
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<p>Let us start with a function, in this case it is <span class="large">f(x) = x<sup>2</sup></span>, but it could be anything:</p>
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<p align="center"><img src="images/function-square.svg" alt="Square function" /></p>
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<p align="center"><span class="larger"> f(x) = x<sup>2</sup></span></p>
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<p>Here are some simple things we can do to move or scale it on the graph:</p>
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<h3>We can move it up or down by adding a constant to the y-value:</h3>
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<p align="center"><img src="images/function-translation-a.gif" alt="Translation" width="219" height="190" /></p>
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<p align="center" src="images/function-translation-a.gif"><span class="larger">g(x) = x<sup>2</sup> + C</span></p>
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<p>Note: to move the line <b>down</b>, we use a <b>negative</b> value for C. </p>
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<ul>
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<li>C > 0 moves it up</li>
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<li>C < 0 moves it down</li>
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</ul>
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<h3> </h3>
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<h3>We can move it left or right by adding a constant to the x-value:</h3>
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<p align="center"><img src="images/function-translation-e.gif" alt="Translation" width="246" height="221" /></p>
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<p align="center"><span class="larger">g(x) = (x+C)<sup>2</sup></span></p>
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<div class="center80">
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<p><span class="larger"><b>Adding C</b> moves the function to the <b>left</b> (the negative direction).</span> </p>
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<p><b>Why?</b> Well imagine you will inherit a fortune when your <b>age=25</b>. If you change that to <b>(age+4) = 25</b> then you will get it when you are 21. Adding 4 made it happen earlier.</p>
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</div>
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<ul>
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<li>C > 0 moves it left</li>
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<li>C < 0 moves it right</li>
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</ul>
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<br>
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<p>BUT we must <b>add C wherever x appears</b> in the function (we are <a href="../algebra/substitution.html">substituting</a> x+C for x).</p>
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<div class="example">
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<h3>Example: the function v(x) = x<sup>3</sup> - x<sup>2</sup> + 4x</h3>
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<p>To move C spaces to the left, add C to x <b>wherever x appears</b>: </p>
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<p class="center"><b>w(x) = (x <span class="hilite">+ C</span>)<sup>3</sup> − (x <span class="hilite">+ C</span>)<sup>2</sup> + 4(x <span class="hilite">+ C</span>)</b></p>
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</div>
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<p> </p>
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<div class="center80">
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<p>An easy way to remember what happens to the graph when we add a constant:</p>
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<p align="center" class="larger">add to <b>y</b> to go <b>high</b>
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<br /> add to <b>x</b> to go <b>left</b></p>
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</div>
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<p> </p>
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<h3>We can stretch or compress it in the y-direction by multiplying the whole function by a constant.</h3>
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<p align="center"><img src="images/function-scaling-a.gif" alt="Scaling" width="217" height="222" /></p>
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<p align="center"><span class="larger">g(x) = 0.35(x<sup>2</sup>)</span></p>
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<ul>
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<li>C > 1 stretches it</li>
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<li>0 < C < 1 compresses it</li>
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</ul>
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<h3> </h3>
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<h3>We can stretch or compress it in the x-direction by multiplying x by a constant.</h3>
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<p align="center"><img src="images/function-scaling-b.gif" alt="Scaling" width="161" height="221" /></p>
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<p align="center"><span class="larger">g(x) = (2x)<sup>2</sup></span></p>
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<ul>
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<li>C > 1 compresses it</li>
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<li>0 < C < 1 stretches it</li>
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</ul>
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<p>Note that (unlike for the y-direction), <b>bigger</b> values cause more <b>compression</b>.</p>
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<h3> </h3>
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<h3>We can flip it upside down by multiplying the whole function by −1:</h3>
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<p align="center"><img src="images/function-scaling-c.gif" alt="Scaling" width="217" height="248" /></p>
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<p align="center"><span class="larger">g(x) = −(x<sup>2</sup>)</span></p>
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<p>This is also called <b>reflection about the x-axis</b> (the axis where y=0)</p>
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<p>We can combine a negative value with a scaling:</p>
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<div class="example">
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<p>Example: multiplying by −2 will flip it upside down AND stretch it in the y-direction.</p>
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</div>
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<h3> </h3>
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<h3>We can flip it left-right by multiplying the x-value by −1:</h3>
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<p align="center" src="images/function-scaling-d.gif"><img src="images/function-scaling-d.gif" alt="Scaling" width="246" height="221" /></p>
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<p align="center"><span class="larger">g(x) = (−x)<sup>2</sup></span></p>
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<p><b>It really does flip it left and right!</b> But you can't see it, because <span class="larger">x<sup>2</sup></span> is <a href="../algebra/equation-symmetry.html">symmetrical about the y-axis</a>. So here is another example using <span class="larger">√(x)</span>:</p>
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<p align="center" src="images/function-scaling-d.gif"><img src="images/function-scaling-e.gif" alt="Scaling" width="221" height="137" /></p>
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<p align="center"><span class="larger">g(x) = √(−x)</span></p>
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<p>This is also called <b>reflection about the y-axis</b> (the axis where x=0)</p>
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<h2>Summary</h2>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td class="larger">y = f(x) + C</td>
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<td>
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<ul>
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<li>C > 0 moves it up</li>
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<li>C < 0 moves it down</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td class="larger">y = f(x + C)</td>
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<td>
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<ul>
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<li>C > 0 moves it left</li>
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<li>C < 0 moves it right</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td class="larger">y = Cf(x)</td>
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<td>
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<ul>
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<li>C > 1 stretches it in the y-direction</li>
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<li>0 < C < 1 compresses it</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td class="larger">y = f(Cx)</td>
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<td>
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<ul>
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<li>C > 1 compresses it in the x-direction</li>
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<li>0 < C < 1 stretches it</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td class="larger">y = −f(x)</td>
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<td>
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<ul>
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<li>Reflects it about x-axis</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td class="larger">y = f(−x)</td>
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<td>
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<ul>
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<li>Reflects it about y-axis</li>
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</ul>
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</td>
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</tr>
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</table>
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</div>
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<p> </p>
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<h2>Examples</h2>
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<div class="example">
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<h3>Example: the function g(x) = 1/x</h3>
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<p><i>Here are some things we can do:</i></p>
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<div class="tbl">
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<div class="row"><span class="left">Move 2 spaces up:</span><span class="right"><b>h(x) = 1/x + 2</b></span></div>
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<div class="row"><span class="left">Move 3 spaces down:</span><span class="right"><b>h(x) = 1/x − 3</b></span></div>
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<div class="row"><span class="left">Move 4 spaces right:</span><span class="right"><b>h(x) = 1/(x−4) </b><a href="../data/function-grapher15b4.html?func1=1/(x-4)&xmin=-12&xmax=12&ymin=-8&ymax=8">graph</a></span></div>
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<div class="row"><span class="left">Move 5 spaces left:</span><span class="right"><b>h(x) = 1/(x+5)</b></span></div>
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<div class="row"><span class="left">Stretch it by 2 in the y-direction:</span><span class="right"><b>h(x) = 2/x</b></span></div>
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<div class="row"><span class="left">Compress it by 3 in the x-direction:</span><span class="right"><b>h(x) = 1/(3x)</b></span></div>
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<div class="row"><span class="left">Flip it upside down:</span><span class="right"><b>h(x) = −1/x</b></span></div>
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</div>
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</div>
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<div class="example">
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<h3>Example: the function v(x) = x<sup>3</sup> − 4x</h3>
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<p><i>Here are some things we can do:</i></p>
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<div class="tbl">
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<div class="row"><span class="left">Move 2 spaces up:</span><span class="right"><b>w(x) = x<sup>3</sup> − 4x + 2</b></span></div>
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<div class="row"><span class="left">Move 3 spaces down:</span><span class="right"><b>w(x) = x<sup>3</sup> − 4x − 3</b></span></div>
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<div class="row"><span class="left">Move 4 spaces right:</span><span class="right"><b>w(x) = (x−4)<sup>3</sup> − 4(x−4)</b></span></div>
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<div class="row"><span class="left">Move 5 spaces left:</span><span class="right"><b>w(x) = (x+5)<sup>3</sup> − 4(x+5)</b> <a href="../data/function-grapherf7a5.html?func1=x^3-4x&func2=(x+5)^3-4(x+5)&xmin=-12&xmax=12&ymin=-8&ymax=8">graph</a></span></div>
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<div class="row"><span class="left">Stretch it by 2 in the y-direction:</span><span class="right"><b>w(x) = 2(x<sup>3</sup> − 4x)<br>
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= 2x<sup>3</sup> − 8x</b></span></div>
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<div class="row"><span class="left">Compress it by 3 in the x-direction:</span><span class="right"><b>w(x) = (3x)<sup>3</sup> − 4(3x)<br>
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= 27x<sup>3</sup> − 12x</b></span></div>
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<div class="row"><span class="left">Flip it upside down:</span><span class="right"><b>w(x) = −x<sup>3</sup> + 4x</b></span></div>
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</div>
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</div>
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<h2>All In One ... !</h2>
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<p>We can do all transformation <b>in one go</b> using this:</p>
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<p align="center" class="larger"><img src="images/function-transform-all.gif" alt="af( b(x + c) ) + d " width="295" height="62" />
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<br /> </p>
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<div class="dotpoint">
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<p><span class="largest">a</span><span class="larger"> is vertical stretch/compression </span></p>
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<ul>
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<li>|a| > 1 stretches
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<br /> </li>
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<li>|a| < 1 compresses
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<br /> </li>
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<li>a < 0 flips the graph upside down</li>
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</ul>
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</div>
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<div class="dotpoint">
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<p><span class="largest">b</span><span class="larger"> is horizontal stretch/compression</span></p>
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<ul>
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<li>|b| > 1 compresses
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<br /> </li>
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<li>|b| < 1 stretches
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<br /> </li>
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<li>b < 0 flips the graph left-right</li>
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</ul>
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</div>
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<div class="dotpoint">
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<p> <span class="largest">c</span><span class="larger"> is horizontal shift </span></p>
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<ul>
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<li>c < 0 shifts to the right
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<br /> </li>
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<li>c > 0 shifts to the left</li>
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</ul>
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</div>
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<div class="dotpoint">
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<p><span class="largest">d</span><span class="larger"> is vertical shift</span></p>
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<ul>
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<li>d > 0 shifts upward
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<br /> </li>
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<li>d < 0 shifts downward</li>
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</ul>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: 2√(x+1)+1</h3>
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<p align="center">a=2, c=1, d=1</p>
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<p>So it takes the square root function, and then</p>
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<ul>
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<li>Stretches it by 2 in the y-direction </li>
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<li>Shifts it left 1, and</li>
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<li>Shifts it up 1 </li>
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</ul>
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<p><a href="../data/function-grapher77bb.html?func1=sqrt(x)&func2=2*sqrt(1*(x+1))+1&xmin=-12&xmax=12&ymin=-8&ymax=8">Play with this graph</a></p>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">
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getQ(7260, 7261, 7262, 7267, 7268, 555, 556, 557, 558, 1191);
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</script> </div>
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<div class="related"><a href="function.html">What is a Function?</a> <a href="../algebra/index.html">Algebra Index</a></div>
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