Respond to each of the following questions. be sure to include and thoroughly describe each transformation.
31. describe the transformations necessary to transform the graph of
f(x) to that of g(x).
hint: be sure to include and thoroughly describe each transformation.
you may use the gizmo to check your answer.
f(x) = x² and g(x) = 3(x - 1)² + 4
32. given the parent function f(x) = x², define the function g(x) that
transforms f(x) as follows:
reflection over the x-axis
horizontal shift left 3 units
vertical shift up 7 units
hint: you may use the gizmo to check your answer.

31. 1) Vertical shift up 4 units.

     2) Horizontal shift right 1 unit.

     3) Vertically stretched by a factor of 3.

32.

Step-by-step explanation:

Some transformations for a function f(x) are shown below:

- If , the function is shifted up "k" units.

- If , the function is shifted down "k" units.

- If , the function is shifted left "k" units.

- If , the function is shifted right "k" units.

- If , the function is reflected over the x-axis.

- If and , the function is stretched vertically by a factor of "b".

31. Given the function and the function , we can notice that the transformations from the graph of f(x) to the graph of g(x) are:

1)  

2)  

3)  , being

Therefore, we can conclude that the transformations necessary to transform the graph of f(x) to the graph g(x) are:

1) Vertical shift up 4 units.

2) Horizontal shift right 1 unit.

3) Vertical stretch by a factor of 3.

32. Knowing the parent function:

And given the transformations:

1) Reflection over the x-axis.

2) Horizontal shift left 3 units.

3) Vertical shift up 7 units.

We can define that the function g(x) is: