a)

Parent function is the basic function of a family of functions that preserves the definitions, shape of its graph and properties of the entire family.

Parent function used in this question is the absolute value function \(f{{\left({x}\right)}}={\left[{x}\right]}\)

To identify the parent function, strip all the arithmetic operations on the function to leave behind one higher order operation in just x.

So, remove the arithmetic operation of multiplication by -1 and then addition of 4 to x and addition of 8 to it from the given funcion to get the parent function.

Conclusion:

So, remove the arithmetic operation of subtractionof 1 from x and multiplication by -2 and then subtraction of g

\(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4},\ \text{the parent function is}\ f{{\left({x}\right)}}={\left[{x}\right]}.\)

b)

The sequence of transformations from f to g depicts the steps followed and the transformations used to reach from the parent function f to g.

Types of shifts used in function transformation:

1. Vertical shift: If c is a real number which is also positive, then the graph of \(f (x)\ +\ c\ \text{is the graph of}\ y = f (x)\) shifted upward by c units.

If c is a real number which is also positive, thenthe graph of \(f (x)\ -\ c\ \text{is the graph of} y = f (x)\) shifted downwards by c units.

2. Horizontal Shift: If c is a real number which is also positive then, the graph of \(f (x\ +\ c)\ \text{is the graph of} y = f (x)\) shifted left by c units.

If c is a real number which is also positive then, the graph of \(f (x\ -\ c)\ \text{is the graph of}\ y = f (x)\) shifted right by c units.

3. Reflection: The graph for the function say \(y = f (-x)\ \text{is the graph of}\ y = f (x)\) is the reflection in y-axis.

The graph for the function say \(y =\ -f (x)\ \text{is the graph of} y = f(x)\) is the reflection in x-axis.

4. Vertical Stretching and Shrinking: If c succ 1 then, the graph of \(y = cf (x)\ \text{is the graph of}\ y = f (x)\) stretched vertically by c units.

If 0 prec c prec 1 then, the graph of \(y = cf (x)\ \text{is nothing but the graph of}\ y = f(x)\) shrunk vertically by c units.

5. Horizontal Stretching and Shrinking: If c succ 1 then, the graph of \(y = cf (x)\ \text{is nothing but the graph of}\ y = f (x)\) shrunk horizontally by c units.

If 0 prec c prec 1 then, the graph of \(y = cf (x)\ \text{is nothing but the graph of}\ y = f (x)\) stretched horizontally by c units.

The shape of \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\) is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then downward by 4 units.

The sequence of transformations from f to g depicts the steps followed and the transformations used to reach from the parent function f to g.

Conclusion:

The shape of is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then shifted downward by 4 units is the required sequence of transfomations from f to g.

c)

Use the sequence of tranformation to plot the graph of the function.

Obtain the graph of \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\)

d)

Multiply \(f (x)\ \by\ -2\) and then subtract from the parent function to get g(x) in terms of f(x).

Parent function is the basic function of a family of functions that preserves the definitions, shape of its graph and properties of the entire family.

Parent function used in this question is the absolute value function \(f{{\left({x}\right)}}={\left[{x}\right]}\)

To identify the parent function, strip all the arithmetic operations on the function to leave behind one higher order operation in just x.

So, remove the arithmetic operation of multiplication by -1 and then addition of 4 to x and addition of 8 to it from the given funcion to get the parent function.

Conclusion:

So, remove the arithmetic operation of subtractionof 1 from x and multiplication by -2 and then subtraction of g

\(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4},\ \text{the parent function is}\ f{{\left({x}\right)}}={\left[{x}\right]}.\)

b)

The sequence of transformations from f to g depicts the steps followed and the transformations used to reach from the parent function f to g.

Types of shifts used in function transformation:

1. Vertical shift: If c is a real number which is also positive, then the graph of \(f (x)\ +\ c\ \text{is the graph of}\ y = f (x)\) shifted upward by c units.

If c is a real number which is also positive, thenthe graph of \(f (x)\ -\ c\ \text{is the graph of} y = f (x)\) shifted downwards by c units.

2. Horizontal Shift: If c is a real number which is also positive then, the graph of \(f (x\ +\ c)\ \text{is the graph of} y = f (x)\) shifted left by c units.

If c is a real number which is also positive then, the graph of \(f (x\ -\ c)\ \text{is the graph of}\ y = f (x)\) shifted right by c units.

3. Reflection: The graph for the function say \(y = f (-x)\ \text{is the graph of}\ y = f (x)\) is the reflection in y-axis.

The graph for the function say \(y =\ -f (x)\ \text{is the graph of} y = f(x)\) is the reflection in x-axis.

4. Vertical Stretching and Shrinking: If c succ 1 then, the graph of \(y = cf (x)\ \text{is the graph of}\ y = f (x)\) stretched vertically by c units.

If 0 prec c prec 1 then, the graph of \(y = cf (x)\ \text{is nothing but the graph of}\ y = f(x)\) shrunk vertically by c units.

5. Horizontal Stretching and Shrinking: If c succ 1 then, the graph of \(y = cf (x)\ \text{is nothing but the graph of}\ y = f (x)\) shrunk horizontally by c units.

If 0 prec c prec 1 then, the graph of \(y = cf (x)\ \text{is nothing but the graph of}\ y = f (x)\) stretched horizontally by c units.

The shape of \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\) is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then downward by 4 units.

The sequence of transformations from f to g depicts the steps followed and the transformations used to reach from the parent function f to g.

Conclusion:

The shape of is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then shifted downward by 4 units is the required sequence of transfomations from f to g.

c)

Use the sequence of tranformation to plot the graph of the function.

Obtain the graph of \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\)

d)

Multiply \(f (x)\ \by\ -2\) and then subtract from the parent function to get g(x) in terms of f(x).