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79 lines
2.6 KiB
HTML
79 lines
2.6 KiB
HTML
<div class="book-prose">
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<p>
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Variation describes how one quantity changes with another. It turns word statements into useful
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algebraic models.
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</p>
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<h2 id="direct-variation" class="book-prose__heading">Direct variation</h2>
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<p>
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\(y\) varies directly as \(x\):
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\[
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y\propto x \quad\Rightarrow\quad y=kx
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\]
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where \(k\) is the constant of variation.
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</p>
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<p><strong>Example:</strong> If \(y=18\) when \(x=6\), then \(k=3\), so \(y=3x\).</p>
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<h2 id="inverse-variation" class="book-prose__heading">Inverse variation</h2>
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<p>
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\(y\) varies inversely as \(x\):
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\[
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y\propto\frac1x \quad\Rightarrow\quad y=\frac{k}{x}.
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\]
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</p>
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<p><strong>Example:</strong> If \(y=5\) when \(x=4\), then \(k=20\), so \(y=\frac{20}{x}\).</p>
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<h2 id="power-variation" class="book-prose__heading">Variation with powers</h2>
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<p>
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You may see statements like:
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</p>
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<ul>
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<li>\(y\propto x^2 \Rightarrow y=kx^2\)</li>
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<li>\(y\propto \sqrt{x} \Rightarrow y=k\sqrt{x}\)</li>
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<li>\(y\propto \frac{1}{x^2} \Rightarrow y=\frac{k}{x^2}\)</li>
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</ul>
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<h2 id="joint-variation" class="book-prose__heading">Joint variation</h2>
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<p>
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If \(y\) varies jointly as \(x\) and \(z\):
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\[
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y\propto xz \Rightarrow y=kxz.
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\]
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</p>
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<p><strong>Example:</strong> \(y=24\) when \(x=2,z=3\Rightarrow k=4\), so \(y=4xz\).</p>
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<h2 id="partial-variation" class="book-prose__heading">Partial variation</h2>
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<p>
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If \(y\) varies partly as \(x\), model has constant and variable part:
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\[
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y=mx+c,\quad c\ne0.
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\]
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</p>
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<p>
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This appears in pricing: fixed fee + per-unit charge.
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</p>
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<h2 id="solving-workflow" class="book-prose__heading">Solving workflow</h2>
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<ol>
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<li>Translate words into proportional equation with \(k\).</li>
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<li>Use given data to find \(k\).</li>
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<li>Substitute required values to answer question.</li>
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<li>Check units and reasonableness.</li>
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</ol>
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<h2 id="word-problems" class="book-prose__heading">Common word-problem forms</h2>
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<ul>
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<li>“\(A\) is directly proportional to \(B\)”</li>
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<li>“\(A\) varies inversely as \(B\)”</li>
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<li>“\(A\) varies jointly as \(B\) and \(C\)”</li>
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<li>“\(A\) varies directly as square/cube of \(B\)”</li>
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</ul>
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<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
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<ul>
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<li>Always introduce \(k\) first before substituting numbers.</li>
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<li>Keep symbolic form until \(k\) is found.</li>
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<li>For inverse models, domain often excludes zero.</li>
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<li>State final answer with units when applicable.</li>
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</ul>
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</div>
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