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Enhance chapter content and metadata: Update SKILL.md to include requirements for chart axes in all widgets. Implement axis labels, arrowheads, and graduations in the simultaneous graph explorer widget. Update math curriculum JSON to reflect the status change of multiple chapters to live. Add last modified timestamps to chapter templates and generate corresponding metadata for improved site navigation and SEO.

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title = "Functions & graphs"
description = "Model input-output rules, work with domain and range, and sketch key function graphs and transformations."
template = "chapter.html"
weight = 15
[extra]
subject = "math"
chapter_id = "functions-graphs"
strand = "Number & Algebra"
+++
<div class="book-prose">
<p>
A <strong>function</strong> is a rule that assigns exactly one output to each allowed input.
In high school, functions help you connect equations, tables, and graphs.
</p>
<h2 id="what-is-a-function" class="book-prose__heading">What is a function?</h2>
<p>
If \(f\) is a function, then \(f(x)\) means “the output when input is \(x\).”
Example:
</p>
<p class="book-formula">
\[
f(x)=2x+3.
\]
</p>
<p>
Then \(f(1)=5\), \(f(4)=11\), and \(f(-2)=-1\).
</p>
<div class="book-callout book-callout--formula" role="note">
<p class="book-callout__label">Key idea</p>
<p class="book-formula">
\[
\text{One input} \rightarrow \text{one output}.
\]
</p>
</div>
<h2 id="domain-range" class="book-prose__heading">Domain and range</h2>
<p>
The <strong>domain</strong> is the set of allowed inputs. The <strong>range</strong> is the set of
outputs produced from that domain.
</p>
<p><strong>Example 1.</strong> \(f(x)=\frac{1}{x-2}\)</p>
<ul>
<li>Domain: all real \(x\) except \(x=2\).</li>
<li>Range: all real \(y\) except \(y=0\).</li>
</ul>
<p><strong>Example 2.</strong> \(g(x)=\sqrt{x+1}\)</p>
<ul>
<li>Domain: \(x \ge -1\) (square root needs non-negative inside).</li>
<li>Range: \(y \ge 0\).</li>
</ul>
<h2 id="function-vs-relation" class="book-prose__heading">Function vs relation</h2>
<p>
Not every relation is a function. If one input gives two outputs, it is not a function.
</p>
<p>
On a graph, use the <strong>vertical line test</strong>: if any vertical line crosses the curve
more than once, it is not a function.
</p>
<h2 id="graphs-basics" class="book-prose__heading">Graph basics</h2>
<p>
To sketch a graph:
</p>
<ol>
<li>Choose key \(x\)-values and make a value table.</li>
<li>Plot points \((x,f(x))\) on labeled axes.</li>
<li>Connect appropriately (line, curve, or separate branches).</li>
<li>Identify intercepts and turning features.</li>
</ol>
<h2 id="key-families" class="book-prose__heading">Important function families</h2>
<h3 id="linear" class="book-prose__heading">Linear: \(y=mx+c\)</h3>
<p>
Straight line with slope \(m\) and \(y\)-intercept \(c\). A larger \(|m|\) means steeper line.
</p>
<h3 id="quadratic" class="book-prose__heading">Quadratic: \(y=ax^2+bx+c\)</h3>
<p>
Parabola. If \(a>0\), opens upward; if \(a<0\), opens downward.
</p>
<h3 id="absolute-value" class="book-prose__heading">Absolute value: \(y=|x|\)</h3>
<p>
V-shaped graph with vertex at the origin before transformations.
</p>
<h3 id="rational" class="book-prose__heading">Rational: \(y=\frac{1}{x}\)</h3>
<p>
Hyperbola with asymptotes \(x=0\) and \(y=0\). Graph has two separate branches.
</p>
<h2 id="transformations" class="book-prose__heading">Transformations</h2>
<p>
Start from \(y=f(x)\):
</p>
<ul>
<li>\(y=f(x)+k\): shift up by \(k\).</li>
<li>\(y=f(x-h)\): shift right by \(h\).</li>
<li>\(y=af(x)\): vertical stretch/compression by factor \(|a|\), and reflect if \(a&lt;0\).</li>
<li>\(y=f(bx)\): horizontal scale by factor \(\frac{1}{|b|}\), and reflect in \(y\)-axis if \(b&lt;0\).</li>
</ul>
<p><strong>Example.</strong> From \(y=x^2\) to \(y=(x-3)^2+2\): move right 3, up 2.</p>
<h2 id="intercepts-and-meaning" class="book-prose__heading">Intercepts and meaning</h2>
<ul>
<li><strong>\(y\)-intercept:</strong> set \(x=0\).</li>
<li><strong>\(x\)-intercepts:</strong> solve \(f(x)=0\).</li>
</ul>
<p>
In applications, intercepts often represent starting value and break-even/zero-output points.
</p>
<h2 id="piecewise" class="book-prose__heading">Piecewise functions</h2>
<p>
Some rules change in different intervals. Example:
</p>
<p class="book-formula">
\[
f(x)=
\begin{cases}
x+2, & x&lt;1\\
3, & x\ge 1
\end{cases}
\]
</p>
<p>
Graph each rule only on its stated interval, and mark endpoints correctly (open/closed points).
</p>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>State domain restrictions before graphing.</li>
<li>Use a neat value table near turning points or asymptotes.</li>
<li>Label key coordinates: intercepts, vertex, asymptote equations.</li>
<li>Check whether a requested equation is function form or relation form.</li>
</ul>
</div>
<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>

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<div class="book-prose">
<p>
A <strong>function</strong> is a rule that assigns exactly one output to each allowed input.
In high school, functions help you connect equations, tables, and graphs.
</p>
<h2 id="what-is-a-function" class="book-prose__heading">What is a function?</h2>
<p>
If \(f\) is a function, then \(f(x)\) means “the output when input is \(x\).”
Example:
</p>
<p class="book-formula">
\[
f(x)=2x+3.
\]
</p>
<p>
Then \(f(1)=5\), \(f(4)=11\), and \(f(-2)=-1\).
</p>
<div class="book-callout book-callout--formula" role="note">
<p class="book-callout__label">Key idea</p>
<p class="book-formula">
\[
\text{One input} \rightarrow \text{one output}.
\]
</p>
</div>
<h2 id="domain-range" class="book-prose__heading">Domain and range</h2>
<p>
The <strong>domain</strong> is the set of allowed inputs. The <strong>range</strong> is the set of
outputs produced from that domain.
</p>
<p><strong>Example 1.</strong> \(f(x)=\frac{1}{x-2}\)</p>
<ul>
<li>Domain: all real \(x\) except \(x=2\).</li>
<li>Range: all real \(y\) except \(y=0\).</li>
</ul>
<p><strong>Example 2.</strong> \(g(x)=\sqrt{x+1}\)</p>
<ul>
<li>Domain: \(x \ge -1\) (square root needs non-negative inside).</li>
<li>Range: \(y \ge 0\).</li>
</ul>
<h2 id="function-vs-relation" class="book-prose__heading">Function vs relation</h2>
<p>
Not every relation is a function. If one input gives two outputs, it is not a function.
</p>
<p>
On a graph, use the <strong>vertical line test</strong>: if any vertical line crosses the curve
more than once, it is not a function.
</p>
<h2 id="graphs-basics" class="book-prose__heading">Graph basics</h2>
<p>
To sketch a graph:
</p>
<ol>
<li>Choose key \(x\)-values and make a value table.</li>
<li>Plot points \((x,f(x))\) on labeled axes.</li>
<li>Connect appropriately (line, curve, or separate branches).</li>
<li>Identify intercepts and turning features.</li>
</ol>
<h2 id="key-families" class="book-prose__heading">Important function families</h2>
<h3 id="linear" class="book-prose__heading">Linear: \(y=mx+c\)</h3>
<p>
Straight line with slope \(m\) and \(y\)-intercept \(c\). A larger \(|m|\) means steeper line.
</p>
<h3 id="quadratic" class="book-prose__heading">Quadratic: \(y=ax^2+bx+c\)</h3>
<p>
Parabola. If \(a>0\), opens upward; if \(a<0\), opens downward.
</p>
<h3 id="absolute-value" class="book-prose__heading">Absolute value: \(y=|x|\)</h3>
<p>
V-shaped graph with vertex at the origin before transformations.
</p>
<h3 id="rational" class="book-prose__heading">Rational: \(y=\frac{1}{x}\)</h3>
<p>
Hyperbola with asymptotes \(x=0\) and \(y=0\). Graph has two separate branches.
</p>
<h2 id="transformations" class="book-prose__heading">Transformations</h2>
<p>
Start from \(y=f(x)\):
</p>
<ul>
<li>\(y=f(x)+k\): shift up by \(k\).</li>
<li>\(y=f(x-h)\): shift right by \(h\).</li>
<li>\(y=af(x)\): vertical stretch/compression by factor \(|a|\), and reflect if \(a&lt;0\).</li>
<li>\(y=f(bx)\): horizontal scale by factor \(\frac{1}{|b|}\), and reflect in \(y\)-axis if \(b&lt;0\).</li>
</ul>
<p><strong>Example.</strong> From \(y=x^2\) to \(y=(x-3)^2+2\): move right 3, up 2.</p>
<h2 id="intercepts-and-meaning" class="book-prose__heading">Intercepts and meaning</h2>
<ul>
<li><strong>\(y\)-intercept:</strong> set \(x=0\).</li>
<li><strong>\(x\)-intercepts:</strong> solve \(f(x)=0\).</li>
</ul>
<p>
In applications, intercepts often represent starting value and break-even/zero-output points.
</p>
<h2 id="piecewise" class="book-prose__heading">Piecewise functions</h2>
<p>
Some rules change in different intervals. Example:
</p>
<p class="book-formula">
\[
f(x)=
\begin{cases}
x+2, & x&lt;1\\
3, & x\ge 1
\end{cases}
\]
</p>
<p>
Graph each rule only on its stated interval, and mark endpoints correctly (open/closed points).
</p>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>State domain restrictions before graphing.</li>
<li>Use a neat value table near turning points or asymptotes.</li>
<li>Label key coordinates: intercepts, vertex, asymptote equations.</li>
<li>Check whether a requested equation is function form or relation form.</li>
</ul>
</div>

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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>