Scaling functions vertically: examples | Transformations of functions | Algebra 2 | Khan Academy


– [Instructor] So, we’re told that this is the graph of function
f right over here. And then they tell us that
function g is defined as g of x is equal to one third f of x. What is the graph of g? And if we were doing this on Khan Academy, this is a screenshot from our mobile app, it has multiple choices, but I thought we could
just try to sketch it. So pause this video, maybe in your mind, imagine what you think the graph
of g is going to look like, or at least how you would tackle it. All right, so g of x is
equal to one third f of x. So, for example, we can
see here that f of three is equal to negative three. So, g of three should be one third that, so it should be negative one. Likewise, so, g of three
would be right over there, and likewise, g of negative
three, what would that be? Well, f of negative three is three, so g of negative three is
going to be one third that, or it’s going to be equal to one. F of zero is zero, one
third of that is still zero, so g of zero is still going
to be right over there. And we know that’s going to
happen there and there as well, and so we already have a sense of what this graph is going to look like. The function g is going
to look something like, something like this. I’m just connecting the dots and they did give us some dots that we can use as reference points, so the graph of g is going
to look something like this. It gets a little bit flattened out or a little bit squooshed
or smooshed a little bit to look something like that and you would pick the
choice that looks like that. Let’s do another example. So, here we are told
this the graph of f of x and it’s defined by this expression. What is the graph of g of x, and g of x is this. So pause this video and
think about it again. All right, now the key realization is, is it looks like g of x is, if you were to take
all the terms of f of x and multiply it by two, or at least if you were to multiply the absolute value by two, and then if you were to multiply
this negative two by two. So it looks like g of x
is equal to two times, two times f of x. And we could even set
up a little table here, this is another of the way
that we can think about it. We can think about x, we
can think about f of x, and now we can think about g of x, which should be two times that. So we can see that when
x is equal to zero, f of x is equal to one, so g of x should be equal to two because it’s two times f of x. So g of x is going to be equal to– Or g of zero, I should say,
is going to be equal to two. What about when at x equals, we’ll say when x equals three. When x equals three, f
of x is negative two. G of x is going to be two times that, ’cause it’s two times f of x, so it’s going to be negative four. So, g of x, or I should say g of three is going to be negative four. It’s going to be right over there. And then maybe let’s think
about one more point. So, f of five is equal to zero. G of five is going to be two times that, which is still going to be equal to zero, so it’s going to be right over there. And so the graph is going
to look something like this, I’m just really just connecting, I’m just connecting the dots, trying to draw some straight lines. It’s going to look something like this, you can see it’s kind of stretched in the vertical direction. So, if you were doing
this on Khan Academy, it’d be multiple choice, you’d look for the graph
that looks like that. Let’s do a few more examples. So, here we’re given a function g is a vertically scaled version of f. So we can see that g is a
vertically scaled version of f. The functions are graphed
where f is a solid and g is dashed. Yeah, we see that. What is the equation of g in terms of f? So, pause this video and
try to think about it. Well, the way that I would
tackle this is once again, let’s do it with a table and let’s see the
relationship between f and g. So, this column is x, this column is f of x, and then this column is g of x. I’ll make another column right over here. And so, let’s see some interesting points. So, when, actually, I could pick zero, but zero is maybe less interesting than this point over here. So, this is when x is
equal to negative three. F of negative three is negative three. What is g of negative three? It looks like it is negative nine. When f is, when x is zero, f of zero is negative two. What is g of zero? It is equal to two negative six. And so we already see a pattern forming. Whatever f is, g is three times that. Whatever f is, g is three times that. And so we don’t even actually
need these big columns, but we can see that g of x is
equal to three times f of x. So that is the equation
of g in terms of f. Let’s do one more example. So, once again, they give us f of x, this time, they’re telling
us the expression for f of x, and they’re telling g is a
vertically scaled version of f. The functions are graphed where
f is solid and g is dashed just like before. What is the equation of g? So, pause the video again. Try to work on it. All right, well, I’ll
tackle it the same way that we did the last one. I’m going to make a table, so x, and then I’m going to
have another column for f of x, and then I’m going to have
another column for g of x. Now let’s pick some interesting values. So, when x is equal to one, f of one is equal to two, g of one is equal to eight. Interesting. All right, let’s pick
another value, let’s see, when x is equal to four, g, or I should say f of four
is equal to negative one. When x is equal to four, f of
four is equal to negative one, yeah, I got that right. What is g of four? It’s equal to negative four. So it looks like, and I could
try it with other points, f of zero, when x is
zero, f of zero’s zero, g is zero as well. And so, it’s clear that from these points that g of x is four times f of x, in all of these cases, to
go from f of x to g of x, I multiply by four. I am multiplying by four, zero times four is still zero. So we could write that g of x
is equal to four times f of x. But we aren’t done. They’re asking what is the equation of g. And I think on Khan
Academy, if you do this, they might give some multiple choice, or you actually you might
be able to type it in, but either way, I think
they want the expression in terms of an actual
algebraic expression, not just in terms of f of x. So we could rewrite this as g of x is equal to four times what is f of x? It’s all of this business. Negative six log base two of x plus eight, and so we distribute that for g of x is equality to four times negative six is negative 24 log base two of x plus four times eight is 32. And we are done.

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