Question

# a) Find the sequence of transformation from f to g Given information: g{{left({x}right)}}=frac{1}{{2}}{left|{x}-{2}right|}-{3}{quadtext{and}quad} f{{l

Transformation properties
a) Find the sequence of transformation from f to g
Given information: $$g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}{\quad\text{and}\quad} f{{\left({x}\right)}}={x}^{3}$$
b) To sketch the graph of g.
Given information: $$f{{\left({x}\right)}}={\left|{x}\right|}$$
c) To write g in terms of f.
Given information: $$g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}{\quad\text{and}\quad} f{{\left({x}\right)}}={\left|{x}\right|}$$

2021-02-09
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).
$$g{{\left({x}\right)}}=\frac{1}{{2}} f{{\left({x}\right)}}-{3}$$