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Discuss/Explain how the graph F(x) = -2f(x+1) -3 can be obtained from the graph f(x). If (0,5), (6,7), and (-9,-4) are on the graph of f. where do they end up on the graph of F? 

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Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
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F(x) = -2f(x+1) -3
 
-f(x) is f(x) reflected over the x-axis
-2f(x) is -f(x) stretch 2 times in the y direction
-2f(x+1) - 3 is -2f(x) translated (shifted) 1 left and 3 down
 
Putting it all together:
 
F(x) is f(x) reflected over the x-axis, then stretched 2 times in the y direction, and then  translated (shifted) 1 left and 3 down.
 
If P(a,b) is a point on f(x), then
R(a,-b) is P reflected over the x-axis,
S(a,-2b) is R stretched 2 times in the y direction, and 
T(a-1,-2b-3) is S translated (shifted) 1 left and 3 down.
 
So if P(a,b) is a point on f(x), then T(a-1,-2b-3) is the corresponding point on F(x).
 
E.g.:
(0,5) → (0-1,-2(5)-3) = (-1,-13)
(6,7) → (6-1,-2(7)-3) = (5,-17)
(-9,-4) → (-9-1,-2(-4)-3) = (-10,5)