How do you sketch the function f(x)=ln(2x+1) for x≥0?

[Kulla_9289]

How do you sketch the function f(x)=ln(2x+1) for x≥0? I know the plotting method. To save time, I am trying to learn how you would sketch these types of graphs. The asymptote of this function is x=-0.5, which is not in the domain, so it is irrelevant. Any help will be appreciated.

 

[Harry_the_cat]

What "plotting method" do you know? Do you mean drawing up a table and plotting the points?

Do you know about transformations of graphs? That is, how does he graph of f(x) = ln(2x+1) = ln (2(x + 1/2)) compare to the graph of the basic natural log function f(x) = ln (x)?

 

[Dr.Peterson]

How do you sketch the function f(x)=ln(2x+1) for x≥0? I know the plotting method. To save time, I am trying to learn how you would sketch these types of graphs. The asymptote of this function is x=-0.5, which is not in the domain, so it is irrelevant. Any help will be appreciated.

Actually, the asymptote is highly relevant: It is the first thing I'd sketch. Of course that isn't part of the graph itself, but it guides you.

After that, if you don't want to just plot a few points, I'd use transformations. How do you see ln(x) being transformed? This is a relatively challenging case (with two "horizontal" transformations), so we'll want to see whatever attempt you can make, as a starting point.

 

[Steven G]

Start by graphing y=ln(x). Then find the vertex of your function, y=ln(2x+1), and graph y=ln(2x+1).

 

[Dr.Peterson]

Start by graphing y=ln(x). Then find the vertex of your function, y=ln(2x+1), and graph y=ln(2x+1).

Where is that?? Do you mean the transformations? Or the asymptote?

 

[Harry_the_cat]

Start by graphing y=ln(x). Then find the vertex of your function, y=ln(2x+1), and graph y=ln(2x+1).

The graph of a basic log function doesn't have a vertex??

 

[Steven G]

The graph of a basic log function doesn't have a vertex??

I like to think that it does but formally you are correct.

 

[Steven G]

Where is that?? Do you mean the transformations? Or the asymptote?

I consider that the vertex is at (0,0).

 

[Harry_the_cat]

I consider that the vertex is at (0,0).

What is YOUR definition of a vertex?

 

[Dr.Peterson]

I consider that the vertex is at (0,0).

I would perhaps call that the "relative origin", in the sense that in shifting the graph, you can identify where the origin moves to, and draw the new graph relative to that point. That's how I often picture such transformations, drawing a new graph relative to shifted axes.

But it absolutely is not the vertex of the graph by any definition I've ever heard of; it is not even on the graph!

 

[Steven G]

I draw a new set of axes by what Dr Peterson calls the relative origin and then tell the student to graph a regular curve (in this case a ln graph)

You teach students to graph many graphs (sqrt, x^2, x^3, |x|,...) by using the term vertex. In my opinion, why bother to change the name of the process when you want the student to graph functions like 1/x, 1/x^2, lnx, e^x...

I just think that it is not a big deal calling it a vertex rather than relative origin.

 

[Dr.Peterson]

I draw a new set of axes by what Dr Peterson calls the relative origin and then tell the student to graph a regular curve (in this case a ln graph)

You teach students to graph many graphs (sqrt, x^2, x^3, |x|,...) by using the term vertex. In my opinion, why bother to change the name of the process when you want the student to graph functions like 1/x, 1/x^2, lnx, e^x...

I just think that it is not a big deal calling it a vertex rather than relative origin.

Except that no one else calls it a vertex, and the word vertex has a different meaning. It means a turning point of a curve (among other related things). It does not mean the process of translating a graph; nor is it a name for the origin (though the vertices of your examples are all at the origin).

Why not use the proper name: origin? Tell them, "Just as you can move the vertex of some curves from the origin to a new location, you can move the origin of the coordinate system for these other "parent functions" and draw the graph relative to that."

Words are meant to communicate, and that requires a shared vocabulary. What happens if your students look up "vertex" and get confused, or use the word in a subsequent class and are told they are wrong?

 

[Kulla_9289]

What is transformations of graphs?

 

[Dr.Peterson]

What is transformations of graphs?

When a function f(x) is modified to f(2x+1), its graph is shifted (translated) left 1 unit, and then compressed horizontally by a factor of 2. These are transformations of the graph. And as Harry said (and I prefer), you can rewrite it as f(2(x + 1/2)) and view it as a compression followed by a shift by 1/2 unit, which is more natural.

If you haven't learned this concept, then you aren't expected to use it; but it makes it much easier to graph a function whose basic form ("parent function") you are familiar with. And in my experience, this is commonly taught before teaching about logarithms.

Please tell us what you have learned about graphing, as we asked. The more you tell us (including showing any work you've done), the more effectively we can help.

 

[Kulla_9289]

I know logarithm and the laws of it including natural logs and Euler's constant. Only this sketching seems tough

 

[JeffM]

I know logarithm and the laws of it including natural logs and Euler's constant. Only this sketching seems tough

@Kulla_9289

This reiterates Dr. Peterson’s advice: learn about transformations of graphs. To start:

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:shift/v/shifting-functions-intro

and

Scaling functions introduction (video) | Khan Academy

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