# If the following graph is of a function #f(x)#, how will the graph of (i) #f(x+3)#, (ii) #f(x)+3#, (iii) #-f(x)# and (iv) #1-f(x-3)# appear?

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(i) The graph of f(x+3) will shift the original graph of f(x) to the left by 3 units.

(ii) The graph of f(x)+3 will shift the original graph of f(x) vertically upwards by 3 units.

(iii) The graph of -f(x) will reflect the original graph of f(x) over the x-axis.

(iv) The graph of 1-f(x-3) will shift the original graph of f(x) to the right by 3 units and then vertically flip it over the x-axis and shift it upwards by 1 unit.

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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.

- Circle A has a radius of #2 # and a center at #(5 ,1 )#. Circle B has a radius of #1 # and a center at #(3 ,2 )#. If circle B is translated by #<-2 ,6 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A triangle has corners at #(3, 9 )#, ( 6, -5)#, and #( 4, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?
- Circle A has a radius of #3 # and a center of #(8 ,5 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Points A and B are at #(6 ,1 )# and #(3 ,5 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?
- A triangle has corners at #(2, 7 )#, #( 6, 3 )#, and #( 2 , 5 )#. If the triangle is dilated by # 2 x# around #(2, 5)#, what will the new coordinates of its corners be?

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