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eb915bae-f79f-44ca-ad25-082b0b456ed8.html
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<h4>Activity</h4>
<p>Here is a graph of \(y=f(x)\) where \(f(x)=2x+5\).</p>
<p><img alt="Graph of an increasing linear function, x y plane, origin O." src="https://k12.openstax.org/contents/raise/resources/e22c1ae4cc52cd736d0fa6fa34869c1f9ff0e9d9"></p>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="79afb7d5-e724-4c7f-bfd0-f099878524f4" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li> How do the values of \(f\) change whenever \(x\) increases by 1, for instance, when it increases from 1 to 2, or from 19 to 20? Be prepared to explain or show how you know. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>The value of \(f\) increases by 2 whenever the input increases by 1. For example: </p>
<ol class="os-raise-noindent" type="a">
<li> From the graph, we can see the \(y\)-value increasing by 2 when the \(x\)-value increases by 1. </li>
<li> \(f(1)=7\) and \(f(2)=9\). The difference between \(f(1)\) and \(f(2)\) is 2. Likewise, \(f(20)=45\) and \(f(19)=43\), and \(f(20)−f(19)=2\). </li>
</ol>
</div>
</div>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="f2763e93-9d9c-4908-8a90-0ad03fa23dfa" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li> Here is an expression we can use to find the difference in the values of \(f\) when the input changes from \(x\) to \(x+1\). </li>
</ol>
<blockquote>
<p>\(\lbrack 2(x+1)+5 \rbrack − \lbrack 2x+5 \rbrack \)</p>
<p>Does this expression have the same value as what you found in the previous questions? Show your reasoning.</p>
</blockquote>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Yes. \(2(x+1)=2x+2\). Replacing \(2(x+1)\) with \(2x+2\) and performing arithmetic gives \(f(x+1)−f(x)=2\). </p>
</div>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="2102bc77-0928-4612-9d18-0405ce5504c7" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li> How do the values of \(f\) change whenever \(x\) increases by 4? Explain or show how you know. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> The values of \(f\) change by 8 if the input is increased by 4. Increasing the input by 4 is the same as increasing the input by 1 four successive times, and each time the value of \(f\) increased by 2. </p>
</div>
</div>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="16694d15-5b9e-425f-96d6-aeaa9f53dce8" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li> Write an expression that shows the change in the values of \(f\) when the input value changes from \(x\) to \(x+4\). </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> \(f(x+4)−f(x)\) can be expressed as \(\lbrack 2(x+4)+5 \rbrack− \lbrack 2x+5 \rbrack \). </p>
</div>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="51baf309-482d-4568-82d9-73dfec163a0c" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li> Show or explain how that expression has a value of 8. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Applying the distributive property to \( \lbrack 2(x+4)+5 \rbrack − \lbrack 2x+5 \rbrack \) gives us: \(2x+8+5−2x−5\) or \(2x−2x+8+5−5\), which equals 8. </p>
</div>
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