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### Course: Algebra 2>Unit 9

Lesson 5: Putting it all together

# Identifying function transformations

Sal walks through several examples of how to write g(x) implicitly in terms of f(x) when g(x) is a shift or a reflection of f(x). Created by Sal Khan.

## Want to join the conversation?

• What is f(x) = |x| - 3

The fact that x is in between the absolute value sign confuses me. I know -3 would mean that we're going to the left on the horizontal plane, is that technically it?
• f(x)=|x|-3. It's like f(x)=x-3 except the 3 is inside absolute value brackets. The only difference is that you will take the absolute value of the number you plug into x.
Remember that x just represents an unknown number.
To find f(x) (you can think of f(x) as being y), you need to plug a number into x.
f(x)=|x|-3
x=-2
Plug -2 into x
|-2|-3
The absolute value of any number is positive. Thus, -2 will become 2. Then subtract. 2-3=-1.
When x=-2 y=-1
(-2, -1)
• Are there more detailed videos that focus specifically on horizontal and vertical shifting and shrinking? Thanks
• I use this reference formula g(x)=a*f((1/b)x-h)+k
a is for vertical stretch/compression and reflecting across the x-axis.
b is for horizontal stretch/compression and reflecting across the y-axis. *It's 1/b because when a stretch or compression is in the brackets it uses the reciprocal aka one over that number.
h is the horizontal shift. *It's the opposite sign because it's in the brackets.
k is the vertical shift.
• At , Why is it f(x-2) instead of f(x+2)? If you do minus 2, the values will get more negative, (from -3 to -5) but if you do plus two, then you would get the values of g...
Do you normally do the opposite when going left to right?
• ayo did you figure it out? cause i am wondered too
• What would the transformation do if g(x)=(x+6)^2-10 and g(x) is in absolute value bars? Like this: |g(x)|.
• Taking the absolute value of a function reflects the negative parts over the x-axis, and leaves the positive parts unchanged. So a central segment of your parabola will be reflected so that it opens downward, with sharp corners at the roots.
• can some one help me?
What happens to the graph for f(x)=x when compared to the graph f(x)=x-5?
• f(x)=x is equal to f(x)=x+0, just written in a more abstract way. This is useful when comparing to another linear functions such as your example.

f(x)=x-5 is simply just f(x)=x brought down by 5 units, hence the "-5" for the b term. I recommend using desmos for a more visual interpretation, as writing down explanations can be more convoluted.

Hopefully that helps !
• Could anyone ennumerate all the ways a function can be transformed? Thank you!
• Well, a function can be transformed the same way any geometric figure can:
They could be shifted/translated, reflected, rotated, dilated, or compressed. So that's pretty much all you can do with a function, in terms of transformations. Hope that answered your question!
• f(x)=x,g(x)=x+1
would the transformation of the problem be translation
• Yep, for linear functions of the form mx+b m will stretch or shrink the function (Or rotate depending on how you look at it) and b translates. Then if m is negative you can look at it as being flipped over the x axis OR the y axis.

For all other functions, so powers, roots, logs, trig functions and everything else, here is what is hopefully an easy guide.

a*f(b(x+c))+d

so for example if f(x) is x^2 then the parts would be a(b(x+c))^2+d

a will stretch the graph by a factor of a vertically. so 5*f(x) would make a point (2,3) into (2,15) and (5,7) would become (5,35)

b will shrink the graph by a factor of 1/b horizontally, so for f(5x) a point (5,7) would become (1,3) and (10,11) would become (2,11)

c translates left if positive and right if negative so f(x-3) would make (4,6) into (7,6) and (6,9) into (9,9)

d translates up if positive and down if negative, so f(x)-8 would make the points (5,5) and (7,7) into (5,-3) and (7,-1)

Also should note -a flips the graph around the x axis and -b flips the graph around the y axis. Hope I didn't over explain, just proud of what I made tbh
• When could you use this in a real life situation?
• You wouldn't really use this kind of things in real life unless you are planning on to a career that involves math, which is just about everything.

Anyways, let's say f(x) is a parabola which models a rocket's path. You could translate the function, shift it to know how the rocket's path will differ.

That is just one example I came up with, of course there are many others.