Given a hyperbola, determine whether any of its points lie within $xin [0,1]mid 0leq yleq 1$
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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.
algebra-precalculus functions algorithms conic-sections hyperbolic-geometry
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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.
algebra-precalculus functions algorithms conic-sections hyperbolic-geometry
2
Interesting way to write $yleq 0$. :D
â 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.
â Arnaud Mortier
Aug 8 at 20:15
But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
â 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
â wjmccann
Aug 9 at 4:02
add a comment |Â
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.
algebra-precalculus functions algorithms conic-sections hyperbolic-geometry
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.
algebra-precalculus functions algorithms conic-sections hyperbolic-geometry
edited Aug 9 at 4:01
asked Aug 8 at 19:00
wjmccann
578117
578117
2
Interesting way to write $yleq 0$. :D
â 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.
â Arnaud Mortier
Aug 8 at 20:15
But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
â 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
â wjmccann
Aug 9 at 4:02
add a comment |Â
2
Interesting way to write $yleq 0$. :D
â 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.
â Arnaud Mortier
Aug 8 at 20:15
But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
â 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
â wjmccann
Aug 9 at 4:02
2
2
Interesting way to write $yleq 0$. :D
â Batominovski
Aug 8 at 19:36
Interesting way to write $yleq 0$. :D
â 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.
â 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.
â Arnaud Mortier
Aug 8 at 20:15
But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
â NickD
Aug 8 at 20:28
But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
â 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
â 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
â wjmccann
Aug 9 at 4:02
add a comment |Â
1 Answer
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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.
add a comment |Â
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.
add a comment |Â
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.
add a comment |Â
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.
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.
answered Aug 8 at 19:55
Andrei
7,5832822
7,5832822
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2
Interesting way to write $yleq 0$. :D
â 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.
â Arnaud Mortier
Aug 8 at 20:15
But in the body of the question, it is made clear that what is meant is $0 le y le 1$.
â 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
â wjmccann
Aug 9 at 4:02