Given a hyperbola, determine whether any of its points lie within $xin [0,1]mid 0leq yleq 1$

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












Given
$$
y=fracc_1 +c_2xc_3+c_4x
$$



Is there any test using the values of $c$ to see if between $xin[0,1]$ there is at least one point where $yin[0,1]$ other than just evaluating the function on the interval?



Examples could include using the distance between the hyperbola points or shifting the hyperbola.







share|cite|improve this question


















  • 2




    Interesting way to write $yleq 0$. :D
    – Batominovski
    Aug 8 at 19:36











  • @batominovski I've encountered that way many times written by $3 $ rd level students - it may come from some remote country.
    – Arnaud Mortier
    Aug 8 at 20:15











  • But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
    – NickD
    Aug 8 at 20:28










  • I made a mistake because I typed it up on my phone. No I am not from a remote country haha
    – wjmccann
    Aug 9 at 4:02














up vote
1
down vote

favorite












Given
$$
y=fracc_1 +c_2xc_3+c_4x
$$



Is there any test using the values of $c$ to see if between $xin[0,1]$ there is at least one point where $yin[0,1]$ other than just evaluating the function on the interval?



Examples could include using the distance between the hyperbola points or shifting the hyperbola.







share|cite|improve this question


















  • 2




    Interesting way to write $yleq 0$. :D
    – Batominovski
    Aug 8 at 19:36











  • @batominovski I've encountered that way many times written by $3 $ rd level students - it may come from some remote country.
    – Arnaud Mortier
    Aug 8 at 20:15











  • But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
    – NickD
    Aug 8 at 20:28










  • I made a mistake because I typed it up on my phone. No I am not from a remote country haha
    – wjmccann
    Aug 9 at 4:02












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Given
$$
y=fracc_1 +c_2xc_3+c_4x
$$



Is there any test using the values of $c$ to see if between $xin[0,1]$ there is at least one point where $yin[0,1]$ other than just evaluating the function on the interval?



Examples could include using the distance between the hyperbola points or shifting the hyperbola.







share|cite|improve this question














Given
$$
y=fracc_1 +c_2xc_3+c_4x
$$



Is there any test using the values of $c$ to see if between $xin[0,1]$ there is at least one point where $yin[0,1]$ other than just evaluating the function on the interval?



Examples could include using the distance between the hyperbola points or shifting the hyperbola.









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 9 at 4:01

























asked Aug 8 at 19:00









wjmccann

578117




578117







  • 2




    Interesting way to write $yleq 0$. :D
    – Batominovski
    Aug 8 at 19:36











  • @batominovski I've encountered that way many times written by $3 $ rd level students - it may come from some remote country.
    – Arnaud Mortier
    Aug 8 at 20:15











  • But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
    – NickD
    Aug 8 at 20:28










  • I made a mistake because I typed it up on my phone. No I am not from a remote country haha
    – wjmccann
    Aug 9 at 4:02












  • 2




    Interesting way to write $yleq 0$. :D
    – Batominovski
    Aug 8 at 19:36











  • @batominovski I've encountered that way many times written by $3 $ rd level students - it may come from some remote country.
    – Arnaud Mortier
    Aug 8 at 20:15











  • But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
    – NickD
    Aug 8 at 20:28










  • I made a mistake because I typed it up on my phone. No I am not from a remote country haha
    – wjmccann
    Aug 9 at 4:02







2




2




Interesting way to write $yleq 0$. :D
– Batominovski
Aug 8 at 19:36





Interesting way to write $yleq 0$. :D
– Batominovski
Aug 8 at 19:36













@batominovski I've encountered that way many times written by $3 $ rd level students - it may come from some remote country.
– Arnaud Mortier
Aug 8 at 20:15





@batominovski I've encountered that way many times written by $3 $ rd level students - it may come from some remote country.
– Arnaud Mortier
Aug 8 at 20:15













But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
– NickD
Aug 8 at 20:28




But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
– NickD
Aug 8 at 20:28












I made a mistake because I typed it up on my phone. No I am not from a remote country haha
– wjmccann
Aug 9 at 4:02




I made a mistake because I typed it up on my phone. No I am not from a remote country haha
– wjmccann
Aug 9 at 4:02










1 Answer
1






active

oldest

votes

















up vote
1
down vote













Yes. You can easily manipulate the above formula to get an expression of the form $$y=A+fracBx+C$$
What does this expression tells you? You can compare with $y=1/x$ hyperbola. The $C$ value just changes the vertical asymptote. The $A$ value just changes the horizontal asymptote. $B$ is a stretching factor in the vertical direction. Note that if $B<0$, you flip the hyperbola upside-down.



What you need to do is to calculate $y$ when $x=0$ and $x=1$, and calculate $x$ when $y=0$ or $y=1$. If none of these four points verify your requirements, there are no other options.






share|cite|improve this answer




















    Your Answer




    StackExchange.ifUsing("editor", function ()
    return StackExchange.using("mathjaxEditing", function ()
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    );
    );
    , "mathjax-editing");

    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',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    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%2f2876450%2fgiven-a-hyperbola-determine-whether-any-of-its-points-lie-within-x-in-0-1%23new-answer', 'question_page');

    );

    Post as a guest






























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote













    Yes. You can easily manipulate the above formula to get an expression of the form $$y=A+fracBx+C$$
    What does this expression tells you? You can compare with $y=1/x$ hyperbola. The $C$ value just changes the vertical asymptote. The $A$ value just changes the horizontal asymptote. $B$ is a stretching factor in the vertical direction. Note that if $B<0$, you flip the hyperbola upside-down.



    What you need to do is to calculate $y$ when $x=0$ and $x=1$, and calculate $x$ when $y=0$ or $y=1$. If none of these four points verify your requirements, there are no other options.






    share|cite|improve this answer
























      up vote
      1
      down vote













      Yes. You can easily manipulate the above formula to get an expression of the form $$y=A+fracBx+C$$
      What does this expression tells you? You can compare with $y=1/x$ hyperbola. The $C$ value just changes the vertical asymptote. The $A$ value just changes the horizontal asymptote. $B$ is a stretching factor in the vertical direction. Note that if $B<0$, you flip the hyperbola upside-down.



      What you need to do is to calculate $y$ when $x=0$ and $x=1$, and calculate $x$ when $y=0$ or $y=1$. If none of these four points verify your requirements, there are no other options.






      share|cite|improve this answer






















        up vote
        1
        down vote










        up vote
        1
        down vote









        Yes. You can easily manipulate the above formula to get an expression of the form $$y=A+fracBx+C$$
        What does this expression tells you? You can compare with $y=1/x$ hyperbola. The $C$ value just changes the vertical asymptote. The $A$ value just changes the horizontal asymptote. $B$ is a stretching factor in the vertical direction. Note that if $B<0$, you flip the hyperbola upside-down.



        What you need to do is to calculate $y$ when $x=0$ and $x=1$, and calculate $x$ when $y=0$ or $y=1$. If none of these four points verify your requirements, there are no other options.






        share|cite|improve this answer












        Yes. You can easily manipulate the above formula to get an expression of the form $$y=A+fracBx+C$$
        What does this expression tells you? You can compare with $y=1/x$ hyperbola. The $C$ value just changes the vertical asymptote. The $A$ value just changes the horizontal asymptote. $B$ is a stretching factor in the vertical direction. Note that if $B<0$, you flip the hyperbola upside-down.



        What you need to do is to calculate $y$ when $x=0$ and $x=1$, and calculate $x$ when $y=0$ or $y=1$. If none of these four points verify your requirements, there are no other options.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 8 at 19:55









        Andrei

        7,5832822




        7,5832822






















             

            draft saved


            draft discarded


























             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2876450%2fgiven-a-hyperbola-determine-whether-any-of-its-points-lie-within-x-in-0-1%23new-answer', 'question_page');

            );

            Post as a guest













































































            這個網誌中的熱門文章

            How to combine Bézier curves to a surface?

            Mutual Information Always Non-negative

            Why am i infinitely getting the same tweet with the Twitter Search API?