Chapter 1.4, Problem 14E

To determine

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

Explanation of Solution

Given:

The given functions are:

  $\begin{array}{l}f(x)=x^2\\ g(x)=\frac{1}{4}{x}^2+2\\ h(x)=-\frac{1}{4}{x}^2\end{array}$

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 $g(x)=\frac{1}{4}{x}^2+2$

Values of x	Values of g (x)
0	2
1	2.25
-1	2.25
2	3
-2	3

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

Calculation for graph:

Consider $h(x)=-\frac{1}{4}{x}^2$

Values of x	Values of h (x )
0	0
1	-0.25
-1	-0.25
2	-1
-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, $f(x)=x^2$ is the base graph.

  $g(x)$ is the non-rigid transformation of base graph with vertical shrink and vertical shift of 2 unit upwards.

  $h(x)$ is the non-rigid transformation of base graph with vertical shrink and reflection of base graph about x -axis.