a945b15e-aac4-48dd-a13d-c6c699025cbd.html

<h4>Horizontal Transformations of Linear Functions</h4>
<p>Horizontal transformations are the result of altering the input values before applying the function rule. Vertical transformations result from altering the output values after the function rule has been applied.</p>

<h4>Example 1</h4>
<p>Describe how \( f(x) \) is transformed into \( f(x - 3) \).</p>
<p><strong>Solution:</strong></p>
<p>First, graph the given function, \( f(x) \), the parent linear function.</p>
<br><img height="400" src="https://k12.openstax.org/contents/raise/resources/61cfb03b386e5266f81b2e3ca27eea9b9ffc4262"><br>
<p>Now, let’s examine an input value of \(x=2\) using \(f(x)\).</p>
<ul>
  <li>For the parent function \(f(x)\), all points on the line are represented by \((x, f(x))\).</li> 
  <li>So, when \(x = 2\), then \(f(x)\) becomes \(f(2)\).</li> 
  <li>That means this point is located at \((2, f(2))\) or \((2, 2)\).</li></ul>
<p>In order to get the same output value for \(f(x-3)\) as we had for \(f(x)\), we need to figure out how to change the input value. 
<ul>
    <li>Because our transformed equation is subtracting three from what we input, we have to increase the prior input value by 3 units to yield the same output as \(f(x)\).</li>
  <li>The input value that is 3 units larger than \(x = 2\) is 5. In the transformed equation, we must use \(x=5\).</li>
</ul>
<p>So, in the original equation where \(x = 2\) resulted in \(f(2) = 2\), in the transformed equation, we have to use \(x = 5\) in order for \(f(x-3) = f(5-3) = f(2)\).</p>

<br><img height="400" src="https://k12.openstax.org/contents/raise/resources/8f9d902064d1fc2e5f2f3ae3446228943b50a051"><br>


<p>On the graph, \(x = 5\) is located 3 units to the right of \(x = 2\). So, the transformed equation of \(f(x-3)\) is 3 units to the right of \(f(x)\).</p> 

<p>Horizontal transformations are represented by the rule \(f(x - c)\) where \(c\) indicates the number of units to the right or left that the graph is shifted.</p>

<h4>Example 2</h4>
<p>Graph \( f(x + 2) \) when \( f(x) = \frac{1}{2}x - 3 \). Then, determine the equation of the transformed function.</p>
<p><strong>Solution:</strong></p>

  <p><strong>Step 1 - </strong>Graph the given function, \( f(x) = \frac{1}{2}x - 3 \).</p>
<p>
    The y-intercept is (0, -3) and the slope is \(\frac{1}{2}\).
    <br>
    For horizontal shifts, it is helpful to identify the x-intercept. On our graph, the x-intercept or zero is located at (6,0). 
</p>
  
<br><p><img height="400" src="https://k12.openstax.org/contents/raise/resources/7c298488a153cf8edefee6e5013ed6689356bb41" alt="RAISE 4.11.5 AR Ex0 graph" title="RAISE 4.11.5 AR Ex0 graph"></p><br>

  <p><strong>Step 2 - </strong>Apply the designated transformation. Remember that the rule for a horizontal shift is \( f(x - c) \). If we rewrite \( f(x + 2) \) in an equivalent form, we can make it is easier to see \( x - c \).</p>

<p>\( f(x + 2) = f(x - (-2)) \) That means \( c = -2 \).</p>
<p>So, \( f(x + 2) \) represents a horizontal shift of two units to the left. On our graph, we move the x-intercept two units left from (6, 0) to (4, 0).</p>
<br><p><img height="400" src="https://k12.openstax.org/contents/raise/resources/7001e55a2d1f2cc14391c9d6c025175e4db07cb5" alt="Transformed Function Graph"></p><br>
<p>The orange dashed line is the graph of the original function and the blue solid line is the transformed function.</p>
<p><strong>Step 3 - </strong>Determine the equation of the transformed function.</p>
<p>The y-intercept of the transformed line is (0, -2) and the slope of the line is \(\frac{2}{4}\) or \(\frac{1}{2}\).</p>
<p>That means the equation \( f(x + 2) = \frac{1}{2}x - 2 \)</p>
<p>Want to check your work? We can check our equation algebraically by substituting \( x + 2 \) into \( f(x) \).</p>
<p>\( f(x) = \frac{1}{2}x - 3 \)</p>
<p>\( f(x+2) = \frac{1}{2}(x + 2) - 3 = \frac{1}{2}x + 1 - 3 = \frac{1}{2}x - 2 \) ✔</p>

<h4>Example 3</h4>
<p>Graph \( f(4x) \) when \( f(x) = \frac{1}{2}x - 3 \). Then, determine the equation of the transformed function.</p>
<p><strong>Solution:</strong></p>
<p><strong>Step 3 - </strong> Graph the given function, \( f(x) = \frac{1}{2}x - 3 \).</p>
  <p>The y-intercept is (0, -3) and the slope is \(\frac{1}{2}\).</p>
  <p>For horizontal shifts, it is helpful to identify the x-intercept. On our graph, the x-intercept or zero is located at (6,0).</p>
  <br>
  <p><img height="400" src="https://k12.openstax.org/contents/raise/resources/7c298488a153cf8edefee6e5013ed6689356bb41" alt="Graph of f(4x)"></p>
<br>
<p><strong>Step 2 - </strong>Apply the designated transformation. Remember that the rule for a horizontal shift is \( f(bx) \). And, when the value of \( b \) is greater than 1, the function experiences a horizontal compression.</p>

<p>As a result of the horizontal compression, the linear function should look steeper because it is being compressed by a scale factor of 4.</p>
<p>On our graph, this means the x-intercept is compressed from (6, 0) to (1.5, 0).</p><br>
<p><img height="400" src="https://k12.openstax.org/contents/raise/resources/927791cf4178250c5bc0dc28647a22ee79115553" alt="Graph of f(4x)"></p><br>
<p>The orange dashed line is the graph of the original function and the blue solid line is the transformed function.</p>
<p><strong>Step 3 - </strong>Determine the equation of the transformed function.</p>
<p>The y-intercept of the transformed line is (0, -3) and the slope of the line is 3/1.5 or 2.</p>
<p>That means the equation \( f(4x) = 2x - 3 \).</p>
<p>Want to check your work? We can check our equation algebraically by substituting 4x into \( f(x) \).</p>
<p>\( f(x) = \frac{1}{2}x - 3 \)</p>
<p>\( f(4x) = \frac{1}{2}(4x) - 3 = 2x - 3 \) ✔</p>

<h4>Try It: Horizontal Transformations of Linear Functions</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>Name the transformation to go from the parent function, \( f(x) = x \), to \( f(x - 1) \).</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 transformation:</p>
  <p>Because \( f(x - c) \) defines a horizontal shift, the transformation \( f(x - 1) \) moves the parent function 1 unit to the right.</p>
</div>