I know the following equation is definitely NOT EVEN, BUT I DO NOT KNOW WHY IT'S NOT ODD!. . .

PLEASE. . .EXPLAIN THOROUGHLY!. . .

y = 2(x - 1)^2 + 3

f(-x) = 2(-x - 1)^2 + 3
f(-x) does not equal f(x), therefore, not even. . .

-f(x) = -[2(x + 1)^2 + 3]
-f(x) = -2(-x - 1)^2 - 3
-f(x) does not equal f(x), therefore, not odd ----- BUT WHY? I DON'T SEE THIS!!!!!. . .

Isn't the equation y = 2(x - 1)^2 + 3 opposite of y = -2(-x + 1)^2 - 3?????????

To determine whether a function is odd or even, we need to compare the function value f(x) with its negative counterpart -f(x) and see if they are equal.

Let's start by analyzing the original function y = 2(x - 1)^2 + 3:

1. Substituting -x for x, we have:
f(-x) = 2(-x - 1)^2 + 3

2. Now, we compare f(-x) with f(x):
f(-x) = 2(-x - 1)^2 + 3
f(x) = 2(x - 1)^2 + 3

As we can see, f(-x) is not equal to f(x). Therefore, the original function y = 2(x - 1)^2 + 3 is not an even function. It does not possess symmetry about the y-axis.

Now, let's analyze the negation of the function -f(x) = -[2(x + 1)^2 + 3]:

1. Comparing -f(x) with f(x):
-f(x) = -[2(x + 1)^2 + 3]
f(x) = 2(x - 1)^2 + 3

2. Again, we see that -f(x) is not equal to f(x). Therefore, the negation of the function -f(x) = -[2(x + 1)^2 + 3] is also not odd.

To understand why the original function y = 2(x - 1)^2 + 3 is not the opposite of y = -2(-x + 1)^2 - 3, let's simplify both functions and compare them:

Original function:
y = 2(x - 1)^2 + 3
y = 2(x^2 - 2x + 1) + 3
y = 2x^2 - 4x + 2 + 3
y = 2x^2 - 4x + 5

Opposite function:
y = -2(-x + 1)^2 - 3
y = -2((-x)^2 - 2(-x) + 1) - 3
y = -2(x^2 + 2x + 1) - 3
y = -2x^2 - 4x - 2 - 3
y = -2x^2 - 4x - 5

As you can see, the two functions are not the exact opposite of each other. The coefficient of x^2 and the constant terms have different signs. Therefore, these functions are not opposites.

In summary, the original function y = 2(x - 1)^2 + 3 is not even because f(-x) does not equal f(x). Additionally, its negation is not odd because -f(x) does not equal f(x). The original function and its negation are not opposites because their coefficients and constant terms have different signs.