Horizontal shifts and translations

[jpanknin]

Can anyone explain why horizontal shifts for functions when a term is added to the argument go the opposite direction of the sign of the additional term? For example:

[MATH]y = sin(x + pi/2)[/MATH]
is shifted [MATH]pi/2[/MATH] to the left instead of to the right (see figure below from Mckeague/Turner Trigonometry 8E).



Why doesn't the shift occur to the right instead? If we have a term added outside the argument, such as [MATH]y = sin (x) + 3[/MATH] then the vertical shift is in the same direction of the sign of the additional term. If we assume that the [MATH]x[/MATH] in the argument starts at the typical place, 0, then why doesn't the [MATH]pi/2[/MATH] get added to the argument of the function, like:



instead of starting the period at [MATH]-pi/2[/MATH] by adding [MATH]-pi/2[/MATH]? Is it because the argument needs to start as a 0?

 

[Dr.Peterson]

Can anyone explain why horizontal shifts for functions when a term is added to the argument go the opposite direction of the sign of the additional term? For example:

[MATH]y = sin(x + pi/2)[/MATH]
is shifted [MATH]pi/2[/MATH] to the left instead of to the right (see figure below from Mckeague/Turner Trigonometry 8E).

View attachment 23865

Why doesn't the shift occur to the right instead? If we have a term added outside the argument, such as [MATH]y = sin (x) + 3[/MATH] then the vertical shift is in the same direction of the sign of the additional term. If we assume that the [MATH]x[/MATH] in the argument starts at the typical place, 0, then why doesn't the [MATH]pi/2[/MATH] get added to the argument of the function, like:

View attachment 23869

instead of starting the period at [MATH]-pi/2[/MATH] by adding [MATH]-pi/2[/MATH]? Is it because the argument needs to start as a 0?

When you add something on the outside of the function, you are adding it to the y coordinate of a point on the old graph, so it moves up.

When you add something on the inside, you are adding it to an x-coordinate on the new graph, before it is used in the function; in order to get the corresponding point on the old graph, you have to subtract from the x-coordinate you want to use.

Here is another way to see it. Suppose you have the graph of [MATH]y = f(x)[/MATH], and you replace [MATH]x[/MATH] in the equation with [MATH]x-a[/MATH], and replace [MATH]y[/MATH] in the equation with [MATH]y-b[/MATH]: [MATH]y - b = f(x - a)[/MATH]. If a point [MATH](x_1, y_1)[/MATH] is on the original graph, so that [MATH]y_1 = f(x_1)[/MATH], then the point [MATH](x_1+a, y_1+b)[/MATH] will be on the new graph, because [MATH](y_1+b)-b=f((x_1+a)-a)[/MATH]. So a shift up and to the right requires subtracting from both [MATH]x[/MATH] and [MATH]y[/MATH] in the equation.

And that equation is equivalent to [MATH]y = f(x - a)+b[/MATH], where we have replaced [MATH]x[/MATH] with [MATH]x-a[/MATH], but added [MATH]b[/MATH] to the function.

 

[Steven G]

Lets look a simpler functions like y=x^3 and y=(x-2)^3

Suppose y = 1, 8 and 27

In the function y = x^3 what should x be so when you cube it you get 1, 8, 27? The answers are 1, 2 and 3. That is the three points are (1,1), (2, 8) and (3,27).

In the function y=(x-2)^3 to get 1, 2 and 3 for y, you need to let x =3, 4 and 5. Do you see this part? If you want to cube 7 for example you need to let x be 2 more than 7, that is 9. Why? Because 9-2 = 7

Plot these points and see that the curve y = (x-2)^3 is to the right of y = x^2

 

[Otis]

Can anyone explain …

Hi jpanknin. I can give a simile, using time. Consider the x-axis as a timeline where x is the current elapsed time, measured from some starting point. Therefore, function sin(x) gives the current sine value. You and I can't travel into the future, but we can bring the future to us! Let's say at time x, when everybody is looking at the current value of sine, you and I actually want to see what sine will be doing Pi/2 units from now. That is, we want to see the future value sin(x+Pi/2) instead of the current value sin(x). Viewing sin(x+Pi/2) at time x allows you and I to get sine values Pi/2 units of time before everyone else does. Anyone using x+Pi/2 is seeing what sine will be doing Pi/2 units from now, so using x+Pi/2 instead of x is like reaching Pi/2 units into the future and pulling that future sine behavior backwards in time to our present. On timelines that point to the right, moving anything backwards in time means shifting it to the left.

?

 

[jpanknin]

Thanks, @Jomo and @Otis. Those help very much. Appreciate it.

 

[Otis]

We're glad to help. I could have mentioned it, but you likely got it: subtracting from x causes "past" behavior (at x-Pi/2) to be "pulled into the present (at x). Hence, the graph of sin(x - Pi/2) is sine shifted Pi/2 units to the right.

By the way, when I'd asked about this in college, the instructor had a somewhat different take. Adding Pi/2 to all x is like shifting the entire coordinate system Pi/2 units to the right, without shifting the graph with it. He'd used a transparency of the sine wave positioned correctly on top of a sheet of paper with the xy-axes. Then, he slid the paper to the right Pi/2 units, while holding the transparency in place. Shifting the coordinate system to the right without shifting the sine graph as well has the effect of shifting the sine wave to the left (with respect to the coordinate system). Just another perspective.

?

 

[apple2357]

We're glad to help. I could have mentioned it, but you likely got it: subtracting from x causes "past" behavior (at x-Pi/2) to be "pulled into the present (at x). Hence, the graph of sin(x - Pi/2) is sine shifted Pi/2 units to the right.

By the way, when I'd asked about this in college, the instructor had a somewhat different take. Adding Pi/2 to all x is like shifting the entire coordinate system Pi/2 units to the right, without shifting the graph with it. He'd used a transparency of the sine wave positioned correctly on top of a sheet of paper with the xy-axes. Then, he slid the paper to the right Pi/2 units, while holding the transparency in place. Shifting the coordinate system to the right without shifting the sine graph as well has the effect of shifting the sine wave to the left (with respect to the coordinate system). Just another perspective.

?

And i guess when you do y= f(x) + pi/2 which can be written as y-pi/2 = f(x) , you are shifting the coordinate system down. so this has the effect of the graph moving up.

This problem arises from trying to apply the same way of thinking about two different issues. When doing y=f(x)+ k, it is easy to see how the y coordinates increases by k, however applying the same reasoning to f(x+k) is not so straightforward, yet we expect it to be. I think this is one of the harder concepts to explain easily without going back to thinking about points, which is what others have done above.

Incidently the transparency is a neat way of illustrating what is happening. I wonder if there is a similar effect you can do with a graph plotter? I would love to see it.