Cubic function

<h2 id="history">History</h2>
Main article: <a href="/facts/Cubic_equation/Eqm3HCMx">Cubic equation § History</a>
<h2 id="critical-and-inflection-points">Critical and inflection points</h2>

The <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical points</a> of a cubic function are its <a href="/facts/Stationary_point/f56GSgBj">stationary points</a>, that is the points where the slope of the function is zero.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Thus the critical points of a cubic function f defined by 

f(x) = ax3 + bx2 + cx + d,
occur at values of x such that the <a href="/facts/Derivative/7Ito70Dh">derivative</a>

3
        a
        
          x
          
            2
          
        
        +
        2
        b
        x
        +
        c
        =
        0
      
    
    {\displaystyle 3ax^{2}+2bx+c=0}

of the cubic function is zero.
The solutions of this equation are the x-values of the critical points and are given, using the <a href="/facts/Quadratic_formula/j5ftMowG">quadratic formula</a>, by

x
          
            critical
          
        
        =
        
          
            
              −
              b
              ±
              
                
                  
                    b
                    
                      2
                    
                  
                  −
                  3
                  a
                  c
                
              
            
            
              3
              a
            
          
        
        .
      
    
    {\displaystyle x_{\text{critical}}={\frac {-b\pm {\sqrt {b^{2}-3ac}}}{3a}}.}

The sign of the expression Δ0 = b2 – 3ac 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 b2 – 3ac = 0, then there is only one critical point, which is an <a href="/facts/Inflection_point/Ev2dv4WS">inflection point</a>. If b2 – 3ac < 0, then there are no (real) critical points. In the two latter cases, that is, if b2 – 3ac is nonpositive, the cubic function is strictly <a href="/facts/Monotonic/MjQK0YL2">monotonic</a>. See the figure for an example of the case Δ0 > 0.
The inflection point of a function is where that function changes <a href="/facts/Second_derivative/3aA6j4EY">concavity</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> An inflection point occurs when the <a href="/facts/Second_derivative/3aA6j4EY">second derivative</a> 
 
 
 
 
 f
 ″
 
 (
 x
 )
 =
 6
 a
 x
 +
 2
 b
 ,
 
 
 {\displaystyle f''(x)=6ax+2b,}
 
 is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at

x
          
            inflection
          
        
        =
        −
        
          
            b
            
              3
              a
            
          
        
        .
      
    
    {\displaystyle x_{\text{inflection}}=-{\frac {b}{3a}}.}

<h2 id="classification">Classification</h2>

The <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of a cubic function is a <a href="/facts/Cubic_curve/FSU8tXAb">cubic curve</a>, though many cubic curves are not graphs of functions.
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="/facts/Similarity_(geometry)/Sbd6hEAN">similar</a> to the graph of a function of the form

y
        =
        
          x
          
            3
          
        
        +
        p
        x
        .
      
    
    {\displaystyle y=x^{3}+px.}

This similarity can be built as the composition of <a href="/facts/Translation/2foI82so">translations</a> parallel to the coordinates axes, a <a href="/facts/Homothecy/WUmuRuzC">homothecy</a> (<a href="/facts/Uniform_scaling/pFm6fvpu">uniform scaling</a>), and, possibly, a <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a> (<a href="/facts/Mirror_image/xkbUhBzI">mirror image</a>) with respect to the y-axis. A further <a href="/facts/Uniform_scaling/pFm6fvpu">non-uniform scaling</a> can transform the graph into the graph of one among the three cubic functions

y
              
              
                
                =
                
                  x
                  
                    3
                  
                
                +
                x
              
            
            
              
                y
              
              
                
                =
                
                  x
                  
                    3
                  
                
              
            
            
              
                y
              
              
                
                =
                
                  x
                  
                    3
                  
                
                −
                x
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}y&=x^{3}+x\\y&=x^{3}\\y&=x^{3}-x.\end{aligned}}}

This means that there are only three graphs of cubic functions <a href="/facts/Up_to/ydz4ztzX">up to</a> an <a href="/facts/Affine_transformation/BD7yDr10">affine transformation</a>.
The above <a href="/facts/Geometric_transformation/qRjrasJf">geometric transformations</a> can be built in the following way, when starting from a general cubic function

y
        =
        a
        
          x
          
            3
          
        
        +
        b
        
          x
          
            2
          
        
        +
        c
        x
        +
        d
        .
      
    
    {\displaystyle y=ax^{3}+bx^{2}+cx+d.}

Firstly, if a < 0, the <a href="/facts/Change_of_variable/vjWdXCDF">change of variable</a> x → –x allows supposing a > 0. After this change of variable, the new graph is the mirror image of the previous one, with respect of the y-axis.
Then, the change of variable x = x1 – ⁠b/3a⁠ provides a function of the form

y
        =
        a
        
          x
          
            1
          
          
            3
          
        
        +
        p
        
          x
          
            1
          
        
        +
        q
        .
      
    
    {\displaystyle y=ax_{1}^{3}+px_{1}+q.}

This corresponds to a translation parallel to the x-axis.
The change of variable y = y1 + q corresponds to a translation with respect to the y-axis, and gives a function of the form

y
          
            1
          
        
        =
        a
        
          x
          
            1
          
          
            3
          
        
        +
        p
        
          x
          
            1
          
        
        .
      
    
    {\displaystyle y_{1}=ax_{1}^{3}+px_{1}.}

The change of variable 
 
 
 
 
 
 x
 
 1
 
 
 =
 
 
 
 x
 
 2
 
 
 
 a
 
 
 
 ,
 
 y
 
 1
 
 
 =
 
 
 
 y
 
 2
 
 
 
 a
 
 
 
 
 
 
 {\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}}
 
 corresponds to a uniform scaling, and give, after multiplication by 
 
 
 
 
 
 a
 
 
 ,
 
 
 {\displaystyle {\sqrt {a}},}
 
 a function of the form

y
          
            2
          
        
        =
        
          x
          
            2
          
          
            3
          
        
        +
        p
        
          x
          
            2
          
        
        ,
      
    
    {\displaystyle y_{2}=x_{2}^{3}+px_{2},}

which is the simplest form that can be obtained by a similarity.
Then, if p ≠ 0, the non-uniform scaling 
 
 
 
 
 
 x
 
 2
 
 
 =
 
 x
 
 3
 
 
 
 
 
 |
 
 p
 
 |
 
 
 
 ,
 
 
 y
 
 2
 
 
 =
 
 y
 
 3
 
 
 
 
 
 |
 
 p
 
 
 |
 
 
 3
 
 
 
 
 
 
 
 {\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}}
 
 gives, after division by 
 
 
 
 
 
 
 
 |
 
 p
 
 
 |
 
 
 3
 
 
 
 
 ,
 
 
 
 {\displaystyle \textstyle {\sqrt {|p|^{3}}},}

y
          
            3
          
        
        =
        
          x
          
            3
          
          
            3
          
        
        +
        
          x
          
            3
          
        
        sgn
        ⁡
        (
        p
        )
        ,
      
    
    {\displaystyle y_{3}=x_{3}^{3}+x_{3}\operatorname {sgn}(p),}

where 
 
 
 
 sgn
 ⁡
 (
 p
 )
 
 
 {\displaystyle \operatorname {sgn}(p)}
 
 has the value 1 or –1, depending on the sign of p. If one defines 
 
 
 
 sgn
 ⁡
 (
 0
 )
 =
 0
 ,
 
 
 {\displaystyle \operatorname {sgn}(0)=0,}
 
 the latter form of the function applies to all cases (with 
 
 
 
 
 x
 
 2
 
 
 =
 
 x
 
 3
 
 
 
 
 {\displaystyle x_{2}=x_{3}}
 
 and 
 
 
 
 
 y
 
 2
 
 
 =
 
 y
 
 3
 
 
 
 
 {\displaystyle y_{2}=y_{3}}
 
).

<h2 id="symmetry">Symmetry</h2>
For a cubic function of the form 
 
 
 
 y
 =
 
 x
 
 3
 
 
 +
 p
 x
 ,
 
 
 {\displaystyle y=x^{3}+px,}
 
 the inflection point is thus the origin. As such a function is an <a href="/facts/Odd_function/tGHWJqAa">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="/facts/Similarity_(geometry)/Sbd6hEAN">similarity</a>, the following is true for all cubic functions.
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.

<h2 id="collinearities">Collinearities</h2>

The tangent lines to the graph of a cubic function at three <a href="/facts/Collinear_points/SxbVBJYR">collinear points</a> intercept the cubic again at collinear points.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> This can be seen as follows.
As this property is invariant under a <a href="/facts/Rigid_motion/0KzWYQXl">rigid motion</a>, one may suppose that the function has the form

f
        (
        x
        )
        =
        
          x
          
            3
          
        
        +
        p
        x
        .
      
    
    {\displaystyle f(x)=x^{3}+px.}

If α is a real number, then the tangent to the graph of f at the point (α, f(α)) is the line

{(x, f(α) + (x − α)f ′(α)) : x ∈ R}.
So, the intersection point between this line and the graph of f can be obtained solving the equation f(x) = f(α) + (x − α)f ′(α), that is

x
          
            3
          
        
        +
        p
        x
        =
        
          α
          
            3
          
        
        +
        p
        α
        +
        (
        x
        −
        α
        )
        (
        3
        
          α
          
            2
          
        
        +
        p
        )
        ,
      
    
    {\displaystyle x^{3}+px=\alpha ^{3}+p\alpha +(x-\alpha )(3\alpha ^{2}+p),}

which can be rewritten

x
          
            3
          
        
        −
        3
        
          α
          
            2
          
        
        x
        +
        2
        
          α
          
            3
          
        
        =
        0
        ,
      
    
    {\displaystyle x^{3}-3\alpha ^{2}x+2\alpha ^{3}=0,}

and factorized as

(
        x
        −
        α
        
          )
          
            2
          
        
        (
        x
        +
        2
        α
        )
        =
        0.
      
    
    {\displaystyle (x-\alpha )^{2}(x+2\alpha )=0.}

So, the tangent intercepts the cubic at

(
        −
        2
        α
        ,
        −
        8
        
          α
          
            3
          
        
        −
        2
        p
        α
        )
        =
        (
        −
        2
        α
        ,
        −
        8
        f
        (
        α
        )
        +
        6
        p
        α
        )
        .
      
    
    {\displaystyle (-2\alpha ,-8\alpha ^{3}-2p\alpha )=(-2\alpha ,-8f(\alpha )+6p\alpha ).}

So, the function that maps a point (x, y) of the graph to the other point where the tangent intercepts the graph is

(
        x
        ,
        y
        )
        ↦
        (
        −
        2
        x
        ,
        −
        8
        y
        +
        6
        p
        x
        )
        .
      
    
    {\displaystyle (x,y)\mapsto (-2x,-8y+6px).}

This is an <a href="/facts/Affine_transformation/BD7yDr10">affine transformation</a> that transforms collinear points into collinear points. This proves the claimed result.

<h2 id="cubic-interpolation">Cubic interpolation</h2>
Main article: <a href="/facts/Spline_interpolation/AowdpQwm">Spline interpolation</a>
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="/facts/Cubic_Hermite_spline/S29nwRUF">cubic Hermite spline</a>.
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 interpolate the function with a <a href="/facts/Continuously_differentiable_function/jQqTgLk1">continuously differentiable function</a>, which is a <a href="/facts/Piecewise/nlQwptpe">piecewise</a> cubic function.
If the value of a function is known at several points, <a href="/facts/Cubic_interpolation/S29nwRUF">cubic interpolation</a> consists in approximating the function by a <a href="/facts/Continuously_differentiable_function/jQqTgLk1">continuously differentiable function</a>, which is <a href="/facts/Piecewise/nlQwptpe">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="/facts/Curvature/RFCAEsj8">curvature</a> at the endpoints.

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Cubic functions.

<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Cardano_formula">"Cardano formula"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a 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="/facts/MacTutor_archive/xtwLmN5q">MacTutor archive</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bostock, Linda; Chandler, Suzanne; Chandler, F. S. (1979). Pure Mathematics 2. Nelson Thornes. p. 462. ISBN 978-0-85950-097-5. Thus a cubic equation has either three real roots... or one real root... <a href="978-0-85950-097-5" target="_blank">978-0-85950-097-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weisstein, Eric W. "Stationary Point". mathworld.wolfram.com. Retrieved 2020-07-27. <a href="https://mathworld.wolfram.com/StationaryPoint.html" target="_blank">https://mathworld.wolfram.com/StationaryPoint.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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). Applied Calculus. John Wiley & Sons. p. 181. ISBN 978-1-119-27556-5. A point at which the graph of the function f changes concavity is called an inflection point of f <a href="978-1-119-27556-5" target="_blank">978-1-119-27556-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Whitworth, William Allen (1866), "Equations of the third degree", Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions, Cambridge: Deighton, Bell, and Co., p. 425, retrieved June 17, 2016 <a href="/wiki/William_Allen_Whitworth" target="_blank">/wiki/William_Allen_Whitworth</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Cubic function open-in-new

Cubic function