# Thread: Three Graphs Include y = x

1. ## Three Graphs Include y = x

I want to know why the following three graphs include the line y = x?

(1) g(x) = x + [4/(x^2+1)

(2) f(x) = (x^2+1)/x

(3) y = (x^2-6x+12)/(x-4)

I am trying to learn curve sketching from my single variable calculus textbook. I am not asking you to graph all three. I just want to know why the above functions include the line y = x as part of the graph.

2. ## Re: Three Graphs Include y = x

I'm not entirely sure what you mean by "includes y=x as part of the graph" but from what I think you mean.

1) $\displaystyle g(x)=x+\frac{4}{x^2+1}$

this function tends towards x as the second term tends towards zero so asymptotically g(x)=x

2) $\displaystyle f(x)=\frac{x^2+1}{x}=x+\frac{1}{x}$

this function also tends towards x for the same reason, the second term tends to 0 as x grows.

3) $\displaystyle y=\frac{x^2-6x+12}{x-4}=(x-2)+\frac{4}{x-4}$

y tends to (x-2) here as x grows so once again asymptotically y looks like x albeit offset by the constant 2.

Is this what you are after?

3. ## u

Hello, nycmath!

I want to know why the following three graphs include the line $\displaystyle y = x.$

$\displaystyle (1)\;g(x) \:=\: x + \frac{4}{x^2+1}$

$\displaystyle (2)\;f(x) \:=\:\frac{x^2+1}{x}$

$\displaystyle (3)\;y \:=\;\frac{x^2-6x+12}{x-4}$

That line isn't a part of the graph.
It is an asymptote; the graph approaches that slanted line.

Consider: .$\displaystyle (1)\;y \:=\:x + \frac{4}{x^2+1}$

As $\displaystyle x\to\infty$, the fraction goes to 0.
Hence, the graph gets closer and closer to $\displaystyle y = x.$

In Quadrant 1, the graph looks like this:
Code:
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|*
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|                   .
| *              *.
|              *.
|  *         *.
|   *     * .
|     *   .
|       .
|     .
|   .
| .
- - + - - - - - - - - - - - -
|

4. ## Re: Three Graphs Include y = x

I will practice more curve sketching using calculus 1 techniques. Soroban, check out your inbox. Have a great weekend guys.

5. ## Re: Three Graphs Include y = x

Originally Posted by nycmath
I want to know why the following three graphs include the line y = x?
(1) g(x) = x + [4/(x^2+1)
(2) f(x) = (x^2+1)/x
(3) y = (x^2-6x+12)/(x-4)
For the life of me. I cannot understand why you do not use free on line tools?
Look at this.

You can change the input there to see other graphs.

6. ## Re: Three Graphs Include y = x

Plato: I am trying to learn curve sketching without the use of technology. Thanks for the link.

7. ## Re: Three Graphs Include y = x

I use wolframalpha.com all the time. However, it's a lot more fun trying to graph these functions using curve sketching techniques from calculus 1 textbooks.

8. ## Re: Three Graphs Include y = x

Originally Posted by nycmath
Plato: I am trying to learn curve sketching without the use of technology. Thanks for the link.
O.K. But why not goto WFA and look at the graph? As you look at it, ask yourself "why does that work?"

Does any of that actually work?

If you can answer that question, then pass it on.