What is a dilation, or scaling around a point?

Answer 1

When scaling a plane around a point, the result is a plane of a different size but the same shape.

It is highly-recommended to draw everything out.

SCALING FROM A CENTERPOINT

Scaling is commonly done about #(0,0)#, in which case it looks like this:

and you can see that the #z#-axis (into/out of the screen, or in the #hatk# direction) would pass through the exact centers of both triangles #DeltaABC# and #DeltaA'B'C'#.

In this scenario, it is clearer what has happened to each point. #A("-2,-2")# has been scaled outwards by a factor of #2# to achieve #A'("-4,-4")#, #B("1,1")# has been scaled outwards by a factor of #2# to achieve #B'("2,2")#, and #C("0,2")# has been scaled outwards by a factor of #2# to achieve #C'("0,4")#.

In general, scaling about a central point is given as:

#color(blue)(P'(x,y) = P(kx,ky) " where " k in RR " and " k > 0)#

SCALING OFF-CENTER

Scaling might also be done from an arbitrary non-central point. For instance, scaling a triangular plane #D_k# outwards from point #O# by non-zero scalar #k in RR# where #k = (OA')/(OA) = (OB')/(OB) = (OC')/(OC)# gives the transformation

for which #A#, #B#, and #C# were points on the green triangle. In this case, the scaling shrunk #DeltaABC# into #DeltaA'B'C'# by a factor of #k# where #0 < k < 1#.

You can see that #DeltaABC# and #DeltaA'B'C'# expand along the axes that pass through points #O#, as well as #A#, #B#, #C#, #A'#, #B'#, or #C'#.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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