# Transformations of e^x and lnx

• Mar 15th 2010, 10:34 PM
200001
Transformations of e^x and lnx
Hi
I am getting conflicting suggestions in class about the transformations of graphs and (i) the order to carry out transformations and (ii) the effect it has on the graph

example e^(3x+2) -1

I though its was first a stretch of 1/3 parallel to the x axis, then a translation of 2 units to the left and FINALLY the whole graph moved down one unit.

Understanding that e^(3x+2) is e^2e^3x suggests that this is two stretches?

Second part...the order in whcih the lnx transformations occur has foxed me too.
Am I stretching first and then applying translations?

Thanks
• Mar 16th 2010, 12:33 AM
hollywood
Let's take the easier equation y=3x+2 and see what happens. This is a line with slope 3 and y-intercept 2.

You said stretch by 1/3 parallel to x-axis (so x'=3x and y'=y). The y-intercept will not change, but the slope changes from 3 to 1, and the equation is y=x+2. (I'm being sloppy and dropping the primes as I go.) Now you translate 2 units to the left (so x'=x+2 and y'=y), and the equation is now y=x.

If you had translated by 2 units first, you would have y=3x+2 going to y=3(x-2)+2=3x-4, and then the stretch would give y=x-4, which is not what you want. However, if you recognize y=3x+2=3(x+(2/3)), and you translate by 2/3, then stretch, you'll get the right answer.

I hope this clarifies the order of stretching and translating. Post again if you are still having trouble.
• Mar 16th 2010, 04:33 AM
HallsofIvy
Quote:

Originally Posted by 200001
Hi
I am getting conflicting suggestions in class about the transformations of graphs and (i) the order to carry out transformations and (ii) the effect it has on the graph

example e^(3x+2) -1

I though its was first a stretch of 1/3 parallel to the x axis, then a translation of 2 units to the left and FINALLY the whole graph moved down one unit.

Understanding that e^(3x+2) is e^2e^3x suggests that this is two stretches?

For the logarithm graph, the two are equivalent.

Quote:

Second part...the order in whcih the lnx transformations occur has foxed me too.
Am I stretching first and then applying translations?

Thanks
If you were to calculate for a specific x, "3x+ 2" means "first multiply by 3, then add 2" so "first stretch by 3 then translate by 2" is correct. "First translate by 2 then stretch by 3" would be 3(x+ 2)= 3x+ 6.
• Mar 16th 2010, 04:42 PM
hollywood
I see what you are saying, HallsofIvy. You think the example he gave is the transformation, and I was interpreting it as the graph being transformed.

So for the transformation e^(3x+2)-1, the order in which the simple transformations are performed is the same as the order of operations: multiply by 3, add 2, exponentiate, then subtract 1.

So you can change this transformation to be e^2e^3x-1, and now the order of operations is multiply by 3, exponentiate, multiply by e^2, then subtract 1. They're different transformations in a different order, but they have the same effect - just like the two expressions are different but mean the same thing.

So, 200001, hopefully the two of us have helped you. Fell free to post again if you are still having trouble.