Math help

[Chaos_Sorcerer]

We were supposed to fill out these tables for homework, but I am having trouble understanding them. 

[Teddy07]

come on man. Why not spin the last picture?

[Chaos_Sorcerer]

Just now, Teddy07 said:

come on man. Why not spin the last picture?

Didn't even notice. Sorry.

[Dash Lambda]

Once you sketch the graphs you basically just have to look at them. One will be a symmetrical shape, the other won't (it'll have a tail), one will touch the x axis and the other won't, both will touch the y axis but in different spots, etc.

[Chaos_Sorcerer]

Just now, Dash Lambda said:

Once you sketch the graphs you basically just have to look at them. One will be a symmetrical shape, the other won't (it'll have a tail), one will touch the x axis and the other won't, both will touch the y axis but in different spots, etc.

What about the patterns in the first and second differences? I'm pretty sure that my answers for linear and quadratic growth are correct, but is there a pattern in the first or second differences of exponential growth? I noticed that the difference always tends to be the opposite of a multiple of the minuend.

 

Also, would my description for the graph of exponential growth be correct? What about the other asymptote questions? 

[Dash Lambda]

8 minutes ago, Chaos_Sorcerer said:

What about the patterns in the first and second differences? I'm pretty sure that my answers for linear and quadratic growth are correct, but is there a pattern in the first or second differences of exponential growth? I noticed that the difference always tends to be the opposite of a multiple of the minuend.

 

Also, would my description for the graph of exponential growth be correct? What about the other asymptote questions? 

I would say growing for quadratic first differences, but being 'different' is right too.

That basically is the pattern for exponential differences, I can't remember if they even talked about that in algebra. Each successive set of differences is a constant multiple of the set before it. In fact, it's the same constant each time -So the differences don't reduce to 0 (unless you consider the limit at infinity, then it can).

 

And yes, exponential growth is characterized by an asymptote. Specifically, it's an asymptote towards negative infinity, since it had to be growing with increasing x.

Parabolas, however, do not have asymptotes.

 

For a simple exponential function (that is, b^x), the asymptote is horizontal on the x axis. This is because in one direction the function becomes vanishingly small.

[Chaos_Sorcerer]

2 hours ago, Dash Lambda said:

I would say growing for quadratic first differences, but being 'different' is right too.

That basically is the pattern for exponential differences, I can't remember if they even talked about that in algebra. Each successive set of differences is a constant multiple of the set before it. In fact, it's the same constant each time -So the differences don't reduce to 0 (unless you consider the limit at infinity, then it can).

 

And yes, exponential growth is characterized by an asymptote. Specifically, it's an asymptote towards negative infinity, since it had to be growing with increasing x.

Parabolas, however, do not have asymptotes.

 

For a simple exponential function (that is, b^x), the asymptote is horizontal on the x axis. This is because in one direction the function becomes vanishingly small.

 

So...would this be right? What would the y-intercept of the exponential function be? 

 

*y increases in the negative direction if x > 0, and y increases in the positive direction if x < 0

*y also increases in the negative direction, but is never below 0

*y increases in the positive direction if x > 0, and y increases in the negative direction if x < 0

*y increases in the positive direction

[Dash Lambda]

7 minutes ago, Chaos_Sorcerer said:

-snip-

-Infinity is not a number. An asymptote is not a value the function reaches at infinity, it's a value the function approaches but never hits. Infinity just describes the fact that you're talking about how the function proceeds without end. (This affects two boxes.)

 

And it crosses the y axis at x=0 (by virtue of rectangular coordinates). What is b⁰?

[Chaos_Sorcerer]

3 hours ago, Dash Lambda said:

-Infinity is not a number. An asymptote is not a value the function reaches at infinity, it's a value the function approaches but never hits. Infinity just describes the fact that you're talking about how the function proceeds without end. (This affects two boxes.)

 

And it crosses the y axis at x=0 (by virtue of rectangular coordinates). What is b⁰?

Hmm...so no exponential function will ever cross the x-axis, and will instead stretch along it forever? And (0, 1) would be its y-intercept?

 

Thanks for the help, by the way. 

 

EDIT: Is the rest of the table correct?

[Dash Lambda]

6 minutes ago, Chaos_Sorcerer said:

Hmm...so no exponential function will ever cross the x-axis, and will instead stretch along it forever? And (0, 1) would be its y-intercept?

Is the rest of the table correct?

None of the form b^x, as in order to make a product (exponentials just being products) resolve to 0 you must multiply by 0, and to do that you need b=0.

And (0, 1) would be the y-intercept, except for possibly when b actually is 0. There isn't really good consensus on 0⁰.

 

The rest of the table is correct, given that you also change the range for exponentials to reflect the fact that y cannot equal 0.