Question 189532This question is from textbook Finite Mathematics and Calculus With Applications
: What is the difference between y=-f(x) and y=f(-x)?
The text gives an explanation on page 580, but I don't know how it would look with an actual equation. I can certainly read that y=-f(x) reflects y=f(x) vertically across the x-axis, and y=f(-x) reflects horizontally across the y-axis.
What I'm having difficulty understanding is, for a quadratic equation ax^2+bx+c, does -f(x) make it -ax^2+bx+c, or does it make it -ax^2-bx-c?
What about f(-x)?
This question is from textbook Finite Mathematics and Calculus With Applications
Answer by feliz1965(151)   (Show Source):
You can put this solution on YOUR website! First of all, it is important for you to know that y and f(x) mean exactly the same thing. They are interchangeable. So, y = f(x) and f(x) = y.
Here is an example:
y = 2x + 3 can also be written f(x) = 2x + 3. They BOTH mean the same thing.
What is the difference between y = -f(x) and y = f(-x)?
y = -f(x) represents an EVEN FUNCTION.
y = f(-x) represents an ODD FUNCTION.
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I can certainly read that y=-f(x) reflects y=f(x) vertically across the x-axis, and y=f(-x) reflects horizontally across the y-axis.
To reflect means that the given function creates a mirror-like image of itself about x-axis or y-axis.
You were given the function f(x) = -ax^2 + bx + c.
Simply replace every x that you see in the function with -x. If the result is an even function, then the graph of the function is symmetric with respect to the y-axis. If the function is odd, it will be symmetric with respect to the origin.
In other words, if AFTER replacing every x with -x you get the same ORIGINAL function, the function is even. If you get a function that is the opposite of the original function given, then it will be an odd function. If you get a function that is TOTALLY DIFFERENT (every term is different) from the original function, then the function will be NEITHER odd or even.
Let x become -x.
f(-x) = -a(-x)^2 + b(-x) + c
f(-x) = -ax^2 - bx + c
Do you see a difference in the answer? Notice that the second term changed from bx to -bx. This tells me that it is not the ORIGINAL FUNCTION as given to you.
Since it is NOT the same as the orginal function given, then we know it is NOT even and so, not symmetric with respect to the y-axis.
We now apply -f(x) to the answer.
-f(x) = -(our answer)
-f(x) = -(-ax^2 - bx + c)
-f(x) = ax^2 + bx - c
We just learned that the given function is not odd either.
This means that it is NEITHER odd or even.
Is this clear?
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