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<h4>Vertical Transformations of Linear Functions</h4>
<p>Given the equation of a linear function, use transformations to graph the linear function in the form \(f(x)=mx+b\).</p>
<p><strong>Example</strong></p>
<p>Graph \(f(x)= \frac12 x&minus;3\) using transformations.</p>
<p><strong>Solution:</strong></p>
<p><strong>Step 1 - </strong> Graph the parent function, \(f(x)=x\). </p>
<p><strong>Step 2 - </strong> Vertically stretch or compress the graph by a factor \(m\). The equation for the function shows that \(m=\frac12\), so the identity function is vertically compressed by \(\frac12\). </p>
<p><img alt="This graph shows two functions on an x, y coordinate plane. One shows an increasing function of y = x divided by 2 that runs through the points (0, 0) and (2, 1). The second shows an increasing function of y = x and runs through the points (0, 0) and (1, 1)." src="https://k12.openstax.org/contents/raise/resources/d5621b67f6db03da1c64ec9b78f685850687e2a2"></p>
<p>This graph shows two functions on an \((x, y)\) coordinate plane. One shows an increasing function of \(y = x\) divided by 2 that runs through the points \((0, 0)\) and \((2, 1)\). The second shows an increasing function of \(y = x\) and runs through the points \((0, 0)\) and \((1, 1)\).</p>
<p><strong>Step 3 - </strong> Shift the graph up or down \(b\) units.  </p>
<p> The equation for the function also shows that \(b=&minus;3\), so the parent function is vertically shifted down 3 units.</p>

<p>Now, show the vertical shift:</p>
<p><img alt="This graph shows two functions on an x, y coordinate plane. The first is an increasing function of y = x divided by 2 and runs through the points (0, 0) and (2, 1).  The second shows an increasing function of y = x divided by 2 minus 3 and passes through the points (0, 3) and (2, -2).  An arrow pointing downward from the first function  to the second function reveals the vertical shift." src="https://k12.openstax.org/contents/raise/resources/fa6330d94e6f7cd4b4fcc0bd7d135a06f1c34dd2"></p>
<p>This graph shows two functions on an \((x, y)\) coordinate plane. The first is an increasing function of \(y = x\) divided by 2 and runs through the points \((0, 0)\) and \((2, 1)\). The second shows an increasing function of \(y = x\) divided by 2 minus 3 and passes through the points \((0, 3)\) and \((2, -2)\). An arrow pointing downward from the first function to the second function reveals the vertical shift.</p>
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<h4>Try It: Vertical Transformations of Linear Functions</h4>
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<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>Name the transformations to go from the parent function, \(f(x)=x\), to \(f(x)=4+2x\).</p>
  </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answer, then select the <strong>solution</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 the transformations:</p>
  <p>First, the slope \(m=2\) tells that there is a vertical stretch of 2.</p>
  <p>Next, the \(y\)-intercept is 4, so there is a vertical shift up 4.</p>
</div>