Consider the graph given below.

determine which sequences of transformations could be applied to the parent function, f(x) = x, to obtain the graph above.

shift left 2 units, reflect over the y-axis, and then vertically stretch by a factor of 6

reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units

shift right 3 units, reflect over the y-axis, and then vertically stretch by a factor of 2

reflect over the y-axis, vertically stretch by a factor of 2, and then shift up 6 units

shift left 3 units, reflect over the y-axis, and then vertically stretch by a factor of 2

shift up 6 units, reflect over the x-axis, and then vertically stretch by a factor of 2

Reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units.

Justification:

1) Given: parent function f(x) = x

2) Information obtained from f(x) = x:
   
    x = 0 , y = 0 => (0,0)
    slope = Δy / Δx = 1

3) Observe these special points from the transformed function given by the graph:

     x = 0, y = 6
     x = 3, y = 0
     slope = Δy / Δx = - 2 

Then, you can determine the point - slope form of the function:

            y - 6 = -2(x - 0)

From which, you can obtain the slope-intercept form of the function graphed:
                          
                         y -6 = -2x

                         y = -2x + 6

There are many diffferent combinations of transformations that drive from y = x to y = -2x + 6, so I will follow the order first suggested by the expression y = -2x + 6

4) When you compare y = x with y = -2x + 6, you know the graph of the second can be obtained from the first by:

     - relfecting the f(x) over the x-axis (which is indicated by the change of sign of the coefficient of x)

     - scaling (stretching) by a factor of 2 (which is indicated by the coefficient 2 in front of x)

     - and shifting 6 units upward (which is indiicating by the constant +6 after x).

That set of transformations is what the second choice says: Reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units. So that is the answer.

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