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### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 4

Lesson 14: Topic C: Lessons 20-22: Scaling and transforming graphs

# 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
• I like how everyone is asking about certain math questions and the typical "where in real life would this be useful" kinda thing, and yet I seem to be the only one who's wondering about that fifth graph down at the bottom. What's that doing down there? Why did Sal not do any problems on that one but still did problems on the other four? These are legitimate questions.
• Maybe he had been planning to use it, but then he ran over time or something. Or it could be for another video.