Chapter 1.4, Problem 12E

To determine

To sketch the graph of the given functions on same co-ordinate system

Explanation of Solution

Given:

The given functions are:

  $\begin{array}{}$ f(x)={(x-2)}^2$$ g(x)={(x+2)}^2+2$$ h(x)=-{(x-2)}^2-1$\end{array}$

Calculation for graph:

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

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

By taking different values of x , the graph can be plotted.

Calculation for graph:

Consider $g(x)={(x+2)}^2+2$

Values of x	Values of g (x)
0	6
1	11
-1	3
2	18
-2	2

By taking different values of x , the graph can be plotted.

Calculation for graph:

Consider $h(x)=-{(x-2)}^2-1$

Values of x	Values of h (x )
0	-5
1	-2
-1	-10
2	-1
-2	17

By taking different values of x , the graph can be plotted.

Graphs on the same co-ordinate system:

  

Verification:

Using graphing utility to draw the graph of given functions,

  

Interpretation:

From above graph, it is clear that, the drawn sketch and above graph are identical to each other.

Here, $y=x^2$ is the base graph.

  $f(x)$ is the rigid transformation of base graph with a horizontal shift 2 units to the right.

  $g(x)$ is the rigid transformation of base graph with a horizontal shift of 2 units to the left and vertical shift of 2 units upwards.

  $h(x)$ is the rigid transformation of base graph and is reflection of the same about x-axis with a horizontal shift of 2 units to the right and vertical shift of 1 unit downward..