Graphing: Horizontal Translations

[L3]

First off, sorry if I posted this question in the wrong place, I'm not always sure where to ask them.

Okay, so on to the question.

I'm learning about vertical and horizontal translations in graphing, and I understand the vertical but I'm quite confused by the horizontal. So the site I'm using says,

Points on the graph of y=f(x ) are of the form (x,f( x)) .
Points on the graph of y=f(x +3) are of the form (x,f( x+3)) .

How can we locate these desired points (x,f( x+3)) ?
First, go to the point (x+3, f( x+3)) on the graph of y=f(x ) .
This point has the y-value that we want, but it has the wrong x-value.
Move this point 3 units to the left.
Thus, the y-value stays the same, but the x-value is decreased by 3 .
This gives the desired point (x,f( x+3)) .
Thus, the graph of y=f(x +3) is the same as the graph of y=f(x ) , shifted LEFT three units.
Thus, replacing x by x+3 moved the graph LEFT (not right, as might have been expected!)

I don't understand this at all. If we went to the point (x+3, f( x+3)) and then shifted left three units, we end up with the point (x, f( x+3)). But how then is the line moving to the left at all? Since x is still at x. Isn't it the y line that would be the one to move, depending on what the funtion is and what it will do to the 3?

I know I'm not understanding something since the site says,

Start with the equation y=f(x ) .
Replace every x by x+p to give the new equation y=f(x +p) .
This shifts the graph LEFT p units.
A point (a,b) on the graph of y=f(x ) moves to a point (a-p,b) on the graph of y=f(x +p) .

But I don't understand how they get that.

 

[royhaas]

A horizontal translation involves measuring the distance from one point to another along a number line, so \(\displaystyle x+p = x-(-p)\). Moving a function to the left is the same as moving the axis to the right, if that is easier to understand.

 

[mmm4444bot]

... I don't understand this at all. If we went to the point (x+3, f( x+3)) and then shifted left three units, we end up with the point (x, f( x+3)). But how then is the line moving to the left at all? ...

It moves because f(x + 3) is not the same value as f(x), in general.

Yes, following those instructions, you end up at the same value of x, but now the corresponding y value at that x has changed. It becomes the y value that used to be three units away. And the y value there changes to become the y value that is three units away farther still. All of the y values are shifting to new x positions three units to the left, so the entire graph shifts.

-

... I don't understand how they get [A point (a, b) on the graph of y = f(x) moves to a point (a - p, b) on the graph of y = f(x + p)]

Well, they should also state that p is a positive number.

If p is a positive number, then x + p is "looking ahead to the right" from x to see what the graph is doing p units away. When the value of y there gets "pulled back" to x, it moves to the left p units.

If you cannot picture this shift in your head -- in conjunction with the function notation and these examples that you posted -- then make up some simple examples using functions with which you're familiar and try various real numbers for p. Draw graphs. Use real values of x to experiment.

You could try the following.

f(x) = x

f(x) = x^2

p = 4

p = 1

Draw the graph of f. Then pick a value of x. Look at the y value at x + 4. Now imagine that this particular y value is at x, instead of x + 4.

x is four units to the left of x + 4.

Put a dot there. "There" is at coordinates (x, f(x + 4)).

Do the same thing with other values of x, nearby. Soon, you will see that these dots are forming the same graph of f(x), except that this new graph is shifted four units to the left of f(x).

Do enough examples using real numbers and graphs until it sinks in.

You also need to have a very good grasp of function notation when trying to understand the examples and steps that you posted from your resource.

Make sure that you understand the relative differences between the following four symbols -- each of which represents a number that is either an x- or y-coordinate.

x

x + 4

f(x)

f(x + 4)

Cheers,

~ Mark

 

[L3]

Oh thank-you very muchlies I understand it now