openfreebooks-bookgenerator/content/math/functions-graphs/supplement.html

<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>