Provide answers to all tasks using the information provided. a) Find the parent function f. Given Information: g{{left({x}right)}}=-{2}{left|{x}-{1}right|}-{4} b) Find the sequence of transformation from f to g. Given information: f{{left({x}right)}}={left[{x}right]} c) To sketch the graph of g. Given information: g{{left({x}right)}}=-{2}{left|{x}-{1}right|}-{4} d) To write g in terms of f. Given information: g{{left({x}right)}}=-{2}{left|{x}-{1}right|}-{4}{quadtext{and}quad} f{{left({x}right)}}={left[{x}right]}

Provide answers to all tasks using the information provided. a) Find the parent function f. Given Information: g{{left({x}right)}}=-{2}{left|{x}-{1}right|}-{4} b) Find the sequence of transformation from f to g. Given information: f{{left({x}right)}}={left[{x}right]} c) To sketch the graph of g. Given information: g{{left({x}right)}}=-{2}{left|{x}-{1}right|}-{4} d) To write g in terms of f. Given information: g{{left({x}right)}}=-{2}{left|{x}-{1}right|}-{4}{quadtext{and}quad} f{{left({x}right)}}={left[{x}right]}

Question
Transformation properties
asked 2021-02-25
Provide answers to all tasks using the information provided.
a) Find the parent function f.
Given Information: \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\)
b) Find the sequence of transformation from f to g.
Given information: \(f{{\left({x}\right)}}={\left[{x}\right]}\)
c) To sketch the graph of g.
Given information: \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\)
d) To write g in terms of f.
Given information: \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}{\quad\text{and}\quad} f{{\left({x}\right)}}={\left[{x}\right]}\)

Answers (1)

2021-02-26
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 \(g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}\)
image
d)
Multiply \(f (x)\ \by\ -2\) and then subtract from the parent function to get g(x) in terms of f(x).
0

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