order of transformations - graphing techniques (1 Viewer)

[iloveeggs]

this for 2u graphing techniques. i feel kind of dumb for asking this but i suddenly realised that whenever i get the following type of question i get them wrong and its probably because i dont understand order of transformations. i dont know why either bc i when i did this topic in term 1 y12 i did really well on it so im not sure why i dont get it anymore

given (transformed function), find what transformations that (parent function) has gone through

the funny part is that i can do this type of question in reverse, so if you gave me the parent function and the transformations in the exact order they are done in, i can usually find the transformed function. i also understand that the order matters when you have multiple transformations but i don't really understand what order i should go in.

is there any specific logic/reasoning that helps you understand this concept better? i have tried asking people in my y12 cohort and they say they kind of just know or that they wing these questions and dont really get it either. and when my teacher explains it i just dont get it

 

[cossine]

this for 2u graphing techniques. i feel kind of dumb for asking this but i suddenly realised that whenever i get the following type of question i get them wrong and its probably because i dont understand order of transformations. i dont know why either bc i when i did this topic in term 1 y12 i did really well on it so im not sure why i dont get it anymore

given (transformed function), find what transformations that (parent function) has gone through

the funny part is that i can do this type of question in reverse, so if you gave me the parent function and the transformations in the exact order they are done in, i can usually find the transformed function. i also understand that the order matters when you have multiple transformations but i don't really understand what order i should go in.

is there any specific logic/reasoning that helps you understand this concept better? i have tried asking people in my y12 cohort and they say they kind of just know or that they wing these questions and dont really get it either. and when my teacher explains it i just dont get it

If you add couple of example questions and your attempt that will be useful

 

[iloveeggs]

sure give me a minute

 

[cossine]

this for 2u graphing techniques. i feel kind of dumb for asking this but i suddenly realised that whenever i get the following type of question i get them wrong and its probably because i dont understand order of transformations. i dont know why either bc i when i did this topic in term 1 y12 i did really well on it so im not sure why i dont get it anymore

given (transformed function), find what transformations that (parent function) has gone through

the funny part is that i can do this type of question in reverse, so if you gave me the parent function and the transformations in the exact order they are done in, i can usually find the transformed function. i also understand that the order matters when you have multiple transformations but i don't really understand what order i should go in.

is there any specific logic/reasoning that helps you understand this concept better? i have tried asking people in my y12 cohort and they say they kind of just know or that they wing these questions and dont really get it either. and when my teacher explains it i just dont get it

I pasted an example below although it is more related trig sketching. See if you can understand it.

Question: How to sketch Cos(2x-pi/3)

There two transformations going on, the horizontal stretch and the phase shift (i.e. horizontal shift).

To stretch a function horizontally by factor of n the transformation is just f(x/n).

So let f(x) = cos(x)

=> f(x/(1/2)) = cos(x /(1/2) ) = cos(2x)

So the horizontal stretch is by factor of 1/2.

Since the horizontal stretch is affecting the phase shift pi/3 the actual phase shift is pi/6 to the right as the horizontal stretch is 1/2.

cos(2x-pi/3) = cos(2(x-pi/6))

Let say you now want to sketch cos(-2x+pi/3). Remember that cos theta is even function. A function is even if f(-x) = f(x). Substitute u = 2x-pi/3 =>
cos(-2x+pi/3) = cos(-u) = cos(u) = cos(2x-pi/3)

Similarly you can sketch sin(2x-pi/3) the same way just beware sin is an odd function not an even function. Note a function is odd if f(-x) = -f(x).

If you wanted to sketch an equation tan theta remember that period of tan theta is pi and tan is an odd function.

Basic notes
y= asin(bx + c) + d

Midline:
Vertical shift is just d. This result in the midline y=d.

Amplitude:
The amplitude is the maximum distance from the midline. i.e. amplitude = |a|. The || (absolute value function) makes everything that is negative become positive. As 'a' can be negative we take the absolute value to make sure it is positive. e.g. |-6| = 6, |-pi| = pi, |-2.78| = 2.78. Note that anything that is positive remains positive i.e. |4| = 4, |6| = 6, |pi| = pi. The reason for the absolute value is we are interested in the distance from the midline which is non-negative.

Period:
Note when equation is in the form asin(bx+c) + d the period is just 2pi/|b|. While the period was originally 2pi, the period has been altered by the horizontal stretch. The formula 2pi/|b| for the period might be easier to use than thinking of horizontal stretches.

 

[iloveeggs]

wtf im trying to do a question wrong and i keep getting the right answer

 

[iloveeggs]

but i dont understand why its th right answer and im just doing smth random

 

[iloveeggs]

this is smth we used a while back and here are the solutions. i got the right answers for all of these questions (identical to the answers pretty much) but i literally dont understand how i did it. if you ask me to explain the order of transformations i cant. this is so frustrating

 

[iloveeggs]

I pasted an example below although it is more related trig sketching. See if you can understand it.

Question: How to sketch Cos(2x-pi/3)

There two transformations going on, the horizontal stretch and the phase shift (i.e. horizontal shift).

To stretch a function horizontally by factor of n the transformation is just f(x/n).

So let f(x) = cos(x)

=> f(x/(1/2)) = cos(x /(1/2) ) = cos(2x)

So the horizontal stretch is by factor of 1/2.

Since the horizontal stretch is affecting the phase shift pi/3 the actual phase shift is pi/6 to the right as the horizontal stretch is 1/2.

cos(2x-pi/3) = cos(2(x-pi/6))

Let say you now want to sketch cos(-2x+pi/3). Remember that cos theta is even function. A function is even if f(-x) = f(x). Substitute u = 2x-pi/3 =>
cos(-2x+pi/3) = cos(-u) = cos(u) = cos(2x-pi/3)

Similarly you can sketch sin(2x-pi/3) the same way just beware sin is an odd function not an even function. Note a function is odd if f(-x) = -f(x).

If you wanted to sketch an equation tan theta remember that period of tan theta is pi and tan is an odd function.

Basic notes
y= asin(bx + c) + d

Midline:
Vertical shift is just d. This result in the midline y=d.

Amplitude:
The amplitude is the maximum distance from the midline. i.e. amplitude = |a|. The || (absolute value function) makes everything that is negative become positive. As 'a' can be negative we take the absolute value to make sure it is positive. e.g. |-6| = 6, |-pi| = pi, |-2.78| = 2.78. Note that anything that is positive remains positive i.e. |4| = 4, |6| = 6, |pi| = pi. The reason for the absolute value is we are interested in the distance from the midline which is non-negative.

Period:
Note when equation is in the form asin(bx+c) + d the period is just 2pi/|b|. While the period was originally 2pi, the period has been altered by the horizontal stretch. The formula 2pi/|b| for the period might be easier to use than thinking of horizontal stretches.

i know how to do this actually im pretty okay with trig graphing and transformations

 

[iloveeggs]

WAIT I GET IT OGM I GET IT

SORRY IDEK WHAT THAT WAS ALL ABOUT LMAOO IK HOW TO DO THESE QUESTIONS

 

[cossine]

this is smth we used a while back and here are the solutions. i got the right answers for all of these questions (identical to the answers pretty much) but i literally dont understand how i did it. if you ask me to explain the order of transformations i cant. this is so frustrating

So if you want horizontally stretch a function let say f(x) by a factor of n. The transformation is f(x/n)

A vertical stretch is done by replace y with y/n. So if you want vertically stretch something by factor n then let y/n = f(x). => y=nf(x)

Likewise performing horizontal shift works as follow. f(x-k) shift the function k unit to the right where k is positive. f(x+k) shifts the function k units to the left where k again is positive.

vertical shift are similar. f(x) + k shift the function k units up down depending on if k is positive or negative.

An important note that a horizontal stretch can impact horizontal shift. You will see this in trig example where we discuss the phase shift (note the phase shift is the horizontal shift)

Note there is no precise answers to the questions.

So to give an example from the work sheet. 1/5*(e^(-4x)+2) =1/5*e^(-4x)+2/5. So if we want to go from e^(x) to the other function we first perform horizontal stretch by 1/4 and flip the graph on the x-axis notice the sign of -4. We then perform a vertical stretch by 1/5. This give us 1/5*e^(-4x). We then perform a vertical shift by 2/5.

So this is an alternative answer.

 

[iloveeggs]

So if you want horizontally stretch a function let say f(x) by a factor of n. The transformation is f(x/n)

A vertical stretch is done by replace y with y/n. So if you want vertically stretch something by factor n then let y/n = f(x). => y=nf(x)

Likewise performing horizontal shift works as follow. f(x-k) shift the function k unit to the right where k is positive. f(x+k) shifts the function k units to the left where k again is positive.

vertical shift are similar. f(x) + k shift the function k units up down depending on if k is positive or negative.

An important note that a horizontal stretch can impact horizontal shift. You will see this in trig example where we discuss the phase shift (note the phase shift is the horizontal shift)

Note there is no precise answers to the questions.

So to give an example from the work sheet. 1/5*(e^(-4x)+2) =1/5*e^(-4x)+2/5. So if we want to go from e^(x) to the other function we first perform horizontal stretch by 1/4 and flip the graph on the x-axis notice the sign of -4. We then perform a vertical stretch by 1/5. This give us 1/5*e^(-4x). We then perform a vertical shift by 2/5.

So this is an alternative answer.

yk how for y=(3x-7)^3 its just horizontal translation before dilation. i think i was under the assumption that you ALWAYS had to do dilations first or smth stupid like that. i can still get an answer that way but when it ccomes to questions where there were vertical and horizontal dilations and translations all at the same time, i kept trying to do both of the dilations first and then do both of the translations which doesnt work

 

[iloveeggs]

this is why you have to revise content as you go so you dont get stupid ideas like this on a test