Name: rational functions with slant asymptotes
Uploaded: 2026-01-12T12:24:40.000Z
Duration: 1 min 27 s
Channel: JoVE

Une asymptote inclinée est une droite inclinée que le graphe d’une fonction tend lorsque x devient très grand ou très petit. Dans une fonction rationnelle, une asymptote inclinée existe lorsque le degré du numérateur est exactement un plus haut que celui du dénominateur.Pour déterminer l’asymptote inclinée, on applique une division longue polynomiale, divisant le numérateur par le dénominateur.Le quotient est une expression linéaire, et le reste forme une fraction. Lorsque x tend vers l’infini positif ou négatif, cette fraction tend vers zéro.La fonction tend vers la droite donnée par le quotient, qui est l’asymptote inclinée.Les asymptotes verticales sont déterminées en égalissant le dénominateur à zéro, et les interceptions indiquent où le graphique croise les axes.L’asymptote inclinée définit cependant le comportement à long terme du graphe.En raison du degré plus élevé dans le numérateur, il n’existe pas d’asymptote horizontale. Le graphique courbe fortement près de l’asymptote verticale et s’aligne avec l’asymptote inclinée aux deux extrémités.Dans la réalité, une asymptote inclinée pourrait modéliser une fonction de coût moyen. Si des facteurs comme l’usure de la machine font augmenter le coût total de façon quadratique, le coût moyen par article tend vers une asymptote inclinée.

asymptotes obliques

Slant Asymptotes

Le comportement d&#x2019;une fonction est souvent r&#xE9;gi par des contraintes asymptotiques, o&#xF9; un terme domine un autre, d&#xE9;finissant ainsi une tendance asymptotique. Dans le cas pr&#xE9;sent, la structure math&#xE9;matique correspond &#xE0; une fonction rationnelle : un terme cubique au num&#xE9;rateur est divis&#xE9; par un terme quadratique au d&#xE9;nominateur. Il en r&#xE9;sulte une fonction pr&#xE9;sentant des caract&#xE9;ristiques distinctes, notamment une asymptote oblique, des points critiques et des intervalles o&#xF9; elle n&#x2019;est pas d&#xE9;finie.
Le domaine de d&#xE9;finition de la fonction est d&#xE9;termin&#xE9; par le d&#xE9;nominateur, qui doit &#xEA;tre non nul. Cette contrainte scinde le domaine en intervalles disjoints, &#xE9;vitant ainsi les singularit&#xE9;s o&#xF9; la fonction n&#x2019;est pas d&#xE9;finie. Les points critiques apparaissent lorsque la d&#xE9;riv&#xE9;e premi&#xE8;re s&#x2019;annule, signalant un changement de sens de variation. &#xC9;tant donn&#xE9; que la fonction coupe l&#x2019;axe des abscisses lorsque le num&#xE9;rateur est nul, les racines r&#xE9;elles du terme cubique d&#xE9;finissent ces points. En revanche, l&#x2019;absence d&#x2019;intersection avec l&#x2019;axe des ordonn&#xE9;es indique que la fonction ne passe pas par l&#x2019;origine.
Une caract&#xE9;ristique majeure de cette fonction est la pr&#xE9;sence d&#x2019;une asymptote oblique, qui d&#xE9;coule de la division de polyn&#xF4;mes. Le num&#xE9;rateur &#xE9;tant de degr&#xE9; sup&#xE9;rieur &#xE0; celui du d&#xE9;nominateur, la division euclidienne met en &#xE9;vidence un quotient lin&#xE9;aire qui repr&#xE9;sente la tendance asymptotique. La fonction se rapproche de cette asymptote pour des valeurs tendant vers l&#x2019;infini, sans toutefois l&#x2019;intersecter ni la d&#xE9;passer.
Le taux de variation, donn&#xE9; par la d&#xE9;riv&#xE9;e premi&#xE8;re, confirme une d&#xE9;croissance sur un intervalle donn&#xE9; jusqu&#x2019;&#xE0; l&#x2019;atteinte d&#x2019;un minimum local, suivie d&#x2019;une croissance. La transition progressive entre ces comportements traduit la continuit&#xE9; de la fonction. Par ailleurs, la d&#xE9;riv&#xE9;e seconde reste positive, ce qui indique que la fonction conserve une concavit&#xE9; tourn&#xE9;e vers le haut. Cette propri&#xE9;t&#xE9; garantit que la fonction n&#x2019;oscille pas de mani&#xE8;re impr&#xE9;visible, mais qu&#x2019;elle conserve une courbure stable et coh&#xE9;rente. De telles fonctions apparaissent fr&#xE9;quemment en physique et en ing&#xE9;nierie, mod&#xE9;lisant des ph&#xE9;nom&#xE8;nes o&#xF9; une limite structurelle sous-jacente contraint l&#x2019;&#xE9;volution. L&#x2019;asymptote oblique repr&#xE9;sente alors une borne th&#xE9;orique, &#xE0; l&#x2019;image de situations r&#xE9;elles dans lesquelles une grandeur se rapproche d&#x2019;une limite sans jamais l&#x2019;atteindre.

Asymptotes obliques

Le comportement d&#x2019;une fonction est souvent r&#xE9;gi par des contraintes asymptotiques, o&#xF9; un terme domine un autre, d&#xE9;finissant ainsi ...

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rational functions with slant asymptotes

Rational functions can follow a slant asymptote when the numerator grows faster than the denominator. In this case, a cubic numerator is divided by a squared denominator. That degree difference creates a linear trend that guides the graph at large input values.

The function is only defined where the denominator is not zero. Those excluded x-values split the domain into separate regions and avoid singular points. The graph can still cross the x-axis where the numerator equals zero, so the real roots of the cubic give the horizontal intercepts. There is no vertical intercept, so the graph does not pass through the origin.

Long division shows the slant asymptote, which is the linear quotient that the graph follows as x gets very large or very small. The function may approach that line, but it does not intersect or go past it. The graph also has critical points, where the first derivative equals zero. These points mark a change in direction.

The rate of change shows a decreasing region that reaches a local minimum and then rises again. The function is smooth through this transition, so it stays continuous. Its second derivative stays positive, which means the graph has upward concavity. That steady curve helps explain why the function does not wobble unpredictably.

Functions like this are useful in physics and engineering. They can model situations where growth is limited by a structural constraint. In that sense, the slant asymptote acts like a theoretical bound. The graph keeps getting close to the bound, but it never quite reaches it.

Rational Functions with Slant Asymptotes

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key terms and concepts

Explore how a rational function with a cubic numerator and squared denominator creates a slant asymptote, critical points, and vertical restrictions.

Rational Functions with Slant Asymptotes | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Rational Functions with Slant Asymptotes

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