Respuesta :

Parent function: f(x)=x
The graph is of a linear function y=mx+b
Intercept with y-axis: b=6
Slope: m=?
Intercept with x-axis: a=3
m=-b/a=-6/3→m=-2
The graph is of the function y=-2x+6

1) Shift left 2 units (x=x+2): y=x+2
Reflect over the y-axis (x=-x): y=-x+2
Vertically stretch by a factor of 6 (y=6y): y=6(-x+2)→y=-6x+12 different to the function of the graph y=-2x+6

2) Reflect over the x-axis (y=-y): y=-x
Vertically stretch by a factor of 2 (y=2y): y=2(-x)→y=-2x
Shift up 6 units (y=y+6): y=-2x+6 = Function of the graph  y=-2x+6

3) Shift right 3 units (x=x-3): y=x-3
Reflect over the y-axis (x=-x): y=-x-3
Vertically stretch by a factor of 2 (y=2y): y=2(-x-3)→y=-2x-6 different to the function of the graph y=-2x+6

4) Reflect over the y-axis (x=-x): y=-x
Vertically stretch by a factor of 2 (y=2y): y=2(-x)→y=-2x
Shift up 6 units (y=y+6): y=-2x+6 = Function of the graph  y=-2x+6

5) Shift left 3 units (x=x+3): y=x+3
Reflect over the y-axis (x=-x): y=-x+3
Vertically stretch by a factor of 2 (y=2y): y=2(-x+3)→y=-2x+6 = Function of the graph  y=-2x+6

6) Shift up 6 units (y=y+6): y=x+6
Reflect over the x-axis (y=-y): y=-(x+6)→y=-x-6
Vertically stretch by a factor of 2 (y=2y)→y=2(-x-6)→y=-2x-12 different to the function of the graph y=-2x+6

Answers: 3 Options:
Option 2: Reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units.
Option 4: Reflect over the y-axis, vertically stretch by a factor of 2, and then shift up 6 units.
Option 5: Shift left 3 units, reflect over the y-axis, and then vertically stretch by a factor of 2.
mathmate
Recall that
f(x)->f(x-h)  shift right h units
f(x)->-f(x)    reflect over x-axis
f(x)-> f(-x)   reflect over y-axis
f(x)-> 2f(x)  vertically stretch by 2 units
and
f(x)=x
g(x) has a slope of -2, and a y-intercept of 6, so
g(x)=-2x+6

A. 
f(x)->f(x+2) -> f(-x+2) -> 6f(-x+2) =6(-x+2)=-6x+12  [ does not equal g(x) ]

B. 
f(x)->-f(x) -> -2f(x) -> -2f(x)+6 = -2x+6    [ equals g(x), so YES ]

C.
f(x)->f(x-3)->f(-x-3)-> 2f(-x-3) = 2(-x-3)=-2x-6     [ does not equal g(x) ]

D. 
f(x)->f(-x)->2f(-x)->2f(-x)+6=-2x+6            [ equals g(x), so YES ]

E. 
f(x)->f(x+3)->f(-x+3)->2f(-x+3)=-2x+6       [ equals g(x), so YES ]

F.
f(x)->f(x)+6->-f(x)-6->-2f(x)-12 = -2x-12    [ does not equal g(x) ]

Following images show the sequence of transformations for cases A through E, of which cases B,D and E result in the given graph.

Sorry, I am only allowed 5 images, so F will not be illustrated.
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