Chapter 1.4, Problem 76E

To determine

To identify which of the following given equations are probable of representing the given graph.

Answer to Problem 76E

Option (b) and (d).

Explanation of Solution

Given:

  

Calculation:

The best approach for solving this problem is by plotting the graph of each given function and compare with the given graph.

1. $f(x)=-{(x-4)}^3+2$

Calculation for graph:

Consider $f(x)=-{(x-4)}^3+2$

Values of x	Values of f (x )
2	10
3	3
4	2
5	1
6	-6

Using the above table the graph can be plotted.

Graph:

  

Comparison:

Now, compare above graph and given graph side by side.

(i) (ii)

Interpretation:

After comparing both the graphs, it can be interpreted that both the graph are rigid-transformation of base graph of $y=x^3$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the right and vertical shift upwards after taking reflection of base graph about x -axis.

Graph (ii): It has a horizontal shift towards left and vertical shift upwards after taking reflection of base graph about x -axis.

Hence, both the graphs are not same.

2. $f(x)=-{(x+4)}^3+2$

Calculation for graph:

Consider $f(x)=-{(x+4)}^3+2$

Values of x	Values of f (x )
0	-62
-1	-25
-2	-6
-4	0
-6	10

Using the above table the graph can be plotted.

Graph:

  

Comparison:

Now, compare above graph and given graph side by side.

(i)(ii)

Interpretation:

After comparing both the graphs, it can be interpreted that both the graph are rigid-transformation of base graph of $y=x^3$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the left and vertical shift upwards after taking reflection of base graph about x- axis.

Graph (ii): It has a horizontal shift towards left and vertical shift upwards after taking reflection of base graph about x -axis.

Hence, both the graphs are same.

3. $f(x)=-{(x-2)}^3+4$

   Calculation for graph:

   Consider $f(x)=-{(x-2)}^3+4$

   Values of x	Values of f (x )
   0	12
   1	5
   2	4
   -2	68
   3	3

   Using the above table the graph can be plotted.

   Graph:

     

   Comparison:

   Now, compare above graph and given graph side by side.

   (i) (ii)

   Interpretation:

   After comparing both the graphs, it can be interpreted that both the graph are rigid-transformation of base graph of $y=x^3$ with certain shifts in both horizontal and vertical shift.

   Graph (i): It has horizontal shift to the right and vertical shift upwards after taking reflection of base graph about x- axis.

   Graph (ii): It has a horizontal shift towards left and vertical shift upwards after taking reflection of base graph about x -axis.

   Hence, both the graphs are not same.

4. $f(x)={(-x-4)}^3+2$

Calculation for graph:

Consider $f(x)={(-x-4)}^3+2$

Values of x	Values of f (x )
0	-62
-1	-25
-2	-6
-3	1
-4	2

Using the above table the graph can be plotted.

Graph:

  

Comparison:

Now, compare above graph and given graph side by side.

(i) (ii)

Interpretation:

After comparing both the graphs, it can be interpreted that both the graph are rigid-transformation of base graph of $y=x^3$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the left and vertical shift upwards after taking reflection of base graph about x- axis.

Graph (ii): It has a horizontal shift towards left and vertical shift upwards after taking reflection of base graph about x -axis.

Hence, both the graphs are same.

5. $f(x)={(x+4)}^3+2$

Calculation for graph:

Consider $f(x)={(x+4)}^3+2$

Values of x	Values of f (x )
0	66
1	127
-1	29
-2	10
-3	3

Using the above table the graph can be plotted.

Graph:

  

Comparison:

Now, compare above graph and given graph side by side.

(i) (ii)

Interpretation:

After comparing both the graphs, it can be interpreted that both the graph are rigid-transformation of base graph of $y=x^3$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the left and vertical shift upwards .

Graph (ii): It has a horizontal shift towards left and vertical shift upwards after taking reflection of base graph about x -axis.

Hence, both the graphs are not same.

6. $f(x)={(-x+4)}^3+2$

Calculation for graph:

Consider $f(x)={(-x+4)}^3+2$

Values of x	Values of f (x )
0	66
1	29
-1	127
2	10
3	3

Using the above table the graph can be plotted.

Graph:

  

Comparison:

Now, compare above graph and given graph side by side.

(i) (ii)

Interpretation:

After comparing both the graphs, it can be interpreted that both the graph are rigid-transformation of base graph of $y=x^3$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the right and vertical shift upwards after taking reflection of base graph about x- axis.

Graph (ii): It has a horizontal shift towards left and vertical shift upwards after taking reflection of base graph about x -axis.

Hence, both the graphs are not same.

Conclusion:

Out of all the given functions, only (b) and (d) are the identical functions which can represent the given graph.