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## Algebra (all content)

# Reflecting & compressing functions

Given the graphs of functions f and g, where g is the result of reflecting & compressing f by a factor of 3, Sal finds g(x) in terms of f(x).

## Want to join the conversation?

- Is f(-1/3x) not equal to -1/3f(x)?(16 votes)
- I'm pretty sure that when the -1/3 is placed within the brackets containing "x", it alters the "x" values. Alternatively, if it is like "-1/3f(x)" then the y-values are being changed. I'm not entirely sure what the difference would look like graphically, however, on a table, Khan noticed that the y-values were -1/3 of f(x), so he wrote -1/3f(x).

If you selected two x values and you came up with -1/3, then the answer would be f(-1/3x).

Hope this helped.(20 votes)

- At1:26, how do you just eyeball it

and know that is 1/3? doesn't make any sense for me.. :((12 votes)- When he draws it he is looking at how far below the x axis the line is, and then he finds the corresponding points above the x axis and sketches it as he goes. Eyeballing it just means he didn't carefully mark the points and then draw a super accurate representation.(4 votes)

- Even though I am able to identify shifts in the exercise below, 1) I still don't understand the difference between reflections over x and y axes in terms of how they are written. 2) I have constantly had trouble with the difference between horizontal and vertical compression of functions, their identification, and how their notation works. Could someone help me with this?(6 votes)
- For the first step, when he draws the opposite of the line, flips it, is this how you must solve every problem? Because I can't figure out how to draw the opposite of lines on graphs. Is this something you must practice at?(4 votes)
- For every value of the curve f(x), its reflection has the negative, -f(x). Try it, with, say, y = x, and y = x^2, y = sqrt(x) x>0, y = .5x+1, etc.(3 votes)

- How do you determine how the original equation is altered on the graph?(3 votes)
- How are we supposed to know how much we reflect, or if we should reflect?(2 votes)
- In my experience, knowing where/if to reflect is mainly something you have to build an intuition for. What I like to do is to experiment with transformations in my head until (mentally) I have transformed the original function into the new function.(2 votes)

- May seem off-topic, but why doe Mr. Khan repeat himself so much? Especially at0:37-0:45?(1 vote)
- It could be because he is restating what he said earlier while writing it, just to make sure if you missed what he said the first time, you hear it again. He might also record his videos without editing or in one take, and (from personal experience) if you are talking for a while, you tend to develop a habit of repeating phrases, especially while thinking about what to say next.(2 votes)

- What's the different between the functions f(kx) and kf(x) (k is a real number)? Is f(kx) always equal to kf(x) or they just equal in some case?(1 vote)
- I think I can guess that k is the coefficient in this case, correct? So with f(kx) it acts on and changes the input, or x values, whereas kf(x) acts on and changes the output, or y values. Hopefully this helps.(2 votes)

- May i ask for advise please!

if we have f(x) and we have g(x) in terms of f as g(x)=-0.5f(x) i recognize that it also equal to g(x)=f(-0.5x). i had experience this many times over geogebra (grapjing application) and every time i found both -0.5f(x) and f(-0.5x) are identical. and theoretically both are just a reflection over the x-axis due to the minus sign with some kind of horizontal stretching.

if this is true, i have a problem in the unit test where in some likewise problem i found both selections are available and i have to choose only one as true answer. thank you in advance(1 vote) - It is not loading for me. It is showing the purple monkey with a hammer.(1 vote)

## Video transcript

- [Voiceover] So we're told g of x is a transformation of f of x. The graph below shows f
of x as a solid blue line. So this is the graph of
y is equal to f of x, and the g of x is a dotted red line. So that's the graph of
y is equal to g of x. What is g of x in terms of f of x? And they gave us some choices here, and I encourage you to pause the video and see if you can figure this out. Well there's a couple of ways
that you can think about it. One is if you just eyeball it, it looks like if you flipped f of x over the x-axis, it looks
a little bit like g(x), but g(x) looks like a version of that that's diminished a little bit. So for example, if you were to flip it perfectly over the x-axis, you would get something-- you would get something that looks like-- and I'm just going to sketch it. So if you just flipped
perfectly over the x-axis, you would get something
that looks like this, and trying my best to eyeball it. It would look something like this. If you perfectly just flipped it over. It would look something like that. I might not get it perfectly right. But that would be-- this right over here. So let me be clear. If this is y is equal to
f of x, then this line right over here that I just drew, that would be y is equal
to negative f of x. Because whatever f of x would give you, just take the negative of it
you're flipping over the x-axis and now g of x looks like a
diminished version of that, and if I were to just eyeball it, it looks like it's roughly 1/3 of this. So my initial guess - and we can verify this little bit - is that this right over here is 1/3 the value of this. So I would say it is... So g of x is equal to
1/3 of negative f of x or negative 1/3 of f of x. And that is not one of the choices, which makes me extra cautious, but let me just emphasize
why I like this choice. Because we said this is
negative f of x right there, and it looks like for any x value, what g of x is is 1/3 of that. So instead of getting to four, we're getting to a little bit over one. Instead of getting to
three right over here, we're getting to one. Instead of getting to one right over here, we are only getting to 1/3. So it looks like it is 1/3 of this line that I just hand drew,
which is negative f of x. So it would be 1/3 of negative f of x, which would be negative 1/3 f of x. Once again, not a choice here, but let's actually look at some values. So if we said, "Let's
see some values where "it looks like we're
hitting integer values." So for example, right at this point, right over here, it looks
like f of negative seven is equal to negative one. It looks like g of negative seven... So it looks like g of negative seven is equal to positive 1/3, positive 1/3, if I am just eyeballing it. So that seems consistent with this, because if f of negative
seven is negative one, you take the negative 1/3 times that, and you get positive 1/3. We can do that in a
couple of other places. If we look right over... If we look right over here, it looks like f of negative one. f of negative one is
equal to negative three, and it looks like g of one
is negative 1/3 times that. Negative 1/3 times that is
that point right over there. Negative 1/3 times negative
three is positive one. So g of... g is colored in that red color. So we get for that point right over there, we get g of negative one
is equal to positive one. So once again, this is negative 1/3 times this right over here. So I feel pretty confident in my answer. g of x is equal to
negative 1/3 times f of x. Once again, negative f of
x would just flipped over, and then multiplying it by
the 1/3 diminishes that, which we see right over there. So I feel pretty confident with my none of the above.