Chapter 1.4, Problem 47E

To determine

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

Explanation of Solution

Given:

The given functions are:

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

Calculation:

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

  $\begin{array}{l}h(x)=\frac{1}{2}f(x)\\ \Rightarrow h(x)=\frac{1}{2}(x^3-3x^2)\end{array}$

Calculation for graph:

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

Values of x	Values of f (x )
1	-2
2	-4
3	9
0	0
-1	-4

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

Calculation for graph:

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

Values of x	Values of g (x )
-2	0
-1	-2
0	-4
1	0
2	16

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

Calculation for graph:

Consider $h(x)=\frac{1}{2}(x^3-3x^2)$

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

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

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

  $h(x)$ is the non-rigid transformation of base graph, with vertical shrink of (1/2) times.