What Are the Effects of Different Transformations on the Function y=e^x?

[Samurai44]

Homework Statement

Describe the transformation in :
y=e^6x-2 - 4
y=6e^x-2 - 4 ?

Homework Equations

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The Attempt at a Solution

1)So 6x means stretch parallel to x-axis at s.f of 1/6
and then translation of ( +2 , -4 )
2) the same but the stretch is parallel to y-axis at s.f of 6

but which transformation comes first , and what's the difference if I switched them (in both cases)? Thank you in advance

 

[mfb]

Did you plot the functions?

and then translation of ( +2 , -4 )

The -4 is fine, the +2 is not right.
You can make the description a bit easier if you use a rule for rewriting e^a+b.

2) the same but the stretch is parallel to y-axis at s.f of 6

Right.

 

[Svein]

 

[Samurai44]

Did you plot the functions?
The -4 is fine, the +2 is not right.
You can make the description a bit easier if you use a rule for rewriting e^a+b.

Right.

how +2 isn't correct ? ,,,its like f(x-2) which means translation of +2 in x direction ( to right )

 

[Samurai44]

how do i know if the stretch is carried first or the translation ?
and what's the difference if it was the opposite

 

[mfb]

how +2 isn't correct ? ,,,its like f(x-2) which means translation of +2 in x direction ( to right )

It is not. Subtracting 2 from x would mean $$e^{6(x-2)}-4$$That is not the same as your function.

Did you plot $e^{6x}-4$ and $e^{6x-2}-4$?

 

[SammyS]

Homework Statement

Describe the transformation in :
y=e^6x-2 - 4
y=6e^x-2 - 4 ?

Homework Equations

---

The Attempt at a Solution

1)So 6x means stretch parallel to x-axis at s.f of 1/6
and then translation of ( +2 , -4 )
2) the same but the stretch is parallel to y-axis at s.f of 6

but which transformation comes first , and what's the difference if I switched them (in both cases)?Thank you in advance

There is no single "correct" sequence of transformations to get either of these. Changing the order of the transformations will often require the transformations themselves to be altered, but not always. There often is a "natural" order in which the tranfromations are performed -- this order suggested by the appearance of the function as it's presented to you.

Start with the "parent" function. In both of your examples that is y = e^x.

For example 1) y=e^6x-2 - 4

Start with y = e^x.

Yes. 6x means scaling parallel to the x-axis with a scale factor of 1/6, i.e. a shrink rather than a stretch.. That done first gives y = e^6x.

Next you said translation of ( +2 , -4 ). I assume that refers to x, y respectively.

That's correct for the y, but for the x, that doesn't work.
It literally gives you y = e^6(x-2) - 4.​

Try the translation first, then do the stretch/shrink. Otherwise, is there a different amount ti translate in the x direction?​

 

[Samurai44]

Did you plot $e^{6x}-4$ and $e^{6x-2}-4$?

Try the translation first, then do the stretch/shrink. Otherwise, is there a different amount ti translate in the x direction?​

The problem is these types of questions comes in multiple choice so i won't have enough time sketching or plotting.

I couldn't get the difference between e^6(x-2) -4 and e^6x-2 -4

 

[mfb]

6(x-2) = 6x-12 which is different from 6x-2.

The problem is these types of questions comes in multiple choice so i won't have enough time sketching or plotting.

You have enough time here and it helps to understand what went wrong, so you can get it right in the exam.

 

[Samurai44]

6(x-2) = 6x-12 which is different from 6x-2.
You have enough time here and it helps to understand what went wrong, so you can get it right in the exam.

so in case of 6(x-2) , the translation is +12 , but in 6x-2 , translation is +1/3 ?

 

[Mark44]

so in case of 6(x-2) , the translation is +12 , but in 6x-2 , translation is +1/3 ?

For 6(x - 2), the translation is 2 to the right. For 6(x + 1/3), the translation is 1/3 to the left.

Compare the graphs of $y = e^x$ and $y = e^{6(x - 2)}$. The 6 in the 2nd version causes a compression of the graph of y = e^x toward the y-axis by a factor of 6. If the multiplier happens to be smaller than 1, the transformation is an expansion away from the y-axis.
The x - 2 causes a translation of the compressed graph 2 units to the right.

You can see this by following a point on the graph of y = e^x through both of these transformations

y = e^x
Point (1, e) (or pick any point you like)

y = e^6x
Point (1/6, e) -- Note that this point is 1/6 as far from the y-axis as (1, e) is. IOW, the point (1, e) has been "compressed" toward the y-axis.

y = e^{6(x - 2)}
Point (13/6, e) -- Shift the point in the previous transformation two units right. 1/6 + 2 = 13/6.

If there are compressions/expansions and translations (shifts), you have to do the compressions/expansions before you do the translations. If you don't do them in this order, you don't get the right graph.