On the Graph, For How Many Values of x Does y=x?

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The discussion centers on determining the number of intersection points between the line y=x and a graph resembling an "S" shape. Participants conclude that while the line y=x contains an infinite number of points, the specific intersections with the "S" shape yield three distinct points: (0,0) and the two endpoints of the "S". The confusion arises from the interpretation of the question and the accuracy of the graph representation.

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Homework Statement



On the graph, for how many values of x does y=x?


Homework Equations




The Attempt at a Solution



The obvious answer is (0,0).
There have to be more. I don't know how to go about finding this answer.
 

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If x is 10 can you give me a y such that y = x? If so that's the point (10,y)

If x is 12.3 can you give me a y such that y = x? If so that's the point (12.3,y)

If x is any value can you give me a y such that y = x? If so that's the point (x,y)

Do you see the solution now?
 
Diffy said:
If x is 10 can you give me a y such that y = x? If so that's the point (10,y)

If x is 12.3 can you give me a y such that y = x? If so that's the point (12.3,y)

If x is any value can you give me a y such that y = x? If so that's the point (x,y)

Do you see the solution now?

I don't think so. Would the answer be all of them?

Is there a specific number?

I'm sorry. It's just not clicking.
 
I'm sorry, I didn't see that there was an attachment. I thought your question was a bit easier. Your attachment awaits approval, can you describe it for me?
 
Diffy said:
I'm sorry, I didn't see that there was an attachment. I thought your question was a bit easier. Your attachment awaits approval, can you describe it for me?

An "S" shape is plotted. The points above the graph are symmetrical to the points on the bottom. The only point that is labeled is (1,1). The very center of the "S" is at (0,0).

At (1,1), (2,2) etc, y=x

The possibilities are infinite.

If I am on the right track, what would the answer be?

An infinite number?
 
No No. Again, I thought the question originally was how many points in the x,y plane does y = x. Of course the answer to that question is an infinite number. That however is not the question your were asking.

To answer your real question, consider drawing the graph of the line y=x, through your S shape. Then the number of points at which it intersects your S is the number of values on your graph where y = x.

Again, apologies for the confusion.
 
No worries. I probably didn't phrase it as best as I could have phrased it. I appreciate the help.

If I did it correctly, There are 2 points that touch the "S" on each line. I drew lines at (1,1), (2,2), etc and at (0,0).

At 0, there are 3.

I don't know what the actual number is though. I'm still confused.
 
You've certainly got me confused! What lines did you draw at (1,1), (2,2), etc.?
I have no idea what you are talking about when you say "At 0, there are 3"! What do you mean "at 0"? Do you mean "at (0,0)" or "at x= 0" or "at y=0"? And there are 3 what?

The problem is asking "where does this graph cross the line y= x?" That's difficult to answer here because it depends strongly on how accurate this graph is supposed to be and exactly where each end of the "S" ends. Can you draw the line y= x?
 
HallsofIvy said:
You've certainly got me confused! What lines did you draw at (1,1), (2,2), etc.?
I have no idea what you are talking about when you say "At 0, there are 3"! What do you mean "at 0"? Do you mean "at (0,0)" or "at x= 0" or "at y=0"? And there are 3 what?

The problem is asking "where does this graph cross the line y= x?" That's difficult to answer here because it depends strongly on how accurate this graph is supposed to be and exactly where each end of the "S" ends. Can you draw the line y= x?

Yes, at (0,0)

I don't understand what you mean by the line y=x. I don't know where to draw it, or how/why it's just one line.

I am sorry to confuse you. Things get lost in translation in this format.

I'm not getting the concept but I am trying hard.
 
  • #10
LLS said:
Yes, at (0,0)

I don't understand what you mean by the line y=x. I don't know where to draw it, or how/why it's just one line.

I am sorry to confuse you. Things get lost in translation in this format.

I'm not getting the concept but I am trying hard.

The line y = x consists of all the points where *drumroll...* y = x. For example, (0, 0), (1, 1), (2.439, 2.439), etc. Plot two of these points on the graph (maybe (0, 0) and (1, 1) ). The straight line that connects these points is y = x. It extends out infinitely past both points.
 
  • #11
steelphantom said:
The line y = x consists of all the points where *drumroll...* y = x. For example, (0, 0), (1, 1), (2.439, 2.439), etc. Plot two of these points on the graph (maybe (0, 0) and (1, 1) ). The straight line that connects these points is y = x. It extends out infinitely past both points.

I sort of understand that. But I still don't understand what the answer is.

How many are there? It seems that the number is infinite.
 
  • #12
LLS said:
I sort of understand that. But I still don't understand what the answer is.

How many are there? It seems that the number is infinite.

The number of points on the line y = x is infinite, but the number of points where the line y = x intersects your "S" is not. Draw the line y = x as well as the "S," and the points where the lines cross will be the intersection points.

For example, in http://education.yahoo.com/homework.../1/minialg2gt_12_1_1_27_110/f-438-1-we-1.gif", there are two points where the line intersects the ellipse.
 
Last edited by a moderator:
  • #13
steelphantom said:
The number of points on the line y = x is infinite, but the number of points where the line y = x intersects your "S" is not. Draw the line y = x as well as the "S," and the points where the lines cross will be the intersection points.

For example, in http://education.yahoo.com/homework.../1/minialg2gt_12_1_1_27_110/f-438-1-we-1.gif", there are two points where the line intersects the ellipse.

3?

I don't think that I am drawing y=x correctly.
 
Last edited by a moderator:
  • #14
y=x is a straight line, at an elevation of 45 degrees from the x-axis, working counterclockwise. Plot that, and the answer will be however many times that graph and the line intersect.
 
  • #15
BrendanH said:
y=x is a straight line, at an elevation of 45 degrees from the x-axis, working counterclockwise. Plot that, and the answer will be however many times that graph and the line intersect.

I came up with 2.
 
  • #16
Good! How did you come up with that? I presume you have a clearer picture than we can see on the internet, because I was not at all sure. (Of course, I didn't want to draw on my computer screen!) Obviously, as you said, (0,0) is one point of intersection. Is the other in the lower left (3rd quadrant) or upper right (1st quadrant)?
 
  • #17
HallsofIvy said:
Good! How did you come up with that? I presume you have a clearer picture than we can see on the internet, because I was not at all sure. (Of course, I didn't want to draw on my computer screen!) Obviously, as you said, (0,0) is one point of intersection. Is the other in the lower left (3rd quadrant) or upper right (1st quadrant)?

Would you believe on a wing and a prayer?

Wow, I actually got the answer? Finally??!?
2, 2, 2, 2, 2, 2...!

Upper right?
 
  • #18
If you look at the graph (the image has finally been approved), you will see that the origin and also the two end points of the S all lie on the line y = x, so there are actually 3 points that the line y = x intersects. The end points of that graph should really be labeled, though.
 
  • #19
steelphantom said:
If you look at the graph (the image has finally been approved), you will see that the origin and also the two end points of the S all lie on the line y = x, so there are actually 3 points that the line y = x intersects. The end points of that graph should really be labeled, though.

It's all I have. It might have been easier had the graph been better.

Thanx again

:)
 

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