Equations and graphs of polynomial functions

[littlemisssunshine]

Describe the two ways that the equation of a polynomial function can be used to determine the type of symmetry exhibited by the graph of that function.

 

[Deleted member 4993]

Describe the two ways that the equation of a polynomial function can be used to determine the type of symmetry exhibited by the graph of that function.

Please share with us your work, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[littlemisssunshine]

I have various graphs. One group of them are considered even and the other group is considered odd.

for example: even - f(x)=x*4, f(x)=x*4-8*2, f(x)=-x*4+9x*2 and f(x)=-x*6+7x*4+3x*2
odd - f(x)=x*3, f(x)=x*3-4x, f(x)=-x*5+16x*3, and f(x)=-x*5+5x*3+6x

i need to describe two ways that the equation of a polynomial function can be used to determine the type of symmetry exhibited by the graph of that function.

i have one way and that is that if there is an even degree function, the arms of the function will be on the same side on the axes(either from quadrent 2 to 1 or from 3 to 4) so there will be reflection in the y axis and for odd functions the arms of the function will be in opposite quadrents (from quadrent 3 to 1 or from 2 to 4) therefore making it point symmetry about the origin.

Now i dont know any other way that you can tell the type of symmetry and thats where i need the help.

 

[BigGlenntheHeavy]

Your answer tells all.

If the polynominal is an even function, then it is symmetric with respect to the y axis and if the polynominal is an odd function, then it is symmetric with respect to the origin.

Why can't it be symmetric to the x axis?

 

[mmm4444bot]

One way to use the function definitions (i.e., equations), is to change the sign of the input and see what happens to the output.

Given:

f(x) = polynomial

If f(-x) = -polynomial, f is an odd function

If f(-x) = polynomial, f is an even function

In other words, if changing the sign on x results in no change to the polynomial, the function is even.

If changing the sign on x results in the polynomial changing sign, the function is odd.

EG 1:

f(x) = -x^4 + 9x^2

Now, we change the sign on x, and see what happens.

f(-x) = -(-x)^4 + 9(-x)^2

Since (-x)^4 is the same as x^4 AND (-x)^2 is the same as x^2, the polynomial does not change.

f(-x) = -x^4 + 9x^2

f is even.

EG 2:

f(x) = x^3 - 4x

Now, we change the sign on x, and see what happens.

f(-x) = (-x)^3 - 4(-x)

Since (-x)^3 is the same as -x^3 AND (-x) is the same as -x, the polynomial changes sign.

f(-x) = -x^3 + 4x = -(x^3 - 4x)

Like I said, this is ONE way to use the equations to test for eveness or oddness.

Off the top of my head, I cannot think of a second way to use the equations.

And, when you describe "arms" of polynomials, I don't know what that means, but it sounds like you're making deductions based on graph behavior, instead of using the equations.

BTW: Did you notice that we use the caret symbol ^ to denote exponentiation?

The asterisk * is used for something else (a multiplication sign)

x raised to the fourth power is written: x^4

x times 4 is written: x*4