Graphing Exponential Equations

[Deo3560]

OK, I am having issues with a graphing calculator, because when I zoom out of the graph (containing exponents), the graph changes almost completely. If you could tell me which graph is right, or what is happening, and how to fix it if it isnt right.

The graphing calculator I am using is:
http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

The question I need to solve is:

Make your own table of values for the function y=(-2)?. Use the integers from -3 to +3 for your x values. Graph these points by hand on a sheet of paper. What happens to your graph?

I did the "Table of Values" for the function, but I need help on the graph.

This is the the first graph:
y=(-2)?
x-min. = -10
x-max. = 10
x-scale = 1
y-min. = -10
y-max. = 10
y-scale = 1
http://img148.imageshack.us/img148/7930/pic2nb.png

This is the second graph zoomed out:
y=(-2)?
x-min. = -20
x-max. = 20
x-scale = 1
y-min. = -20
y-max. = 20
y-scale = 1
http://img444.imageshack.us/img444/3918/pic2n.png

And I dont know If this will help but this is an the "Table of Values" the graphing program gave me:
http://img245.imageshack.us/img245/4672/pic2d.png

So if anyone can help me out that would be great

Thanks,
Deo3560

 

[Denis]

y=(-2)?.

Huh?

 

[Deo3560]

y=(-2)^x

the x variable is in the exponent

 

[mmm4444bot]

Make your own table of values for the function y = (-2)^x. Use the integers from -3 to +3 for your x values. Graph these points by hand on a sheet of paper. What happens to your graph?

From looking at your work, it appears that you have not followed any of the three instructions above.

First step: Turn your computer off.

Next steps:

Show us your own table of values, using the required values of x.

Show us your hand-drawn plot of these seven points.

Then you will be in a position to answer the question, "What happens to your graph [when you try to connect these points with a smooth curve]?"

 

[Deo3560]

I did all of that before I used the graphing calculator, I just was checking my answers....
All of my answers were similar... but the graph.
But even though I have the "supposed answer" I would still like to know why that happens for future reference.

 

[mmm4444bot]

I would still like to know why that happens Why what happens?

You have not yet stated any answer to the posted question, so what do you mean when you write the pronoun "that" above?

In other words, what is the "supposed answer" that you claim to have? Tell us!

 

[Deo3560]

The "supposed answer" is what I got myself, like before I used the calculator, and "that" is referring to my original question, of why the 2 graphs are different.

 

[Mrspi]

The "supposed answer" is what I got myself, like before I used the calculator, and "that" is referring to my original question, of why the 2 graphs are different.

Some of us were born at night, but not LAST night.

It would be GREAT if you did as mmm suggested...and described your results. If you really and truly have done that, then you should be able to see (rather easily, I think) why "zooming out" on the graphing calculator changes the APPEARANCE of the graph (but not, of course, the actual values of the coordinates of points on the graph).

When you have DONE THE WORK yourself, as requested, and have described what your results have shown you, we can and will comment further.

 

[Deo3560]

http://img508.imageshack.us/img508/6804/pic2lb.png

There is my work..... there is no need to be rude, I'm asking for help, its your choice if you want to.
I'm not trying to just get a quick cheat cheat on homework, I am trying to figure out why the graph changes it's appearance....

If you really and truly have done that, then you should be able to see (rather easily, I think) why "zooming out" on the graphing calculator changes the APPEARANCE of the graph

So, that was my work and I still don't get it.... All I asked for was for you to explain it to me...

So if you could so kindly tell my why the second graph curved, while the first graph goes back and forth between the x axis.

 

[mmm4444bot]

Those two graphs differ (in part) due to the way in which the software samples values along the chosen, restricted domain. Is the reason why really this important to you? After all, the reason why has nothing to do with the exercise. The exercise does not involve computer-generated graphs.

Both of those machine graphs are WRONG.

Instead of investing your efforts trying to understand the logic errors in this particular computer code, I'm thinking that it would be better for you to learn that machine results cannot be trusted absolutely and to move on.

Try thinking about why functions like this (i.e., with negative bases) cannot be graphed.

Hint: What happens in your table of values, if x is a fraction, like -1/2 or 1/2 or 1/3 or 1/4? In other words, what is happening to the graph "between" the points that you plotted?

 

[Deo3560]

Thank you very much, that was all I wished to hear.

It wasn't that the question was important to me, but what if a similar problem came up in the future, and I didn't understand it. I am only taking precautions because I have had things pop up before that had a slight twist that takes a different turn in what I was being taught.

Even this could be considered as one of those times, because I was only taught y=b?, where b>1 and also another graph where 0<b<1, but wasn't taught negatives. On a lesson a bit after that I did learn about times where b is negative, but this equation wasn't y=-b?, it was y=(-b)?. I was taught when y=-b? it looked like the graph below (though it was mirrored over the y axis because it wasn't a fraction), but then they told me to make a chart for y=(-2)?, where there are parenthesis, and the graph wound up crossing the x axis many times, which I didn't understand since I was not ready for that yet.

But now that I know which graph was right, thanks.

Hint: What happens in your table of values, if x is a fraction, like -1/2 or 1/2 or 1/3 or 1/4? In other words, what is happening to the graph "between" the points that you plotted?

and yeah.... I still dont understand what is going on with the graph because I get what your saying there, but I just graphed it according to whole numbers.

 

[mmm4444bot]

I am only taking precautions because I have had things pop up before that had a slight twist that takes a different turn in what I was being taught.

There is no way to prepare for every curve ball in life.

Instructors do not always explain everything up-front, expecting you to memorize it en masse. Some instructors believe that sometimes it's better for you to discover something on your own, as you will be more likely to remember it that way. One approach to motivate discovery is to throw you a curve ball, because, if you're puzzled, the hope is that you will employ some sort of analysis to "discover" what's going on. Or, at the very least, raise specific questions.

Even this could be considered as one of those times, because I was only taught y=b^x, [where the base b is a positive number other than 1]

Exactly. And, the point of this exercise (I believe) is to get you thinking (analyzing) about specific graphing problems that arise when the base is negative.

But now that I know which graph was right, thanks. None of the graphs are correct for y = (-2)^x because this relationship between x and y cannot be graphed.

(If you were not to have "connected the dots", then your graph would be a correct graph of seven discrete points. But seven points do not comprise a picture of the given function because x takes on infinite values from -3 to 3.)

I still dont understand what is going on with the graph Okay. I'll add some more comments below.

Let's talk about integer values of x, first.

When x is an Integer, it tells us how many factors of the base are multiplied together, yes? (That's what Integer exponents mean.)

Examples:

(-2)^4 means that four factors of -2 are multiplied together.

(-2)^7 means that seven factors of -2 are multiplied together.

(-2)^(-3) means that three factors of -1/2 are multiplied together.

(-2)^(-9) means that nine factors of -1/2 are multiplied together.

Now, if there are an EVEN number of factors, what is the sign of the product?

It has to be positive, yes?

Likewise, if x is an ODD Integer, then the power must be negative.

In other words, as x takes on consecutive Integer values, y is bouncing back and forth infinite times across the x-axis. That is, the sign of y is alternating, as x alternates back and forth between being even and odd).

We cannot simply connect these points, and call it a graph of the function y = (-2)^x, because y is not defined at infinitely many points between the Integers! To understand why, let's talk about some Rational values of x.

Consider what happens halfway between x = 0 and x = 1.

y = (-2)^(1/2)

This is the same as:

y = sqrt(-2)

Square roots of negative numbers do not exist in the Real number system, so y does not exist when x = 1/2.

It gets even worse!

When x is any rational number with an even denominator, y does not exist, because even roots of negative numbers do not exist, in the Real number system.

Therefore, there are "holes" in the graph, when x = 1/2, x = 1/4, x = 1/6, x = 1/8, etc. Same on the other side of zero. There is no graph at the points where x = -1/2, x = -1/4, x = -1/6, etc.

We simply cannot graph a curve that contains an infinite number of holes between all of the Integer values of x AND abruptly jumping back and forth everywhere else.

So, the "discovery" here is that, when the base is NOT a positive number other than 1, the relationship y = b^x is NOT an exponential function.

Questions?