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content/math/functions-graphs/supplement.html

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+<div class="book-prose">
+  <h2 id="history" class="book-prose__heading">History</h2>
+  <p>
+    The modern idea of a function developed over centuries. Early algebra connected formulas to
+    unknown quantities, while analytic geometry (Descartes) linked equations to curves on coordinate
+    axes.
+  </p>
+  <p>
+    Later, Euler and others formalized function notation such as \(f(x)\). Today, function thinking
+    is central in mathematics, science, economics, and computing because it models how one quantity
+    depends on another.
+  </p>
+
+  <h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
+  <h3 id="slope-intercept" class="book-prose__heading">From two points to line equation</h3>
+  <p>
+    If a line passes through \((x_1,y_1)\) and \((x_2,y_2)\), slope is:
+    \[
+    m=\frac{y_2-y_1}{x_2-x_1}.
+    \]
+    Then use \(y=mx+c\) and substitute one point to find \(c\).
+  </p>
+
+  <h3 id="vertex-form" class="book-prose__heading">Why vertex form helps for quadratics</h3>
+  <p>
+    Writing \(y=a(x-h)^2+k\) shows the turning point immediately: vertex \((h,k)\). This is why
+    completing the square is useful for sketching and transformations.
+  </p>
+
+  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
+  <details class="book-question">
+    <summary class="book-question__prompt">
+      For \(f(x)=3x-2\), find \(f(5)\) and \(f(-1)\).
+    </summary>
+    <div class="book-question__solution">
+      <p><strong>Answer:</strong> \(f(5)=13\), \(f(-1)=-5\).</p>
+    </div>
+  </details>
+
+  <details class="book-question">
+    <summary class="book-question__prompt">
+      State domain and range of \(g(x)=\sqrt{x-4}\).
+    </summary>
+    <div class="book-question__solution">
+      <p><strong>Answer:</strong> Domain \(x\ge 4\), range \(y\ge 0\).</p>
+    </div>
+  </details>
+
+  <details class="book-question">
+    <summary class="book-question__prompt">
+      Is \(x=y^2\) a function of \(x\)? Use the vertical line test idea.
+    </summary>
+    <div class="book-question__solution">
+      <p><strong>Answer:</strong> No.</p>
+      <p>
+        For a positive \(x\), there are two \(y\)-values (\(+\sqrt{x}\) and \(-\sqrt{x}\)), so one
+        \(x\) maps to two outputs.
+      </p>
+    </div>
+  </details>
+
+  <details class="book-question">
+    <summary class="book-question__prompt">
+      For \(y=(x+1)^2-3\), state the transformation from \(y=x^2\).
+    </summary>
+    <div class="book-question__solution">
+      <p><strong>Answer:</strong> Shift left 1 and down 3.</p>
+    </div>
+  </details>
+
+  <details class="book-question">
+    <summary class="book-question__prompt">
+      Find intercepts of \(y=x^2-4x+3\).
+    </summary>
+    <div class="book-question__solution">
+      <p><strong>Answer:</strong> \(y\)-intercept: \((0,3)\).</p>
+      <p><strong>Answer:</strong> \(x\)-intercepts: solve \(x^2-4x+3=0\Rightarrow(x-1)(x-3)=0\), so \((1,0)\), \((3,0)\).</p>
+    </div>
+  </details>
+
+  <details class="book-question">
+    <summary class="book-question__prompt">
+      Sketch idea: \(f(x)=\frac{1}{x-2}+1\). What are its asymptotes?
+    </summary>
+    <div class="book-question__solution">
+      <p><strong>Answer:</strong> Vertical asymptote \(x=2\), horizontal asymptote \(y=1\).</p>
+    </div>
+  </details>
+
+  <h2 id="applications" class="book-prose__heading">Applications</h2>
+  <h3 id="finance" class="book-prose__heading">Finance and budgeting</h3>
+  <p>
+    Linear functions model plans with fixed charges plus per-unit costs, such as phone plans or
+    transport fares.
+  </p>
+
+  <h3 id="motion" class="book-prose__heading">Motion and science</h3>
+  <p>
+    Position-time or velocity-time graphs are function graphs. Intercepts, gradients, and curvature
+    describe physical behavior.
+  </p>
+
+  <h3 id="data-models" class="book-prose__heading">Data modeling</h3>
+  <p>
+    Exponential, quadratic, and rational models are used to approximate trends, optimize designs,
+    and make predictions from measured data.
+  </p>
+
+  <h2 id="references" class="book-prose__heading">References</h2>
+  <ul>
+    <li>Standard high-school algebra curricula on functions, graphs, and transformations.</li>
+    <li>Analytic geometry and precalculus references covering domain/range and graph families.</li>
+  </ul>
+</div>