How to find range of a Quadratic/Quadratic function easily without plotting its graph?












0












$begingroup$


Is there any way to find range of a Quadratic/Quadratic function, without plotting its graph?






quadratics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you know how to complete the square?
    $endgroup$
    – Matthew Leingang
    May 9 '18 at 8:22










  • $begingroup$
    Do you mean a rational function with quadratic numerator or denominator ?
    $endgroup$
    – Yves Daoust
    May 9 '18 at 8:52










  • $begingroup$
    This type of problem was extremely common on tests and in textbooks in the late 1800s and very early 1900s, especially in France. I posted some information about it in this 10 December 2008 ap-calculus post archived at Math Forum.
    $endgroup$
    – Dave L. Renfro
    May 9 '18 at 9:42


















0












$begingroup$


Is there any way to find range of a Quadratic/Quadratic function, without plotting its graph?






quadratics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you know how to complete the square?
    $endgroup$
    – Matthew Leingang
    May 9 '18 at 8:22










  • $begingroup$
    Do you mean a rational function with quadratic numerator or denominator ?
    $endgroup$
    – Yves Daoust
    May 9 '18 at 8:52










  • $begingroup$
    This type of problem was extremely common on tests and in textbooks in the late 1800s and very early 1900s, especially in France. I posted some information about it in this 10 December 2008 ap-calculus post archived at Math Forum.
    $endgroup$
    – Dave L. Renfro
    May 9 '18 at 9:42
















0












0








0





$begingroup$


Is there any way to find range of a Quadratic/Quadratic function, without plotting its graph?






quadratics







share|cite|improve this question











$endgroup$




Is there any way to find range of a Quadratic/Quadratic function, without plotting its graph?






quadratics




quadratics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited May 9 '18 at 8:37









José Carlos Santos

176k24137247




176k24137247










asked May 9 '18 at 8:19









ShebSheb

1




1












  • $begingroup$
    Do you know how to complete the square?
    $endgroup$
    – Matthew Leingang
    May 9 '18 at 8:22










  • $begingroup$
    Do you mean a rational function with quadratic numerator or denominator ?
    $endgroup$
    – Yves Daoust
    May 9 '18 at 8:52










  • $begingroup$
    This type of problem was extremely common on tests and in textbooks in the late 1800s and very early 1900s, especially in France. I posted some information about it in this 10 December 2008 ap-calculus post archived at Math Forum.
    $endgroup$
    – Dave L. Renfro
    May 9 '18 at 9:42




















  • $begingroup$
    Do you know how to complete the square?
    $endgroup$
    – Matthew Leingang
    May 9 '18 at 8:22










  • $begingroup$
    Do you mean a rational function with quadratic numerator or denominator ?
    $endgroup$
    – Yves Daoust
    May 9 '18 at 8:52










  • $begingroup$
    This type of problem was extremely common on tests and in textbooks in the late 1800s and very early 1900s, especially in France. I posted some information about it in this 10 December 2008 ap-calculus post archived at Math Forum.
    $endgroup$
    – Dave L. Renfro
    May 9 '18 at 9:42


















$begingroup$
Do you know how to complete the square?
$endgroup$
– Matthew Leingang
May 9 '18 at 8:22




$begingroup$
Do you know how to complete the square?
$endgroup$
– Matthew Leingang
May 9 '18 at 8:22












$begingroup$
Do you mean a rational function with quadratic numerator or denominator ?
$endgroup$
– Yves Daoust
May 9 '18 at 8:52




$begingroup$
Do you mean a rational function with quadratic numerator or denominator ?
$endgroup$
– Yves Daoust
May 9 '18 at 8:52












$begingroup$
This type of problem was extremely common on tests and in textbooks in the late 1800s and very early 1900s, especially in France. I posted some information about it in this 10 December 2008 ap-calculus post archived at Math Forum.
$endgroup$
– Dave L. Renfro
May 9 '18 at 9:42






$begingroup$
This type of problem was extremely common on tests and in textbooks in the late 1800s and very early 1900s, especially in France. I posted some information about it in this 10 December 2008 ap-calculus post archived at Math Forum.
$endgroup$
– Dave L. Renfro
May 9 '18 at 9:42












3 Answers
3






active

oldest

votes


















1












$begingroup$

For $Ane 0$ we have $Ax^2+Bx+C=Ag(x)+D$ where $g(x)=(x+frac {B}{2A})^2$ and $D=C-frac {B^2}{4A}.$ The range of $g(x)$ is $[0,infty).$



If $A>0$ the range of $Ag(x)$ is ${Ay: yin [0,infty)}=[0,infty).$ So the range of $Ax^2+Bx+C$ is ${z+D: zin [0,infty)}=[D,infty).$



If $A<0$ the range of $Ag(x)$ is ${Ay:yin [0,infty)}=(-infty,0].$ So the range of $Ax^2+BX+C$ is ${z+D: zin (-infty,0]}=(-infty,D].$






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    Because of the symmetry of the parabola, its stationary point will always be in the middle of its $x$-intercepts (or zeroes). Using the quadratic formula and taking the average of both roots, the $x$-coordinate of the stationary point of any quadratic function $ax^2+bx+c$ (where $aneq0$) is given by $x=-frac{b}{2a}$.



    When $x=frac{-b}{2a}$, $y=c-frac{b^2}{4a}$.



    Therefore the maximum or minimum value of the quadratic is $c-frac{b^2}{4a}$. Whether it is the maximum or minimum can be determined by examining the sign of $a$.



    If $a$ is positive, then the range is $y geq c-frac{b^2}{4a}$.



    If $a$ is negative, then the range is $y leq c-frac{b^2}{4a}$.






    share|cite|improve this answer











    $endgroup$





















      0












      $begingroup$

      Hint:



      The rational function $dfrac{p(x)}{q(x)}$ has (one or two) vertical asymptotes if the denominator has real roots. It also has an horizontal asymptote $y=dfrac{p_2}{q_2}$.



      The local extrema are given by



      $$left(frac{p(x)}{q(x)}right)'=0$$ or



      $$p'(x)q(x)=p(x)q'(x).$$



      The two members are two cubic polynomial, with the same leading coefficient, so that this is in fact a quadratic equation (can also be linear or constant).



      After solving for the roots, if any, the limits of the range are given by the value at the extrema, at the horizontal asymptote, and $pminfty$ if there are vertical asymptotes.



      A longer analysis is required to exhaust the cases.






      share|cite|improve this answer









      $endgroup$














        Your Answer








        StackExchange.ready(function() {
        var channelOptions = {
        tags: "".split(" "),
        id: "69"
        };
        initTagRenderer("".split(" "), "".split(" "), channelOptions);

        StackExchange.using("externalEditor", function() {
        // Have to fire editor after snippets, if snippets enabled
        if (StackExchange.settings.snippets.snippetsEnabled) {
        StackExchange.using("snippets", function() {
        createEditor();
        });
        }
        else {
        createEditor();
        }
        });

        function createEditor() {
        StackExchange.prepareEditor({
        heartbeatType: 'answer',
        autoActivateHeartbeat: false,
        convertImagesToLinks: true,
        noModals: true,
        showLowRepImageUploadWarning: true,
        reputationToPostImages: 10,
        bindNavPrevention: true,
        postfix: "",
        imageUploader: {
        brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
        contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
        allowUrls: true
        },
        noCode: true, onDemand: true,
        discardSelector: ".discard-answer"
        ,immediatelyShowMarkdownHelp:true
        });


        }
        });














        draft saved

        draft discarded


















        StackExchange.ready(
        function () {
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2773358%2fhow-to-find-range-of-a-quadratic-quadratic-function-easily-without-plotting-its%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        1












        $begingroup$

        For $Ane 0$ we have $Ax^2+Bx+C=Ag(x)+D$ where $g(x)=(x+frac {B}{2A})^2$ and $D=C-frac {B^2}{4A}.$ The range of $g(x)$ is $[0,infty).$



        If $A>0$ the range of $Ag(x)$ is ${Ay: yin [0,infty)}=[0,infty).$ So the range of $Ax^2+Bx+C$ is ${z+D: zin [0,infty)}=[D,infty).$



        If $A<0$ the range of $Ag(x)$ is ${Ay:yin [0,infty)}=(-infty,0].$ So the range of $Ax^2+BX+C$ is ${z+D: zin (-infty,0]}=(-infty,D].$






        share|cite|improve this answer









        $endgroup$


















          1












          $begingroup$

          For $Ane 0$ we have $Ax^2+Bx+C=Ag(x)+D$ where $g(x)=(x+frac {B}{2A})^2$ and $D=C-frac {B^2}{4A}.$ The range of $g(x)$ is $[0,infty).$



          If $A>0$ the range of $Ag(x)$ is ${Ay: yin [0,infty)}=[0,infty).$ So the range of $Ax^2+Bx+C$ is ${z+D: zin [0,infty)}=[D,infty).$



          If $A<0$ the range of $Ag(x)$ is ${Ay:yin [0,infty)}=(-infty,0].$ So the range of $Ax^2+BX+C$ is ${z+D: zin (-infty,0]}=(-infty,D].$






          share|cite|improve this answer









          $endgroup$
















            1












            1








            1





            $begingroup$

            For $Ane 0$ we have $Ax^2+Bx+C=Ag(x)+D$ where $g(x)=(x+frac {B}{2A})^2$ and $D=C-frac {B^2}{4A}.$ The range of $g(x)$ is $[0,infty).$



            If $A>0$ the range of $Ag(x)$ is ${Ay: yin [0,infty)}=[0,infty).$ So the range of $Ax^2+Bx+C$ is ${z+D: zin [0,infty)}=[D,infty).$



            If $A<0$ the range of $Ag(x)$ is ${Ay:yin [0,infty)}=(-infty,0].$ So the range of $Ax^2+BX+C$ is ${z+D: zin (-infty,0]}=(-infty,D].$






            share|cite|improve this answer









            $endgroup$



            For $Ane 0$ we have $Ax^2+Bx+C=Ag(x)+D$ where $g(x)=(x+frac {B}{2A})^2$ and $D=C-frac {B^2}{4A}.$ The range of $g(x)$ is $[0,infty).$



            If $A>0$ the range of $Ag(x)$ is ${Ay: yin [0,infty)}=[0,infty).$ So the range of $Ax^2+Bx+C$ is ${z+D: zin [0,infty)}=[D,infty).$



            If $A<0$ the range of $Ag(x)$ is ${Ay:yin [0,infty)}=(-infty,0].$ So the range of $Ax^2+BX+C$ is ${z+D: zin (-infty,0]}=(-infty,D].$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 24 '18 at 7:37









            DanielWainfleetDanielWainfleet

            35.9k31648




            35.9k31648























                0












                $begingroup$

                Because of the symmetry of the parabola, its stationary point will always be in the middle of its $x$-intercepts (or zeroes). Using the quadratic formula and taking the average of both roots, the $x$-coordinate of the stationary point of any quadratic function $ax^2+bx+c$ (where $aneq0$) is given by $x=-frac{b}{2a}$.



                When $x=frac{-b}{2a}$, $y=c-frac{b^2}{4a}$.



                Therefore the maximum or minimum value of the quadratic is $c-frac{b^2}{4a}$. Whether it is the maximum or minimum can be determined by examining the sign of $a$.



                If $a$ is positive, then the range is $y geq c-frac{b^2}{4a}$.



                If $a$ is negative, then the range is $y leq c-frac{b^2}{4a}$.






                share|cite|improve this answer











                $endgroup$


















                  0












                  $begingroup$

                  Because of the symmetry of the parabola, its stationary point will always be in the middle of its $x$-intercepts (or zeroes). Using the quadratic formula and taking the average of both roots, the $x$-coordinate of the stationary point of any quadratic function $ax^2+bx+c$ (where $aneq0$) is given by $x=-frac{b}{2a}$.



                  When $x=frac{-b}{2a}$, $y=c-frac{b^2}{4a}$.



                  Therefore the maximum or minimum value of the quadratic is $c-frac{b^2}{4a}$. Whether it is the maximum or minimum can be determined by examining the sign of $a$.



                  If $a$ is positive, then the range is $y geq c-frac{b^2}{4a}$.



                  If $a$ is negative, then the range is $y leq c-frac{b^2}{4a}$.






                  share|cite|improve this answer











                  $endgroup$
















                    0












                    0








                    0





                    $begingroup$

                    Because of the symmetry of the parabola, its stationary point will always be in the middle of its $x$-intercepts (or zeroes). Using the quadratic formula and taking the average of both roots, the $x$-coordinate of the stationary point of any quadratic function $ax^2+bx+c$ (where $aneq0$) is given by $x=-frac{b}{2a}$.



                    When $x=frac{-b}{2a}$, $y=c-frac{b^2}{4a}$.



                    Therefore the maximum or minimum value of the quadratic is $c-frac{b^2}{4a}$. Whether it is the maximum or minimum can be determined by examining the sign of $a$.



                    If $a$ is positive, then the range is $y geq c-frac{b^2}{4a}$.



                    If $a$ is negative, then the range is $y leq c-frac{b^2}{4a}$.






                    share|cite|improve this answer











                    $endgroup$



                    Because of the symmetry of the parabola, its stationary point will always be in the middle of its $x$-intercepts (or zeroes). Using the quadratic formula and taking the average of both roots, the $x$-coordinate of the stationary point of any quadratic function $ax^2+bx+c$ (where $aneq0$) is given by $x=-frac{b}{2a}$.



                    When $x=frac{-b}{2a}$, $y=c-frac{b^2}{4a}$.



                    Therefore the maximum or minimum value of the quadratic is $c-frac{b^2}{4a}$. Whether it is the maximum or minimum can be determined by examining the sign of $a$.



                    If $a$ is positive, then the range is $y geq c-frac{b^2}{4a}$.



                    If $a$ is negative, then the range is $y leq c-frac{b^2}{4a}$.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited May 9 '18 at 8:53

























                    answered May 9 '18 at 8:48









                    PKBeamPKBeam

                    31219




                    31219























                        0












                        $begingroup$

                        Hint:



                        The rational function $dfrac{p(x)}{q(x)}$ has (one or two) vertical asymptotes if the denominator has real roots. It also has an horizontal asymptote $y=dfrac{p_2}{q_2}$.



                        The local extrema are given by



                        $$left(frac{p(x)}{q(x)}right)'=0$$ or



                        $$p'(x)q(x)=p(x)q'(x).$$



                        The two members are two cubic polynomial, with the same leading coefficient, so that this is in fact a quadratic equation (can also be linear or constant).



                        After solving for the roots, if any, the limits of the range are given by the value at the extrema, at the horizontal asymptote, and $pminfty$ if there are vertical asymptotes.



                        A longer analysis is required to exhaust the cases.






                        share|cite|improve this answer









                        $endgroup$


















                          0












                          $begingroup$

                          Hint:



                          The rational function $dfrac{p(x)}{q(x)}$ has (one or two) vertical asymptotes if the denominator has real roots. It also has an horizontal asymptote $y=dfrac{p_2}{q_2}$.



                          The local extrema are given by



                          $$left(frac{p(x)}{q(x)}right)'=0$$ or



                          $$p'(x)q(x)=p(x)q'(x).$$



                          The two members are two cubic polynomial, with the same leading coefficient, so that this is in fact a quadratic equation (can also be linear or constant).



                          After solving for the roots, if any, the limits of the range are given by the value at the extrema, at the horizontal asymptote, and $pminfty$ if there are vertical asymptotes.



                          A longer analysis is required to exhaust the cases.






                          share|cite|improve this answer









                          $endgroup$
















                            0












                            0








                            0





                            $begingroup$

                            Hint:



                            The rational function $dfrac{p(x)}{q(x)}$ has (one or two) vertical asymptotes if the denominator has real roots. It also has an horizontal asymptote $y=dfrac{p_2}{q_2}$.



                            The local extrema are given by



                            $$left(frac{p(x)}{q(x)}right)'=0$$ or



                            $$p'(x)q(x)=p(x)q'(x).$$



                            The two members are two cubic polynomial, with the same leading coefficient, so that this is in fact a quadratic equation (can also be linear or constant).



                            After solving for the roots, if any, the limits of the range are given by the value at the extrema, at the horizontal asymptote, and $pminfty$ if there are vertical asymptotes.



                            A longer analysis is required to exhaust the cases.






                            share|cite|improve this answer









                            $endgroup$



                            Hint:



                            The rational function $dfrac{p(x)}{q(x)}$ has (one or two) vertical asymptotes if the denominator has real roots. It also has an horizontal asymptote $y=dfrac{p_2}{q_2}$.



                            The local extrema are given by



                            $$left(frac{p(x)}{q(x)}right)'=0$$ or



                            $$p'(x)q(x)=p(x)q'(x).$$



                            The two members are two cubic polynomial, with the same leading coefficient, so that this is in fact a quadratic equation (can also be linear or constant).



                            After solving for the roots, if any, the limits of the range are given by the value at the extrema, at the horizontal asymptote, and $pminfty$ if there are vertical asymptotes.



                            A longer analysis is required to exhaust the cases.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered May 9 '18 at 9:10









                            Yves DaoustYves Daoust

                            133k676232




                            133k676232






























                                draft saved

                                draft discarded




















































                                Thanks for contributing an answer to Mathematics Stack Exchange!


                                • Please be sure to answer the question. Provide details and share your research!

                                But avoid …



                                • Asking for help, clarification, or responding to other answers.

                                • Making statements based on opinion; back them up with references or personal experience.


                                Use MathJax to format equations. MathJax reference.


                                To learn more, see our tips on writing great answers.




                                draft saved


                                draft discarded














                                StackExchange.ready(
                                function () {
                                StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2773358%2fhow-to-find-range-of-a-quadratic-quadratic-function-easily-without-plotting-its%23new-answer', 'question_page');
                                }
                                );

                                Post as a guest















                                Required, but never shown





















































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown

































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown







                                -

                                Popular posts from this blog

                                Plaza Victoria

                                Image
                                Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l
                                Read more

                                Puebla de Zaragoza

                                Image
                                Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc
                                Read more

                                Musa

                                Image
                                Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo
                                Read more