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?

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)