Stretching/Compressing factor of Secant Function

[xxMsJojoxx]

What does 'stretching/compressing factor' mean?
For example, I have this.
I understand:
- the stretching factor is 4, period is --- > This means the pattern takes place over 6.

- the phase shift is . ----> This means it is moving 1.5 over to the right, form the basic secant function.

But what does 'stretching/compressing factor' mean, in comprison to the basic secant function y= sec x?

 

[Dr.Peterson]

The 4 stretches the graph vertically, so that where the basic graph goes up 1 and down 1 at its peak and valley, the new function goes up 4 and down 4 from its centerline at y=1 (because of the +1). The y value is multiplied by 4, then increased by 1.

 

[xxMsJojoxx]

The 4 stretches the graph vertically, so that where the basic graph goes up 1 and down 1 at its peak and valley, the new function goes up 4 and down 4 from its centerline at y=1 (because of the +1). The y value is multiplied by 4, then increased by 1.

Thank you, Dr. Peterson.

Is there no vertical shift for a secant function then? B/c normally the 'D' variable represents the vertical shifts in other trigonometric functions.

And a vertical shift is how the graph would shift up and down on the plane.



I guess I'm just a bit confused about that. Thank you for your help!

 

[Deleted member 4993]

Thank you, Dr. Peterson.

Is there no vertical shift for a secant function then? B/c normally


in other trigonometric functions.

And a vertical shift is how the graph would shift up and down on the plane.
View attachment 22770


I guess I'm just a bit confused about that. Thank you for your help!

The given equation is:

...................................(1)​

And your equation is:

$\displaystyle y = 4 * sec\left(\frac{\pi}{3}x - \frac{\pi}{2}\right) + 1$..............................................(2)​

And you said:

normally the 'D' variable represents the vertical shifts

Do you see a "1" (in 2) where in (eqn. 1) "D" was?

So then what is the vertical shift?

 

[xxMsJojoxx]

The given equation is:

View attachment 22771 ...................................(1)​

And your equation is:

View attachment 22764 ..............................................(2)​

And you said:

​

Do you see a "1" (in 2) where in (eqn. 1) "D" was?

So then what is the vertical shift?

Ohh yes -- I understand now. The graph moves up and down based on the vertical shift, but we still have to multiply by the stretching factor, to determine how much the graph actually moves up and down.

 

[Dr.Peterson]

Ohh yes -- I understand now. The graph moves up and down based on the vertical shift, but we still have to multiply by the stretching factor, to determine how much the graph actually moves up and down.

I wouldn't quite put it that way, though you probably mean the right thing.

The shift (up 1) tells how far the midline (the original x-axis) moves; the stretch factor determines how far particular points on the graph are from the midline, relatively.

In this case, the low points of the original are at y=1; the shift moves the midline from y=0 up to y=1, and the stretch by 4 means the low points are 4 above that, rather than only 2, making a total of 1+4 = 5. The high points of the original are at y=-1; on the new graph they are at 1 - 4 = -3.

I think of it as stretching first, so those points are at 4 and -4, and then shifting up, moving them to 5 and -3. That's why I am not happy with saying we first shift and then stretch, which would produce different results.