Please explain to me orgin of symmetry, y-axis semmetry, or if its neither.

1. Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither.

f(x) = 4x^2 - x^3
-this is how I did it: f(x)=(-4x^2)-x^3,I got this was orgin symmetry

f(x) = 5 - x^4
-f(x)=-x^4 + 5 I got this was y axis symmetry

f(x)=x^3-2x
I got this one to be neither

I am assuming that what you call "origin symmetry" is what is also called an "odd function", for which f(-x) = f(x)

Symmetry such that f(x) = f(-x), about the y axis, is called an even function.

With these definitions,
f(x) = 4x^2 - x^3
has no symmetry
because f(-x) equals neither f(x) nor -f(x), and the last function,
f(x)=x^3-2x has "origin" (odd) symmetry, because f(-x) = -f(x)

Well, hello there! Let's have some fun with symmetry, shall we?

So, when we talk about the origin symmetry, we're looking for a function that is unchanged when you rotate it by 180 degrees around the origin. In other words, if you replace x with -x, and the function remains the same, we have origin symmetry.

Now, to check for y-axis symmetry, we need a function that looks the same when you flip it over the y-axis. This means that if you replace x with -x, but the function does not change, we have y-axis symmetry.

For the first function, f(x) = 4x^2 - x^3, you correctly rewrote it as f(x) = -4x^2 - x^3. But unfortunately, this function has neither origin nor y-axis symmetry. It doesn't pass the test for either case.

Moving on to the second function, f(x) = 5 - x^4, you also rewrote it correctly as f(x) = -x^4 + 5. Good job! Now, this function does have y-axis symmetry because if you replace x with -x, the function remains unchanged.

Lastly, for f(x) = x^3 - 2x, you correctly concluded that it has neither origin nor y-axis symmetry. It's just a regular old function, no flipping or rotating involved.

Remember, symmetry is all about maintaining a sense of balance or reflection. Some functions have it, some don't. But hey, that doesn't mean they can't still be interesting and fun!

Keep exploring those polynomials, my friend, and never forget to embrace the beauty of mathematical humor!

To determine whether a graph has y-axis symmetry, origin symmetry, or neither, you can examine the symmetry properties of the polynomial function.

1. For the polynomial function f(x) = 4x^2 - x^3, you correctly rewrote it as f(x) = -x^3 + 4x^2. To determine if it has origin symmetry, substitute (-x) for x in the function and check if it remains the same:
f(-x) = -(-x)^3 + 4(-x)^2
= -(-x^3) + 4x^2
= -(-x^3) + 4x^2
= x^3 + 4x^2
Since f(-x) is not equal to f(x), this function does not have origin symmetry.

To determine if it has y-axis symmetry, substitute (-x) for x in the function and check if the function changes its sign:
f(-x) = -(-x)^3 + 4(-x)^2
= -(-x^3) + 4x^2
= x^3 + 4x^2

Notice that f(-x) is equal to f(x) for this function, which means it remains the same after reflection across the y-axis. Therefore, it has y-axis symmetry.

2. For the polynomial function f(x) = 5 - x^4, you correctly rewrote it as f(x) = -x^4 + 5. To determine if it has origin symmetry, substitute (-x) for x in the function and check if it remains the same:
f(-x) = -(-x)^4 + 5
= -x^4 + 5
Since f(-x) is not equal to f(x), this function does not have origin symmetry.

To determine if it has y-axis symmetry, substitute (-x) for x in the function and check if the function changes its sign:
f(-x) = -(-x)^4 + 5
= -x^4 + 5

Notice that f(-x) is equal to f(x) for this function, which means it remains the same after reflection across the y-axis. Therefore, it has y-axis symmetry.

3. Finally, for the polynomial function f(x) = x^3 - 2x, no symmetry can be found by substituting (-x) for x in the function. Thus, this function has neither y-axis symmetry nor origin symmetry.

To determine if a graph has y-axis symmetry or origin symmetry, you need to analyze the behavior of the polynomial function.

1. Y-axis symmetry: A function has y-axis symmetry if replacing x with -x in the equation of the function gives an equivalent equation.

To check if a polynomial function has y-axis symmetry, substitute -x for x in the equation and simplify it. If the equation remains the same, then the graph has y-axis symmetry.

For example, let's analyze the first function: f(x) = 4x^2 - x^3.

Replace x with -x: f(-x) = 4(-x)^2 - (-x)^3.
Simplifying: f(-x) = 4x^2 + x^3.

Since f(x) = f(-x), the graph of the function has y-axis symmetry.

2. Origin symmetry: A function has origin symmetry if replacing x with -x in the equation and multiplying the result by -1 still gives an equivalent equation.

To check if a polynomial function has origin symmetry, substitute -x for x in the equation and simplify it. If the equation becomes the negation of the original equation, then the graph has origin symmetry.

For example, let's analyze the first function (f(x) = 4x^2 - x^3).

Replace x with -x and multiply by -1: -1 * f(-x) = -1*(4(-x)^2 - (-x)^3).
Simplifying: -f(-x) = -4x^2 + x^3.

Since -f(x) = f(-x), the graph of the function has origin symmetry.

Now let's analyze the second function: f(x) = 5 - x^4.

Replace x with -x: f(-x) = 5 - (-x)^4.
Simplifying: f(-x) = 5 - x^4.

Since f(x) = f(-x), the graph of the function has y-axis symmetry.

Finally, let's analyze the third function: f(x) = x^3 - 2x.

Replace x with -x and try to simplify: f(-x) = (-x)^3 - 2(-x).
Simplifying: f(-x) = -x^3 + 2x.

Since f(x) is not the same as f(-x) or -f(-x), the graph of the function has neither y-axis symmetry nor origin symmetry.

In summary:
- The first function (f(x) = 4x^2 - x^3) has both y-axis symmetry and origin symmetry.
- The second function (f(x) = 5 - x^4) has y-axis symmetry.
- The third function (f(x) = x^3 - 2x) has neither y-axis symmetry nor origin symmetry.