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<title>Cubic function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Cubic function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Cubic_equation" title="Cubic equation">Cubic equation</a>.</div>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>cubic function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,}</annotation>

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</math></span><img src="./957eb30228dcd428534d339c2ba5f399e93ad49c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.232ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,}" loading="lazy"></span> that is, a <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> of degree three. In many texts, the <i>coefficients</i> <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>, and <span class="texhtml mvar" style="font-style:italic;">d</span> are supposed to be <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, and the function is considered as a <a href="Real_function" class="mw-redirect" title="Real function">real function</a> that maps real numbers to real numbers or as a complex function that maps <a href="Complex_number" title="Complex number">complex numbers</a> to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its <a href="Codomain" title="Codomain">codomain</a>, even when the <a href="Domain_of_a_function" title="Domain of a function">domain</a> is restricted to the real numbers.

</p><p>Setting <span class="texhtml"><i>f</i>(<i>x</i>) = 0</span> produces a <a href="Cubic_equation" title="Cubic equation">cubic equation</a> of the form

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<annotation encoding="application/x-tex">{\displaystyle ax^{3}+bx^{2}+cx+d=0,}</annotation>

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</math></span><img src="./4c864462f7d6bda7015dffbbfeb51127294147e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.977ex; height:3.009ex;" alt="{\displaystyle ax^{3}+bx^{2}+cx+d=0,}" loading="lazy"></span></dd></dl>

<p>whose solutions are called <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> of the function. The <a href="Derivative" title="Derivative">derivative</a> of a cubic function is a <a href="Quadratic_function" title="Quadratic function">quadratic function</a>.

</p><p>A cubic function with real coefficients has either one or three real roots (<a href="Multiplicity_(mathematics)" title="Multiplicity (mathematics)">which may not be distinct</a>);<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> all odd-degree polynomials with real coefficients have at least one real root.

</p><p>The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a cubic function always has a single <a href="Inflection_point" title="Inflection point">inflection point</a>. It may have two <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a>, a local minimum and a local maximum. Otherwise, a cubic function is <a href="Monotonic" class="mw-redirect" title="Monotonic">monotonic</a>. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. <a href="Up_to" title="Up to">Up to</a> an <a href="Affine_transformation" title="Affine transformation">affine transformation</a>, there are only three possible graphs for cubic functions.

</p><p>Cubic functions are fundamental for <a href="Cubic_interpolation" class="mw-redirect" title="Cubic interpolation">cubic interpolation</a>.

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cubic_equation#History" title="Cubic equation">Cubic equation § History</a></div>

<div class="mw-heading mw-heading2"><h2 id="Critical_and_inflection_points">Critical and inflection points</h2></div>

<p>The <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> of a cubic function are its <a href="Stationary_point" title="Stationary point">stationary points</a>, that is the points where the slope of the function is zero.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Thus the critical points of a cubic function <span class="texhtml"><i>f</i></span> defined by

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<p>occur at values of <span class="texhtml"><i>x</i></span> such that the <a href="Derivative" title="Derivative">derivative</a>

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<annotation encoding="application/x-tex">{\displaystyle 3ax^{2}+2bx+c=0}</annotation>

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</math></span><img src="./3f2483910648ab1431781c509e7b19e338088958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.214ex; height:2.843ex;" alt="{\displaystyle 3ax^{2}+2bx+c=0}" loading="lazy"></span></dd></dl>

<p>of the cubic function is zero.

</p><p>The solutions of this equation are the <span class="texhtml mvar" style="font-style:italic;">x</span>-values of the critical points and are given, using the <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a>, by

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<mtext>critical</mtext>

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<annotation encoding="application/x-tex">{\displaystyle x_{\text{critical}}={\frac {-b\pm {\sqrt {b^{2}-3ac}}}{3a}}.}</annotation>

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</math></span><img src="./3c9b66b16bf99db88813feaba97f580309e610a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.343ex; height:6.176ex;" alt="{\displaystyle x_{\text{critical}}={\frac {-b\pm {\sqrt {b^{2}-3ac}}}{3a}}.}" loading="lazy"></span></dd></dl>

<p>The sign of the expression <span class="texhtml">Δ<sub>0</sub> = </span><span class="texhtml"><i>b</i><sup>2</sup> − 3<i>ac</i></span> inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum. If <span class="texhtml"><i>b</i><sup>2</sup> − 3<i>ac</i> = 0</span>, then there is only one critical point, which is an <a href="Inflection_point" title="Inflection point">inflection point</a>. If <span class="texhtml"><i>b</i><sup>2</sup> − 3<i>ac</i> < 0</span>, then there are no (real) critical points. In the two latter cases, that is, if <span class="texhtml"><i>b</i><sup>2</sup> − 3<i>ac</i></span> is nonpositive, the cubic function is strictly <a href="Monotonic" class="mw-redirect" title="Monotonic">monotonic</a>. See the figure for an example of the case <span class="texhtml">Δ<sub>0</sub> > 0</span>.

</p><p>The inflection point of a function is where that function changes <a href="Second_derivative#Concavity" title="Second derivative">concavity</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> An inflection point occurs when the <a href="Second_derivative" title="Second derivative">second derivative</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(x)=6ax+2b,}">

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<annotation encoding="application/x-tex">{\displaystyle f''(x)=6ax+2b,}</annotation>

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</math></span><img src="./7c59496ac9ca19bd695147a76c36f2c922b97eb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.064ex; height:3.009ex;" alt="{\displaystyle f''(x)=6ax+2b,}" loading="lazy"></span> is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at

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<annotation encoding="application/x-tex">{\displaystyle x_{\text{inflection}}=-{\frac {b}{3a}}.}</annotation>

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</math></span><img src="./17ba266b32c19b7192ee2b40fb7a04edffaee863.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.969ex; height:5.343ex;" alt="{\displaystyle x_{\text{inflection}}=-{\frac {b}{3a}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2></div>

<p>The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a cubic function is a <a href="Cubic_curve" class="mw-redirect" title="Cubic curve">cubic curve</a>, though many cubic curves are not graphs of functions.

</p><p>Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always <a href="Similarity_(geometry)" title="Similarity (geometry)">similar</a> to the graph of a function of the form

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<annotation encoding="application/x-tex">{\displaystyle y=x^{3}+px.}</annotation>

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</math></span><img src="./6ec59763690a7f66b926c7d0f301a834a1b18f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.624ex; height:3.009ex;" alt="{\displaystyle y=x^{3}+px.}" loading="lazy"></span></dd></dl>

<p>This similarity can be built as the composition of <a href="Translation" title="Translation">translations</a> parallel to the coordinates axes, a <a href="Homothecy" class="mw-redirect" title="Homothecy">homothecy</a> (<a href="Uniform_scaling" class="mw-redirect" title="Uniform scaling">uniform scaling</a>), and, possibly, a <a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> (<a href="Mirror_image" title="Mirror image">mirror image</a>) with respect to the <span class="texhtml mvar" style="font-style:italic;">y</span>-axis. A further <a href="Uniform_scaling" class="mw-redirect" title="Uniform scaling">non-uniform scaling</a> can transform the graph into the graph of one among the three cubic functions

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y&=x^{3}+x\\y&=x^{3}\\y&=x^{3}-x.\end{aligned}}}</annotation>

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</math></span><img src="./3be857746f9aa9a9f3ae24cb8085ca588162fb6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.913ex; margin-bottom: -0.258ex; width:12.206ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}y&=x^{3}+x\\y&=x^{3}\\y&=x^{3}-x.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>This means that there are only three graphs of cubic functions <a href="Up_to" title="Up to">up to</a> an <a href="Affine_transformation" title="Affine transformation">affine transformation</a>.

</p><p>The above <a href="Geometric_transformation" title="Geometric transformation">geometric transformations</a> can be built in the following way, when starting from a general cubic function

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<annotation encoding="application/x-tex">{\displaystyle y=ax^{3}+bx^{2}+cx+d.}</annotation>

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</math></span><img src="./201213a1ddb1252c2044dea8a06bf228a360d914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.97ex; height:3.009ex;" alt="{\displaystyle y=ax^{3}+bx^{2}+cx+d.}" loading="lazy"></span>

</p><p>Firstly, if <span class="texhtml"><i>a</i> < 0</span>, the <a href="Change_of_variable" class="mw-redirect" title="Change of variable">change of variable</a> <span class="texhtml"><i>x</i> → −<i>x</i></span> allows supposing <span class="texhtml"><i>a</i> > 0</span>. After this change of variable, the new graph is the mirror image of the previous one, with respect of the <span class="texhtml mvar" style="font-style:italic;">y</span>-axis.

</p><p>Then, the change of variable <span class="texhtml"><i>x</i> = <i>x</i><sub>1</sub> − <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>b</i></span><span class="sr-only">/</span><span class="den">3<i>a</i></span></span>⁠</span></span> provides a function of the form

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<annotation encoding="application/x-tex">{\displaystyle y=ax_{1}^{3}+px_{1}+q.}</annotation>

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</math></span><img src="./135d4d49eff0d65771ef2ce3bd68149017230322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.818ex; height:3.176ex;" alt="{\displaystyle y=ax_{1}^{3}+px_{1}+q.}" loading="lazy"></span></dd></dl>

<p>This corresponds to a translation parallel to the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis.

</p><p>The change of variable <span class="texhtml"><i>y</i> = <i>y</i><sub>1</sub> + <i>q</i></span> corresponds to a translation with respect to the <span class="texhtml mvar" style="font-style:italic;">y</span>-axis, and gives a function of the form

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<annotation encoding="application/x-tex">{\displaystyle y_{1}=ax_{1}^{3}+px_{1}.}</annotation>

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</math></span><img src="./9090a62eb45706b8845b8d705e9ac6f99c611a14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.946ex; height:3.176ex;" alt="{\displaystyle y_{1}=ax_{1}^{3}+px_{1}.}" loading="lazy"></span></dd></dl>

<p>The change of variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}}</annotation>

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</math></span><img src="./0c84c7ff22d4968c34089c83a02deb7385fa6c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.957ex; height:4.343ex;" alt="{\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}}" loading="lazy"></span> corresponds to a uniform scaling, and give, after multiplication by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}},}">

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</math></span><img src="./40e20cbaee2c94e045df8784f4c398de5862e53f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.813ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}},}" loading="lazy"></span> a function of the form

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<annotation encoding="application/x-tex">{\displaystyle y_{2}=x_{2}^{3}+px_{2},}</annotation>

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<p>which is the simplest form that can be obtained by a similarity.

</p><p>Then, if <span class="texhtml"><i>p</i> ≠ 0</span>, the non-uniform scaling <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}}</annotation>

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</math></span><img src="./fde466d364038375d5a8edf19034d8a59a123c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:29.336ex; height:4.843ex;" alt="{\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}}" loading="lazy"></span> gives, after division by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\sqrt {|p|^{3}}},}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {|p|^{3}}},}</annotation>

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</math></span><img src="./21f1ad141a639a935ce04d3a3421daf5557a9018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:6.488ex; height:4.843ex;" alt="{\displaystyle \textstyle {\sqrt {|p|^{3}}},}" loading="lazy"></span>

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{3}=x_{3}^{3}+x_{3}\operatorname {sgn}(p),}</annotation>

</semantics>

</math></span><img src="./2f9f5f6bdd7295fe82d9f22a569dd32c79b8762a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.284ex; height:3.176ex;" alt="{\displaystyle y_{3}=x_{3}^{3}+x_{3}\operatorname {sgn}(p),}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(p)}</annotation>

</semantics>

</math></span><img src="./57144c6bddcc970a72c6e73fbddc0b8bb3846a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(p)}" loading="lazy"></span> has the value 1 or −1, depending on the sign of <span class="texhtml mvar" style="font-style:italic;">p</span>. If one defines <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(0)=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(0)=0,}</annotation>

</semantics>

</math></span><img src="./686d33b703a1445b94c7ddc308ca3fea3c46e112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.251ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(0)=0,}" loading="lazy"></span> the latter form of the function applies to all cases (with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=x_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}=x_{3}}</annotation>

</semantics>

</math></span><img src="./c2b151e4c58c5404cee590c7745100878a7e267d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.009ex;" alt="{\displaystyle x_{2}=x_{3}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2}=y_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{2}=y_{3}}</annotation>

</semantics>

</math></span><img src="./5c945995e350a793fcb5052137b82fb0899ca9e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.485ex; height:2.009ex;" alt="{\displaystyle y_{2}=y_{3}}" loading="lazy"></span>).

</p>

<div class="mw-heading mw-heading2"><h2 id="Symmetry">Symmetry</h2></div>

<p>For a cubic function of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{3}+px,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x^{3}+px,}</annotation>

</semantics>

</math></span><img src="./d0d121150a52123c5af85a10c73fff0b419fc3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.624ex; height:3.009ex;" alt="{\displaystyle y=x^{3}+px,}" loading="lazy"></span> the inflection point is thus the origin. As such a function is an <a href="Odd_function" class="mw-redirect" title="Odd function">odd function</a>, its graph is symmetric with respect to the inflection point, and invariant under a rotation of a half turn around the inflection point. As these properties are invariant by <a href="Similarity_(geometry)" title="Similarity (geometry)">similarity</a>, the following is true for all cubic functions.

</p><p><i>The graph of a cubic function is symmetric with respect to its inflection point, and is invariant under a rotation of a half turn around the inflection point.</i>

</p>

<div class="mw-heading mw-heading2"><h2 id="Collinearities">Collinearities</h2></div>

<p>The tangent lines to the graph of a cubic function at three <a href="Collinear_points" class="mw-redirect" title="Collinear points">collinear points</a> intercept the cubic again at collinear points.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This can be seen as follows.

</p><p>As this property is invariant under a <a href="Rigid_motion" class="mw-redirect" title="Rigid motion">rigid motion</a>, one may suppose that the function has the form

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{3}+px.}</annotation>

</semantics>

</math></span><img src="./2846cba5b1b98499fb19f1180cec676989c10c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.886ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{3}+px.}" loading="lazy"></span></dd></dl>

<p>If <span class="texhtml mvar" style="font-style:italic;">α</span> is a real number, then the tangent to the graph of <span class="texhtml mvar" style="font-style:italic;">f</span> at the point <span class="texhtml">(<i>α</i>, <i>f</i>(<i>α</i>))</span> is the line

</p>

<p>So, the intersection point between this line and the graph of <span class="texhtml mvar" style="font-style:italic;">f</span> can be obtained solving the equation <span class="texhtml"><i>f</i>(<i>x</i>) = <i>f</i>(<i>α</i>) + (<i>x</i> − <i>α</i>)<i>f</i> ′(<i>α</i>)</span>, that is

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{3}+px=\alpha ^{3}+p\alpha +(x-\alpha )(3\alpha ^{2}+p),}</annotation>

</semantics>

</math></span><img src="./c8fd6583f61e90f9b2064260b7dcf6fb218826bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.339ex; height:3.176ex;" alt="{\displaystyle x^{3}+px=\alpha ^{3}+p\alpha +(x-\alpha )(3\alpha ^{2}+p),}" loading="lazy"></span></dd></dl>

<p>which can be rewritten

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{3}-3\alpha ^{2}x+2\alpha ^{3}=0,}</annotation>

</semantics>

</math></span><img src="./12babfd5009505ab1d9beeaccdb04c58523ec581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.711ex; height:3.009ex;" alt="{\displaystyle x^{3}-3\alpha ^{2}x+2\alpha ^{3}=0,}" loading="lazy"></span></dd></dl>

<p>and factorized as

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x-\alpha )^{2}(x+2\alpha )=0.}</annotation>

</semantics>

</math></span><img src="./820443969c0d4fbd6495a9aff5416ef594299dd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.058ex; height:3.176ex;" alt="{\displaystyle (x-\alpha )^{2}(x+2\alpha )=0.}" loading="lazy"></span></dd></dl>

<p>So, the tangent intercepts the cubic at

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-2\alpha ,-8\alpha ^{3}-2p\alpha )=(-2\alpha ,-8f(\alpha )+6p\alpha ).}</annotation>

</semantics>

</math></span><img src="./53a2d66773d8cbc626f3839dedf863218cc613c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.727ex; height:3.176ex;" alt="{\displaystyle (-2\alpha ,-8\alpha ^{3}-2p\alpha )=(-2\alpha ,-8f(\alpha )+6p\alpha ).}" loading="lazy"></span></dd></dl>

<p>So, the function that maps a point <span class="texhtml">(<i>x</i>, <i>y</i>)</span> of the graph to the other point where the tangent intercepts the graph is

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto (-2x,-8y+6px).}</annotation>

</semantics>

</math></span><img src="./d325eb8e15f1ce08720809834157968c51c99e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.361ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto (-2x,-8y+6px).}" loading="lazy"></span></dd></dl>

<p>This is an <a href="Affine_transformation" title="Affine transformation">affine transformation</a> that transforms collinear points into collinear points. This proves the claimed result.

</p>

<div class="mw-heading mw-heading2"><h2 id="Cubic_interpolation">Cubic interpolation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spline_interpolation" title="Spline interpolation">Spline interpolation</a></div>

<p>Given the values of a function and its derivative at two points, there is exactly one cubic function that has the same four values, which is called a <a href="Cubic_Hermite_spline" title="Cubic Hermite spline">cubic Hermite spline</a>.

</p><p>There are two standard ways for using this fact. Firstly, if one knows, for example by physical measurement, the values of a function and its derivative at some sampling points, one can <i>interpolate</i> the function with a <a href="Continuously_differentiable_function" class="mw-redirect" title="Continuously differentiable function">continuously differentiable function</a>, which is a <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> cubic function.

</p><p>If the value of a function is known at several points, <a href="Cubic_interpolation" class="mw-redirect" title="Cubic interpolation">cubic interpolation</a> consists in approximating the function by a <a href="Continuously_differentiable_function" class="mw-redirect" title="Continuously differentiable function">continuously differentiable function</a>, which is <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> cubic. For having a uniquely defined interpolation, two more constraints must be added, such as the values of the derivatives at the endpoints, or a zero <a href="Curvature" title="Curvature">curvature</a> at the endpoints.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBostockChandlerChandler1979" class="citation book cs1">Bostock, Linda; Chandler, Suzanne; Chandler, F. S. (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e2C3tFnAR-wC&q=A+cubic+function+has+either+one+or+three+real+roots&pg=PA462"><i>Pure Mathematics 2</i></a>. Nelson Thornes. p. 462. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-85950-097-5</bdi>. <q>Thus a cubic equation has either three real roots... or one real root...</q></cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/StationaryPoint.html">"Stationary Point"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-27</span></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFHughes-HallettLockGleasonFlath2017" class="citation book cs1">Hughes-Hallett, Deborah; Lock, Patti Frazer; Gleason, Andrew M.; Flath, Daniel E.; Gordon, Sheldon P.; Lomen, David O.; Lovelock, David; McCallum, William G.; Osgood, Brad G. (2017-12-11). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8CeVDwAAQBAJ&q=inflection+point+of+a+function+is+where+that+function+changes+concavity&pg=PA181"><i>Applied Calculus</i></a>. John Wiley & Sons. p. 181. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-119-27556-5</bdi>. <q>A point at which the graph of the function f changes concavity is called an inflection point of f</q></cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFWhitworth1866" class="citation cs2"><a href="William_Allen_Whitworth" title="William Allen Whitworth">Whitworth, William Allen</a> (1866), "Equations of the third degree", <a rel="nofollow" class="external text" href="https://archive.org/details/trilinearcoordin00whit"><i>Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions</i></a>, Cambridge: Deighton, Bell, and Co., p. 425<span class="reference-accessdate">, retrieved <span class="nowrap">June 17,</span> 2016</span></cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Cubic_functions" class="extiw external" title="commons:Category:Cubic functions">Cubic functions</a></span>.</div></div>

</div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Cardano_formula">"Cardano formula"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/history/HistTopics/Quadratic_etc_equations.html">History of quadratic, cubic and quartic equations</a> on <a href="MacTutor_archive" class="mw-redirect" title="MacTutor archive">MacTutor archive</a>.</li></ul>

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</style><div id="Polynomials_and_polynomial_functions_and_polynomial_equations142" style="font-size:114%;margin:0 4em"><a href="Polynomial" title="Polynomial">Polynomials</a> and <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> and <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zero_polynomial" class="mw-redirect" title="Zero polynomial">Zero polynomial (degree undefined or −1 or −∞)</a></li>

<li><a href="Constant_function" title="Constant function">Constant function (0)</a></li>

<li><a href="Linear_function_(calculus)" title="Linear function (calculus)">Linear function (1)</a>

<ul><li><a href="Linear_equation" title="Linear equation">Linear equation</a></li></ul></li>

<li><a href="Quadratic_function" title="Quadratic function">Quadratic function (2)</a>

<ul><li><a href="Quadratic_equation" title="Quadratic equation">Quadratic equation</a></li></ul></li>

<li>

<ul><li><a href="Cubic_equation" title="Cubic equation">Cubic equation</a></li></ul></li>

<li><a href="Quartic_function" title="Quartic function">Quartic function (4)</a>

<ul><li><a href="Quartic_equation" title="Quartic equation">Quartic equation</a></li></ul></li>

<li><a href="Quintic_function" title="Quintic function">Quintic function (5)</a></li>

<li><a href="Sextic_equation" title="Sextic equation">Sextic equation (6)</a></li>

<li><a href="Septic_equation" title="Septic equation">Septic equation (7)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By properties</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Univariate_polynomial" class="mw-redirect" title="Univariate polynomial">Univariate</a></li>

<li><a href="Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">Bivariate</a></li>

<li><a href="Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">Multivariate</a></li>

<li><a href="Monomial" title="Monomial">Monomial</a></li>

<li><a href="Binomial_(polynomial)" title="Binomial (polynomial)">Binomial</a></li>

<li><a href="Trinomial" title="Trinomial">Trinomial</a></li>

<li><a href="Irreducible_polynomial" title="Irreducible polynomial">Irreducible</a></li>

<li><a href="Square-free_polynomial" title="Square-free polynomial">Square-free</a></li>

<li><a href="Homogeneous_polynomial" title="Homogeneous polynomial">Homogeneous</a></li>

<li><a href="Quasi-homogeneous_polynomial" title="Quasi-homogeneous polynomial">Quasi-homogeneous</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools and algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Factorization_of_polynomials" title="Factorization of polynomials">Factorization</a></li>

<li><a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">Greatest common divisor</a></li>

<li><a href="Polynomial_long_division" title="Polynomial long division">Division</a></li>

<li><a href="Horner's_method" title="Horner's method">Horner's method of evaluation</a></li>

<li><a href="Polynomial_identity_testing" title="Polynomial identity testing">Polynomial identity testing</a></li>

<li><a href="Polynomial_resultant" class="mw-redirect" title="Polynomial resultant">Resultant</a></li>

<li><a href="Discriminant" title="Discriminant">Discriminant</a></li>

<li><a href="Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-14" href="https://en.wikipedia.org/wiki/?title=Cubic_function&oldid=1290418164">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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