graph of f(x)

Question

So I have been really struggling trying to figure this out. I guess I am just really trying to find what the graph of f(x) looks like. Because I have to graph these equations:
y=f(x-2)
y=f(x)-2
y=-2f(x)
y=f(-2x)
I know for the first one it would be f(x) shifted 2 to the right and the second one would be f(x) shifted down 2. But I can't figure out what y=f(x) graphs out to be. I tried to look up pictures of it online and some show just a linear line or a quadratic U shape. Others have more curves so I need help graphing these :/

Answer 1

Kari,

 

This is a tricky one since the function f(x) is not given so you can't graph it.  And your answer has to be true whether the unknown function is a simple line, a polynomial in higher powers of x, contains a transcendental function like sin(x), etc.

 

But what you do know is at x=0,  f(x) = f(0) whatever that would evaluate to in the unknown function.  Therefore, you can state what f(0) maps to:  how a single point graphs, not a whole function. So there can be no shifting of the function. But this single-point mapping must be true for any function that someone could specify..

Thus,

y=f(x-2):  f(0) gets mapped to f(-2).

y=f(x)-2:  f(0) gets mapped to f(0) -2.

y=-2f(x):  f(0) gets mapped to -2f(0).

y=f(-2x);  f(0) gets mapped to f(0) and doesn't move.  Other points on the function's graph will move in general.

 

Try the numbers for 2 functions below and graph them to see how it works:

f(x) = x³ + 10   or f(x) = 1/[x² + 4]

 

Answer 2

y=f(x) is just a general statement about a function. There is no way to picture or graph the function. You do not know what it looks like. It is only saying that f(x) is a function. Functions are a subset of all equations and are restricted to certain rules. One rule is the vertical line test. You don't even know the domain or range.

 

y=-2f(x) says the new function is symmetrical about the x axis and stretched vertically by two.

y=f(-2x) depends on if it is an odd or even function, or neither.

if f(x) is an odd function the graph would be symmetrical about the origin and stretched horizontally by two.

if f(x) is an even function the graph would be symmetrical about the y axis and stretched horizontally by two.

if f(x) is neither then f(-2x) may not even exist.