Odd Polynomial Functions: prove algebraically and explain...

[ahlingg]

Prove algebraically and explain grahically why a polynomial that is an odd function is no longer an odd function when a non-zero constant is added. Provide examples.

Thanks in advance

 

[daon]

Re: Odd Polynomial Functions

You have p(-x) = -p(x) for all values x. If q(x) = p(x) + c, where c is nonzero, then compare q(-x) to -q(x).

 

[pka]

Re: Odd Polynomial Functions

One might also note that it is necessary that an odd function contain (0,0) because odd functions are symmetric about the origin. Adding a constant changes that fact.

 

[mmm4444bot]

Re: Odd Polynomial Functions

Hello:

I'm not sure what response you're looking for because you did not ask any question.

Therefore, I have to guess.

It's very easy to explain in words or demonstrate graphically why adding a constant to an ODD function's definition results in a new function that IS NOT odd. Since you cannot do this, I guess the following.

You do not understand the definition of an odd function.

I could type the symbolic definition, but I'm sure you've already seen it, so I doubt it will help you to see it again.

I'm going to explain the concept of an odd function in words.

A function is odd if the result of changing the sign on its INPUT is to only change the sign on the corresponding OUTPUT.

f(2) = 10
f(-2) = -10

g(6) = 1/2
g(-6) = -1/2

h(-5) = 7
h(5) = -7

T(-4/5) = 5000
T(4/5) = -5000

Consider the examples above for functions f, g, h, and T.

If I tell you that function f(x) is odd, and I tell you that f(2) = 10, then you know right away what f(-2) is without having to do any calculations. If you change the sign on the input, the only change to the output is also the sign.

In other words, if we change the sign on 2, then the absolute value of the output will still be 10; only the sign changes.

f(-2) = -10

If I tell you that T(x) is an odd function, and I tell you that T(4/5) equals -5000, then you know right away that T(-4/5) must be 5000.

The output will always be the opposite of what it was before the sign was changed on the input.

The following function is ODD.

f(-8.95) = 22/7

Since you know that f(x) is odd, you also know what the output will be if we change the sign on the input.

f(8.95) = ?

At this point, if you still do not understand what number goes in place of the question mark above, then you've really hit a road block. Please speak with your instructor. If not, then please read on.

Okay. I know that the sine function is not a polynomial, but I'm going to use it as another example, anyway. I hope that you know what the graph of a sine wave looks like.

So, now assuming that you DO understand the relationship between inputs, outputs, and their signs in an ODD function, then I would like you to consider the graph of sin(x).

The sine function is an odd function.

We can see this by looking at the outputs for a few inputs with sign changes.

sin(?/4) = ?2/2

Now change the sign on the input, and see what happens.

sin(-?/4) = -?2/2

The absolute value of the output stays the same; only the sign changes.

sin(-60°) = -?3/2
sin(60°) = ?3/2

sin(?/2) = 1
sin(-?/2) = -1

Now think about what happens to a function if we add a constant to its definition.

F(x) = sin(x) + 3

I want you to investigate whether or not F(x) is odd.

In other words, "when a non-zero constant is added", is the function still odd?

Compare the following values.

sin(?) + 3

sin(-?) + 3

Are they the same? They have to be if we're going to say that sin(x) + 3 is ODD.

Here's something else to think about.

The graph of sin(x) + 3 looks like the graph of sin(x) shifted UP 3 units.

Hopefully, this is enough information for you to explain WHY in words an ODD function is no longer odd when a non-zero constant is added it its definition.

If you still are unable to explain WHY in words, then you are not yet at the point where you could begin to PROVE it algebraically.

There are several people here to help you further, but you NEED TO LET US KNOW WHY you don't understand, so that we don't have to GUESS what it is that you're thinking.

Cheers,

~ Mark

 

[mmm4444bot]

Re: Odd Polynomial Functions

... it is necessary that an odd function contain (0,0) because odd functions are symmetric about the origin.

EXCELLENT POINT! I hope that the original poster eventually gains a solid understanding of the implications of this important concept.