I've been introduced to the idea that the curve on f(x) can be transformed by

[George Gifkins]

simple translations...? And am then told that f(x+a) is a horizontal translation of -a...

But this concept appears to have come out of nowhere... Can someone explain, with regard to f(x)... What the x and the a are?... And what form i would generally see equations in, to Which this rule applies!

Any help will be gratefully received!

 

[joe]

Don't listen to Some1's answer - he's got it wrong.


Try a specific example:
f(x) = x^2 i.e. "x squared"

This function defines a parabola with its vertex at (0,0) and its axis of symmetry lying on the positive y-axis.

Now, change the definition to:
f(x+2) = x^2

The function is still a parabola, but it has shifted to the left because of the +2 in the argument (an argument is just another word for the "input" to a function). The vertex is located at the minimum value of the function, which is where the x^2 is 0, so (x+2) must evaluate to 0 also. In other words, the vertex is now found at the position where x=-2, i.e. at (-2,0) because when x = -2, the argument of the function, (x+2), evaluates to (-2+2), which is 0.

The same goes for every other point on the function - whatever used to be at x is now at x-2 because the whole function has shifted over to the left by 2. Increasing the argument shifts the function to the left; decreasing the argument shifts it to the right.

If we define a constant, a, to represent the amount added to or subtracted from the argument, we get:
f(x+a) = x^2

Positive values of "a" shift the function to the left (more negative) and negative values of "a" shift the function to the right (more positive.)


By the way, if you want to shift a function up or down (as Some1 mentions), you add the constant to the other side of the definition. For example, to move f(x) = x^2 up or down you write:
f(x) = x^2 + b

Positive b moves it up, negative b moves it down.

By using both a and b you can move the function anywhere on the Cartesian plane:
f(x+a) = x^2 + b

Finally, observe that you can move b to the other side:
f(x+a) = x^2 + b => f(x+a) - b = x^2
This gives a another sense of what's going on: a affects x, and b affects f, which is really just y.

 

[Milagros]

just find the rules

 

[Some1]

The general idea is to start with a curve defined by f(x) and then either stretch/compress it or move it to a new position. x is simply there to represent the independent variable of the function. And by now you should probably know how a function is affected by changing x. Regarding "a", it is simply affecting the range of the function by, in a way, altering the domain of the function before it is even processed by the function. All you really need to learn is that, in general, if f(x+a) is positive, and "a" is positive, then the graph of the function will be shifted/moved a units upwards in the Cartesian plane. But if "a" is negative then the converse happens: the graph of f will be shifted |a| units downwards in the Cartesian plane. I am affraid I might have made this sound harder than it actually is. The actual idea behind f(x + a) is simply to illustrate how a curve can shift up and down depending on what "a" is.