a)

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 \(f{{\left({x}\right)}}+c\ \text{is the graph of}\ y = f (x)\) shifted upward by c units.

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

Conclusion:

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.

The shape of \(g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}\ \text{is drawn reflected in the x-axis and then stretched by}\ \frac{1}{2}\) units and then shifted downward by 3 units.

b)

\(g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}\)

Use parent functions and then move them around the coordinate plane throught various types of shifts and thus write one function in terms of the other.

Conclusion:

Obtain the graph of \(g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}\)

\(g{{\left({x}\right)}}=\frac{1}{{2}} f{{\left({x}\right)}}-{3}\)

c)

Multiply \(f (x)\ \text{by}\ 1/2\) and then substract 3 from it to get g(x) in terms of f(x).

\(g{{\left({x}\right)}}=\frac{1}{{2}} f{{\left({x}\right)}}-{3}\)

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 \(f{{\left({x}\right)}}+c\ \text{is the graph of}\ y = f (x)\) shifted upward by c units.

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

Conclusion:

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.

The shape of \(g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}\ \text{is drawn reflected in the x-axis and then stretched by}\ \frac{1}{2}\) units and then shifted downward by 3 units.

b)

\(g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}\)

Use parent functions and then move them around the coordinate plane throught various types of shifts and thus write one function in terms of the other.

Conclusion:

Obtain the graph of \(g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}\)

\(g{{\left({x}\right)}}=\frac{1}{{2}} f{{\left({x}\right)}}-{3}\)

c)

Multiply \(f (x)\ \text{by}\ 1/2\) and then substract 3 from it to get g(x) in terms of f(x).

\(g{{\left({x}\right)}}=\frac{1}{{2}} f{{\left({x}\right)}}-{3}\)