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<h4>Graphing Quadratic Functions Using Transformations</h4>
<div class="os-raise-graybox">
<p><strong>GRAPH A QUADRATIC FUNCTION OF THE FORM </strong>\(f(x)=x^2+k\) <strong>USING A VERTICAL SHIFT </strong></p>
<hr>
<p>The graph of \(f(x)=x^2+k\) shifts the graph of \(f(x)=x^2\) vertically \(k\) units. </p>
<ul>
<li>If \(k > 0\), shift the parabola vertically up \(k\) units. </li>
<li>If \(k < 0\), shift the parabola vertically down \(\vert k\vert\) units.</li>
</ul>
</div>
<br>
<p><img
height="254" src="https://k12.openstax.org/contents/raise/resources/a2fcaa4ed02f164aaf13be28bd7fb200cfa8d2e0"
width="254"></p>
<br>
<p>For the functions graphed above, notice that the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>
<ol>
<li>
<p>The parabola that has been shifted <strong>vertically up</strong> 3 units represents the function \( f(x) = x^2 + 3 \) since \( k = 3 \) and \( 3 \gt 0 \) (orange, dashed graph).</p>
</li>
<li>
<p>The parabola that has been shifted <strong>vertically down</strong> 2 units represents the function \( f(x) = x^2 - 2 \) since \( k = -2 \) and \( -2 \lt 0 \) (green, dotted graph).</p>
</li>
</ol>
<br>
<div class="os-raise-graybox">
<p><strong>GRAPH A QUADRATIC FUNCTION OF THE FORM</strong> \(f(x)=(x−h)^2\) <strong>USING A HORIZONTAL SHIFT </strong></p>
<hr>
<p>The graph of \(f(x)=(x−h)^2\) shifts the graph of \(f(x)=x^2\) horizontally \(h\) units. </p>
<ul>
<li>If \(h > 0\), shift the parabola horizontally right \(h\) units. </li>
<li>If \(h < 0\), shift the parabola horizontally left \(\vert h\vert\) units.</li>
</ul>
</div>
<br>
<p><img
height="254" src="https://k12.openstax.org/contents/raise/resources/91e909c941eca894057108c85b7e2fd7b8e753a4"
width="254"></p>
<br>
<p>Again, for the functions graphed above, the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>
<ol>
<li>
<p>The parabola that has been shifted <strong>horizontally right</strong> 3 units represents the function \( f(x) = (x - 3)^2 \) since \( h = 3 \) and \( 3 \gt 0 \) (orange, dashed graph).</p>
</li>
<li>
<p>The parabola that has been shifted <strong>horizontally left</strong> 2 units represents the function \( f(x) = (x + 2)^2 \) since \( h = -2 \) and \( -2 \lt 0 \) (green, dotted graph).</p>
</li>
</ol>
<br>
<div class="os-raise-graybox">
<p><strong>GRAPH OF A QUADRATIC FUNCTION OF THE FORM</strong> \(f(x)=ax^2\) </p>
<hr>
<p>The coefficient a in the function \(f(x)=ax^2\) affects the graph of \(f(x)=x^2\) by stretching or compressing it. </p>
<ul>
<li>If \(0<|a|<1\), the graph of \(f(x)=ax^2\) will be “wider” than the graph of \(f(x)=x^2\). </li>
<li>If \(|a|>1\), the graph of \(f(x)=ax^2\) will be “skinnier” than the graph of \(f(x)=x^2\).</li>
</ul>
</div>
<br>
<p><img
height="254" src="https://k12.openstax.org/contents/raise/resources/a360a4a00b2e68fe8a1665740e9ca15dbf8c9b18"
width="254"></p>
<br>
<p>Again, the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>
<ol>
<li>
<p>The parabola that has been <strong>vertically compressed</strong> by a scale factor of 2 represents the function \( f(x) = \frac{1}{2}x^2 \) since \( a = \frac{1}{2} \) and \( 0 \lt \frac{1}{2} \lt 1 \) (green, dotted graph).</p>
</li>
<li>
<p>The parabola that has been <strong>vertically stretched</strong> by a scale factor of 3 represents the function \( f(x) = 3x^2 \) since \( a = 3 \) and \( 3 \gt 1 \) (orange, dashed graph).</p>
</li>
</ol>
<br>
<div class="os-raise-graybox">
<p><strong>GRAPH OF A QUADRATIC FUNCTION OF THE FORM</strong> \( f(x) = (bx)^2 \)</p>
<p>The coefficient \( b \) in the function \( f(x) = (bx)^2 \) affects the graph of \( f(x) = x^2 \) by stretching or compressing it.</p>
<ol>
<li>
<p>If \( 0 \lt |b| \lt 1 \), the graph of \( f(x) = (bx)^2 \) will appear “wider” than the graph of \( f(x) = x^2 \).</p>
</li>
<li>
<p>If \( |b| \gt 1 \), the graph of \( f(x) = (bx)^2 \) will appear “skinnier” than the graph of \( f(x) = x^2 \).</p>
</li>
</ol>
</div>
<br>
<p><img
height="254" src="https://k12.openstax.org/contents/raise/resources/434aa81867e0772fd826a5fd3c5cbc67a03ced6c"
width="254"></p>
<br>
<p>Again, the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>
<ol>
<li>
<p>The parabola that has been <strong>horizontally compressed</strong> by a scale factor of 3 represents the function \( f(x) = (3x)^2 \) since \( b = 3 \) and \( 3 \gt 1 \) (orange, dashed graph).</p>
</li>
<li>
<p>The parabola that has been <strong>horizontally stretched</strong> by a scale factor of 2 represents the function \( f(x) = \left(\frac{1}{2}x\right)^2 \) since \( b = \frac{1}{2} \) and \( 0 \lt \frac{1}{2} \lt 1 \) (green, dotted graph).</p>
</li>
</ol>
<br>
<h4>Try It: Graphing Quadratic Functions Using Transformations</h4>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<p>How does the graph of \(f(x)=x^2+3\) change from \(f(x)=x^2\)?</p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer. Then select the <strong>s</strong><strong>olution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Here is how to determine shifts in graphs:</p>
<p>The function \(f(x)=x^2\) is the parent function for quadratics. For \(f(x)=x^2+3\) , when 3 is added to \(f(x)=x^2\), the graph shifts up 3 units. This shift occurs because the value of 3 is being added to the output of the \(x^2\) function. Thus, the shift is seen vertically because it affects the output values.</p>
</div>