Chapter 1.4, Problem 2E

To determine

To match the rigid transformation with the given representation

Explanation of Solution

The correct matching is as follows:

  

  $h(x)=f(x)+c$

To understand this equation assume a function according to the situation and plot them and compare them to understand the situation. Hence,

Let,

  $f(x)=x$

  $h(x)=x+1$

Calculation for graph:

Values of x	Values of f (x )
0	0
1	1
4	4
9	9
16	16

Using the above table the graph can be plotted.

Calculation for graph:

Values of x	Values of h (x )
0	1
1	2
4	5
9	10
16	17

Using the above table the graph can be plotted.

Graph of $f(x)$ and $h(x)$ on the same plane:

  

Interpretation:

Hence, it is clear from the graph that, $h(x)$ is the vertical shift of $f(x)$ by c = 1 units upward.

(b) $h(x)=f(x)+c$

To understand this equation assume a function according to the situation and plot them and compare them to understand the situation. Hence,

Let,

  $f(x)=x$

  $h(x)=x-1$

Calculation for graph:

Values of x	Values of f (x )
0	0
1	1
4	4
9	9
16	16

Using the above table the graph can be plotted.

Calculation for graph:

Values of x	Values of h (x )
0	-1
1	0
4	3
9	8
16	15

Using the above table the graph can be plotted.

Graph of $f(x)$ and $h(x)$ on the same plane:

  

Interpretation:

Hence it is clear from the graph that $h(x)$ is the vertical shift of $f(x)$ by c = 1 units downward.

(c) $h(x)=f(x-c)$

To understand this equation assume a function according to the situation and plot them and compare them to understand the situation. Hence,

Let,

  $f(x)=x^2$

  $h(x)={(x-1)}^2$

Calculation for graph:

Values of x	Values of f (x )
0	0
1	1
2	4
3	9
4	16

Using the above table the graph can be plotted.

Calculation for graph:

Values of x	Values of h (x )
0	1
1	0
2	1
3	4
4	9

Using the above table the graph can be plotted.

Graph of $f(x)$ and $h(x)$ on the same plane:

  

Interpretation:

Hence, it is clear from the graph that $h(x)$ is the horizontal shift of $f(x)$ by c = 1 units towards right.

(d) $h(x)=f(x+c)$

To understand this equation assume a function according to the situation and plot them and compare them to understand the situation. Hence,

Let,

  $f(x)=x^2$

  $h(x)={(x+1)}^2$

Calculation for graph:

Values of x	Values of f (x )
0	0
1	1
2	4
3	9
4	16

Using the above table the graph can be plotted.

Calculation for graph:

Values of x	Values of h (x )
0	1
1	4
2	9
3	16
4	25

Using the above table the graph can be plotted.

Graph of $f(x)$ and $h(x)$ on the same plane:

  

Interpretation:

Hence it is clear from the graph that $h(x)$ is the horizontal shift of $f(x)$ by c= 1 units towards left.

Conclusion:

All the parameters have been matched accordingly with reference to the above example.