Main content

## College Algebra

### Course: College Algebra > Unit 12

Lesson 4: Scaling functions# Scaling functions introduction

The graph y=k⋅f(x) (where k is a real number) is similar to the graph y=f(x), but each point's distance from the x-axis is multiplied by k. A similar thing happens when we graph y=f(k⋅x), only now the distance from the y-axis changes. These operations are called "scaling."

## Want to join the conversation?

- Basically, I had a really hard time understanding this topic, so I am going to write down what I found in terms of differentiating vertical shrinks and stretches from horizontal shrinks and stretches.

Generally, if the point on the y-axis moves, it is a vertical stretch or shrink, and if it doesn't, then it is horizontal. Of course, this only applies if the point on the y-axis is not (0, 0), but that's the case most of the time.

When the graph gets narrower, it is either a vertical stretch or a horizontal shrink; essentially, stretching AWAY from the x-axis or shrinking TO the y-axis.

When the graph gets wider, it is either a vertical shrink or a horizontal stretch: essentially, shrinking TO the x-axis or stretching AWAY from the y-axis.

So, in conclusion:

if the graph moves on the y-axis:

if the graph gets wider: vertical shrink

if the graph gets narrower: vertical stretch

if the graph does not move on the y-axis:

if the graph gets wider: horizontal stretch

if the graph gets narrower: horizontal shrink(15 votes)- I am not sure what you mean by moving and not moving on the y-axis. If you have some function such as g(x)= a f(bx-c) + d, each of a, b, c and d have affects on the parent function. Only c and d actually translate points (which is what is generally referred to as "move"). So the vertical and horizontal stretches and compressions do not move points as much as relate how the points are related to each other/how they are related to the original parent function. a affects the vertical stretch (if a>1) or compression (if a<1<0) as well as the reflection across x (if a is negative). B affects the horizontal stretch (if 1<b<0) and horizontal compression (if b>1) as well as reflection across y (if b is negative). If you leave out the part of moving on the y-axis (which is an effect of translation, not stretches and compressions), your conclusions are correct in that a vertical stretch and a horizontal compression both make a graph get wider (or in the case of a linear equation have a steeper slope). Similarly, a vertical compression or a horizontal stretch make a graph get wider (or in the case of linear, a flatter slope).

So let adding/subtracting things either inside the parentheses with the x or outside the parentheses do the moving of important points, and let multiplying either inside or outside the parentheses affect the stretches and compressions.

As an example, if you have the parent function such as y=x^2, if you change this to a function g(x) = 16(x+2)^2 + 3, you would move the vertex of the parabola to (-2,3) and a vertical stretch by a factor of 16. By taking the √16=4, you could say the same equation could be written as g(x) = (4(x+2)^2+3 and have a horizontal compression by a factor of 4.

While I see where you got the idea of moving along the y axis, if you have f(x) = 2-x^2 and g(x) = k f(x), when you make k=2, you are doing f(x) = 2(2-x^2) or 4-2x^2. If k=-2, you have f(x) = -2(2-x^2) which gives -4 + x^2, so the movement along the y axis is actually still an effect of the d (even though it is in a different order), not the effect of the stretch or compression.

Ask more questions if needed, I hope this makes some sense.(6 votes)

- Summary of this video:

In this case, the output or the dependent value is f(x), so it means that f(x) = y because we often put the dependent value on the y-axis to see the effects of different input/x-values more clearly. Think of the baseline as the one that have the scale factor of 1: 1⋅f(x) or f(x).

• If we were to scale a parabola and put the k (the scale factor) outside of the parentheses (k⋅f(x)), there are two general scenario with each unique property that differentiate them from the scenarios we get if we put the k inside of the parentheses (f(k⋅x)):

1. If you were to scale it by a factor greater than 1 or more negative than -1, the parabola would be narrower because the f(x) or the y-value will increase faster for a given input/x-value compared to the baseline. For example, for x = 1, let's say f(x) = 2x²+3 and compare it with 2⋅f(x). By substituting 1 as the x and calculating it, f(x) = f(1) = 2⋅1²+3 = 2+3 = 5, whereas 2⋅f(x) = 2⋅f(1) = 2⋅(2⋅1²+3) = 2⋅(2+3) = 4+6 (we distribute the 2, which is just multiplying 2 to each of the terms, we do this to show you that the y-intercept changed from 3 to 6 because it is multiplied by the scale factor) = 10.

See? for a same input/value of x, the output/value of y or the f(x) will be 2 times greater because, in this case, we scale up the whole expression (the f(x)) by a scale factor of 2. We know from the slope-intercept form that the y-intercept is the constant number of the expression (not the scale factor, which is the constant number that we use to scale the expression, it's different). So, in this case, the y-intercept of f(1) is 3 and the y-intercept of 2⋅f(1) is 6. For the same input/x-value, the y-intercept depends on the scale factor.

In conclusion, we know that if we put the scale number outside of the parentheses, it means that we have to distribute the scale factor to all of the terms of the expression, thus changing the y-intercept because the y-intercept is one of the factors of the expression. The y-intercept will change based on the scale factor: the greater the scale factor, the greater the y-intercept because we're multiplying the scale factor with the y-intercept, and vice versa. This works if the x is negative, too, because we care about the distance of the scale factor from 0, and if the scale factor is negative, it will just form a parabola that is upside down. Beside that, the more-narrow-or-wide result from the negative scale factor use the same logic as when the scale factor is positive. If the scale factor is more negative than -1, the parabola will be more narrow, and vice versa.

2. If you were to scale it by a factor that is between 0 and 1, the parabola would be wider because the output/y-value or the f(x) is increasing slower for a given input/x-value compared to the baseline. You can think of the scale factor as a fraction because it helps you think of multiplying every terms of the expression by the scale factor that is between 0 and 1 as dividing every terms (because, for example, 1 times 2 is greater than 1/2 times 2). Let's use the same example as before: for x = 1 and f(x) = 2x²+3 and compare it with 0.5⋅f(x). f(x) = f(1) = 2⋅1²+3 = 2+3 = 5, whereas 0.5⋅f(x) = 0.5⋅f(1) = 0.5(2⋅1²+3) = 1 + 1.5 (from distributing the 0.5, we can see that the y-intercept changed from 3 to 1.5) = 2.5.

See? for the same input/x-value, we get an output that is 1/2 times as greater as the baseline because, in this case, the scale factor is 1/2 or 0.5. In this case, the y-intercepts of the two equations also changed (because we put the scale factor outside of the parentheses, which means that we multiply the whole expression by the scale factor): the y-intercept of f(1) = 3, whereas the y-intercept of 0.5⋅f(1) = 1.5.

• Scenarios if we were to scale a parabola with a scale factor that is put inside of the parentheses:

1. If we were to scale a parabola and put the k inside of the parentheses (f(k⋅x)), the greater the k, the narrower the parabola would get, but it wouldn't change the y-intercept. Let's look at the same example: for x = 1, compare f(x) = 2x²+3 to f(2⋅x). After calculating it, f(x) = f(1) = 2⋅1²+3 = 5 and f(2⋅x) = f(2⋅1) = f(2) = 2⋅2²+3 = 11. See that although the output changed, the y-intercept doesn't change because what changed is the input/x-value. The change in input eventually lead to change in output, but what differs is that the y-intercept is not directly affected by the scale factor. The change in output is also not caused directly by the scale factor, rather it's directly affected by the input that is directly affected by the scale factor. In this case, for x = 1, in f(2⋅x), you will get f(2), which have greater value if we compare it with f(x). In f(x), you will only get f(2) if the input/x-value is 2.

So, in this case (where k = 2, which is greater than 1, and we put it inside of the parentheses), everything is happening faster compared to the baseline because the scale factor is directly affecting the input, hence it's more narrow. The same goes if the scale factor is more negative than -1, what differs is that the parabola will be upside down. At4:42and onward, Sal explained it perfectly. Sal also explained it perfectly in a Khan Academy video titled "Scaling functions horizontally; examples", especially at1:38.

2. The same goes if we were to scale a parabola and put the k that is between 0 and 1 inside of the parentheses, the parabola will be more wider, but it will not change the y-intercept. That is because for a given x, everything is happening slower compared to the baseline, hence it's wider. The reason why everything is happening slower is because if the scale factor is between 0 and 1, the scale factor is basically dividing the input/x-value.

In conclusion, if the scale factor is outside of the parentheses, we multiply the whole expression by the scale factor, thus directly changing the output. That is also the reason why the y-intercept changed when the scale factor is outside of the parentheses: because the y-intercept is one of the terms in the expression and we multiply every term in the expression by the scale factor. But, if we put the scale factor inside of the parentheses, we only multiply the x by the scale factor, thus changing the input. The change in input also will result in the change in output, but the change in output is not a direct result of the scale factor. The y-intercept doesn't change if the scale factor is inside of the parentheses because it's not being multiplied by the scale factor.

Oh and I forgot to mention that if the scale factor is negative and is outside of the parentheses, it will be reflected across the x-axis. If the scale factor is negative and is inside of the parentheses, it will be reflected across the y-axis. If there is two scale factor that are negative and are each multiplying inside and outside of the parentheses, it will be reflected across both axes; you can think of it as reflecting it across the y-axis first and then reflecting that reflection across the x-axis or you can do it the other way around. For more of it, check Khan Academy videos on reflecting functions.

Please correct me if I'm wrong and sorry for the bad grammar!(10 votes) - At5:25, when Sal did f(k*x) for the function 2-x^2, was the k only applied to x? So it became 2-k*x^2? I am a bit confused here...(3 votes)
- 𝑓(𝑥) = 2 − 𝑥²

Changing 𝑥 to 𝑘𝑥, we get 𝑓(𝑘𝑥) = 2 − (𝑘𝑥)²

Just like if we changed 𝑥 to 3, we would get 𝑓(3) = 2 − 3²(9 votes)

- this is kind of oversimplifying it, but if the k is inside the parenthesis, it only affects the x. if its outside (multiplying everything), then it affects the whole equation(7 votes)
- where can i find a Desmos graphing calculator such as the one shown in the video

I'll greatly appreciate any help

am i missing any specific link?(3 votes)- just go to a tab and search desmos graphing calculator, it should be easy to find(4 votes)

- I can't understand the difference between k*f(x) and f(k*x). It looks a lil' bit of a bizarre idea!(3 votes)
- Say f(x)=x². If you make k=3, then f(k*x)=(3x)²=9x², while k*f(x)=3x². When the scalar is inside with the x, your replace x with everything inside the parentheses. If the scalar is outside, it's just like multiplying both sides by a number. Do you understand now?(3 votes)

- If f(x)=(x-2)^2 - 1

And y = f(x)

Then what will be the graph of |y|=f(x)(2 votes)- So (x-2)^2 is all positive, but by adding the -1, it shifts it down 1 (vertex at (2,-1)). Since it is negative in the domain of 1 to 3, the equation would be the same when less than or equal to 1 and greater than or equal to 3. However, between 1 and 3, it would reflect the curve across the x axis, so the vertex would flip to (2,1) which then curves to 1 and 3.

It would have a normal quadratic curve (until 1), a small bump in the middle, then back to the normal curve at 3.(4 votes)

- why do you keep changing the scale(2 votes)
- Sal is providing examples, so it becomes intuitive to your understanding. This shows examples of how numbers effect the graph. It can change the slope (rate of change/steepness). It can change the slope and the y-point coordinate as well.(2 votes)

- What if you move the graph left or right, for example, f(x)=|x-3|, when you multiply it by a constant k, would it have an impact on the x intercept? What about f(k*x)?(3 votes)
- f(x)=k|x-3| has the same x intercept at 3 (shifted) but the k factor makes the slopes "steeper". f(kx) in the original expression for f(x), becomes |kx - 3| and the x intercept becomes kx=3 or x=3/k. As before, the slope becomes steeper (slope of +/-k rather than +/- 1)(0 votes)

- In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.[1][self-published source][2][3]

The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a proper rigid transformation, or rototranslation.[citation needed] Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections.

Any object will keep the same shape and size after a proper rigid transformation.

All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformations is called special Euclidean group, denoted SE(n).

In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw displacement.(2 votes)

## Video transcript

- [Tutor] So this is a
screenshot of Desmos, it's an online graphing calculator, what we're gonna do is
use it to understand how we can go about scaling functions and I encourage you to go to
Desmos and try it on your own either during this video or after. So let's start with a
nice, interesting function, let's say f of x is equal
to the absolute value of x, so that's pretty straightforward. Now let's try to create a
scaled version of f of x, so we could say g of x is equal to, well, I'll start with
just absolute value of x, so it's the same as f of x, so we'll just trace the
g of x right on top of f, but now let's multiply it by sum constant, let's multiply it by two. So notice the difference
between g of x and f of x and you can see that g of
x is just two times f of x, in fact we can write it this way, we can write g of x is
equal to two times f of x, we get to the exact same place, but you can see that as our x increases, g of x increases twice as
fast, at least for positive xs on the right-hand side and
actually as x decreases, g of x also increases twice as fast, so is that just a coincidence
that we have a two here and it increased twice as fast? Well, let's put a three here, well now it looks like it's
increasing three times as fast and it does that in both directions. Now what if we were to
put a 0.5 here, 0.5? Well now it looks like it's
increasing half as fast and that makes sense, because
we are just multiplying, we are scaling how much our f of x is. So before when x equals
one, we got to one, but now when x equals one,
we only get to one half, before when x equals five, we got to five, now when we get to x equals
five, we only get to 2.5, so we're increasing half as fast, or we have half the slope. Now an interesting question to think about is what would happen if instead of it just being
an absolute value of x, let's say we were to have
a non-zero y intercept, so let's say, I don't know, plus six, so notice then when we change
this constant out front, it not only changes the slope, but it changes the y intercept, because we're multiplying
this entire expression by 0.5, so if you multiply it by one, we're back to where we got before and now if we multiply it by two, this should increase the y intercept, 'cause remember we're
multiplying both of these terms by two and we see that, it
not only doubles the slope, but it also increases the y intercept. If we go to 0.5, not only did it decrease the
slope by a factor of one half, or I guess you could say
multiple the slope by one half, but it also made our y intercept be half of what it was before and we can see this more generally if we just put a general constant here and we can add a slider and actually let me make the
constant go from zero to 10 with a step of, I don't know, 0.05, that's just how much does it increase every time you change the slider and notice when we increase our constant, not only we're getting narrower, 'cause the magnitude of
the slope is being scaled, but our y intercept increases
and then as k decreases, our y intercept is being scaled down and our slope is being scaled down. Now that's one way that
we could go about scaling, but what if instead of multiplying our entire function by sum constant, we instead just replace the
x with a constant times x, so instead of k times f of x, what if we did it f of k times x? Another way to think about it is g of x is now equal to the absolute
value of kx plus six, what do you think is going to happen? Pause this video and think about it. Well now when we increase k, notice it has no impact
on our y intercept, because it's not scaling the y intercept, but it does have an impact on slope, when k goes from one to two, once again we are now
increasing twice as fast and then when k goes from one to one half, we're now increasing half as fast. Now this is with an
absolute value function, what if we did it with a
different type of function, let's say we did it with a quadratic? So two minus x squared, let me scroll down a little bit and so you can see when k
equals one, these are the same and now if we increase our k, let's say we increase our k to two, notice our parabola is
in this case decreasing as we get further and further from zero at a faster and faster rate, that's because what you would
have seen at x equals two, you're now seeing at x equals one, because you are multiplying two times that and so then if we go between zero and one, notice on either side of zero, our parabola is decreasing
at a lower rate, it's a changing rate, but
it's a lower changing rate, I guess you could put it that way and we could also try just to see what happens with our parabola here, if instead of doing kx, we
once again put the k out front, what is that going to do? And notice that is
changing not only how fast the curve changes at different points, but it's now also
changing the y intercept, because we are now
scaling that y intercept. So I'll leave you there, this is just the beginning
of thinking about scaling, I really want you to
build an intuitive sense of what is going on here and really think about
mathematically why it makes sense and go on to Desmos and
play around with it yourself and also try other types of
functions and see what happens.