Describe the function resulting from reflecting f(x)=x^3+2x^2-x+5 about the y-axis and then shifting to the left by 4. Is this the same as performing the operations in the reverse order
reflect: (x,y)->(-x,y)
f(x) -> -x^3+2x^2+x+5
shift: (x,y)->(x+4,y)
f(x) -> -(x+4)^3+2(x+4)^2+(x+4)+5
= -x^3-10x^2-31x-23
see the graphs here:
http://www.wolframalpha.com/input/?i=plot+y%3Dx^3%2B2x^2-x%2B5%2C+y%3D-x^3-10x^2-31x-23
the shift isn't real clear, but the reflection is.
To reflect the function f(x) = x^3 + 2x^2 - x + 5 about the y-axis, we replace x with -x. So the reflected function, let's call it g(x), can be written as g(x) = (-x)^3 + 2(-x)^2 - (-x) + 5.
Simplifying this, we get g(x) = -x^3 + 2x^2 + x + 5.
Now, to shift the function g(x) to the left by 4 units, we replace x with (x+4) in the equation. The resulting function, let's call it h(x), can be written as h(x) = -(x+4)^3 + 2(x+4)^2 + (x+4) + 5.
Expanding and simplifying this function, we get h(x) = -(x^3 + 12x^2 + 48x + 64) + 2(x^2 + 8x + 16) + x + 4 + 5.
Simplifying further, we get h(x) = -x^3 - 12x^2 - 48x - 64 + 2x^2 + 16x + 32 + x + 4 + 5.
Combining like terms, we have h(x) = -x^3 - 10x^2 - 31x - 23.
Now, to answer the second part of your question, if we perform the operations in reverse order, i.e., first shifting the function f(x) to the left by 4 units and then reflecting it about the y-axis, we would get a different result.
If we shift f(x) to the left by 4 units, we replace x with (x+4) in the equation. So, f(x) becomes f(x) = (x+4)^3 + 2(x+4)^2 - (x+4) + 5.
Expanding this equation, we have f(x) = x^3 + 12x^2 + 48x + 64 + 2(x^2 + 8x + 16) - x - 4 + 5.
Simplifying further, we get f(x) = x^3 + 12x^2 + 48x + 64 + 2x^2 + 16x + 32 - x - 4 + 5.
Combining like terms, we have f(x) = x^3 + 14x^2 + 63x + 97.
As you can see, the function resulting from performing the operations in reverse order is different from the function we obtained earlier. Thus, performing the operations in a different order (shifting first and then reflecting) gives us a different function.