Chapter 1.4, Problem 11E

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}{l}f(x)=-x^2\\ g(x)=-x^2+1\\ $ h(x)=-{(x+3)}^2$\end{array}$

Calculation for graph:

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

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

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

Calculation for graph:

Consider $g(x)=-x^2+1$

Values of x	Values of g (x)
0	1
1	0
-1	0
2	-3
-2	-3

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

Calculation for graph:

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

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

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 and is the reflection of the same about x -axis.

  $g(x)$ is the rigid transformation of base graph and is reflection of the same about x -axis with a vertical shift of 1 unit upward.

  $h(x)$ is the rigid transformation of base graph and is reflection of the same about x -axis with a horizontal shift of 3 units to the left.