Chapter 1.4, Problem 75E

To determine

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

Answer to Problem 75E

Option (c)

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

Calculation for graph:

Consider $f(x)=-\sqrt{x}-4$

Values of x	Values of f (x )
0	2
1	-1
2	0
3	-1
4	-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 $y=x^2$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the right, vertical shift downwards and does not pass through the point $(0,0)$

Graph (ii): It has a horizontal shift towards right, vertical shift downwards and passes though the point $(0,0)$

Hence, both the graphs are not same.

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

Calculation for graph:

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

Values of x	Values of f (x )
0	12
1	21
-1	5
-4	-4
2	32

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 $y=x^2$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the right, vertical shift downwards and does not pass through the point $(0,0)$

Graph (ii): It has a horizontal shift towards right, vertical shift downwards and passes though the point $(0,0)$ .

Hence, both the graphs are not same.

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

Calculation for graph:

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

Values of x	Values of f (x )
0	0
-1	5
2	-4
1	-3
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 $y=x^2$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the right, vertical shift downwards and pass through the point $(0,0)$

Graph (ii): It has a horizontal shift towards right, vertical shift downwards and passes though the point $(0,0)$

Hence, both the graphs are same.

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

Calculation for graph:

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

Values of x	Values of f (x )
0	0
2	12
1	5
3	21
-1	-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 $y=x^2$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the left, vertical shift downwards and pass through the point $(0,0)$

Graph (ii): It has a horizontal shift towards right, vertical shift downwards and passes though the point $(0,0)$

Hence, both the graphs are not same.

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

Calculation for graph:

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

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

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 $y=x^2$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the right, vertical shift upwards after taking reflection of base graph about x- axis and pass through the point $(0,0)$ .

Graph (ii): It has a horizontal shift towards right, vertical shift downwards and passes though the point $(0,0)$ .

Hence, both the graphs are not same.

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

Calculation for graph:

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

Values of x	Values of f (x )
0	0
1	-5
-1	3
2	-12
3	-21

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 $y=x^2$ with certain shifts in both horizontal and vertical shift.

Graph (i): It has horizontal shift to the left, vertical shift upwards after taking reflection of base graph about x- axis and pass through the point $(0,0)$

Graph (ii): It has a horizontal shift towards right, vertical shift downwards and passes though the point $(0,0)$

Hence, both the graphs are not same.

Conclusion:

Out of all the given functions, only (c) is the identical functions which can represent the given graph