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### Course: Integrated math 3>Unit 6

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
• 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.
• 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
• At , 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...
• 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
• 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?
• just go to a tab and search desmos graphing calculator, it should be easy to find
• If f(x)=(x-2)^2 - 1
And y = f(x)
Then what will be the graph of |y|=f(x)
• 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.
• I can't understand the difference between k*f(x) and f(k*x). It looks a lil' bit of a bizarre idea!
• 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?
• thank god for khan i kinda checked out during algebra 2 and now i have precalculus starting in a week and these videos have improved my confidence in starting college
• why do you keep changing the scale
• 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.
• 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)?
• 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)
• 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.