Chapter 1.4, Problem 50E

To determine

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

Explanation of Solution

Given:

The given functions are:

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

Calculation:

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

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

Calculation for graph:

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

Values of x	Values of f (x )
-2	-18
-1	-2
0	2
1	0
2	-2

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

Calculation for graph:

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

Values of x	Values of g (x )
-2	52
-1	18
0	2
1	-2
2	0

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

Calculation for graph:

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

Values of x	Values of h (x )
-2	-110
-1	-18
0	2
1	-2
2	18

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:

Here, $f(x)=x^3-3x^2+2$ is the base graph.

  $g(x)$ is the rigid transformation of base graph which is the reflection of base graph about x -axis.

  $h(x)$ is the non-rigid transformation of base graph, with horizontal shrink of (2) times.