[Maple Math] and Translations

In this section we will examine the graphs of [Maple Math] for various values of n.

Example 1

Graph [Maple Math] and [Maple Math] on the same coordinate system. Note any similarities and differences.

Solution .

[Maple Plot]

The graph of the fourth degree equation is also symmetric about the y-axis ( both are even functions) but is flatter than the parabola. Both functions touch the x-axis at the x-intercept of (0,0). The range for both functions is from zero to infinity.

The figure below shows the same graph with the axis removed.

[Maple Plot]

From the graphs we also note that for -1 < x < 1, [Maple Math] . When [Maple Math] , [Maple Math] .

Example 2

Graph [Maple Math] and [Maple Math] on the same coordinate system. Note any similarities and differences.

Solution .

[Maple Plot]

Both functions are odd (i.e. symmetric about the origin) and cross the x-axis at the x-intercept of (0, 0). The range for both functions is the set of real numbers. Near the origin the fifth degree polynomial is flatter than the cubic. For 0 <x <1, [Maple Math] . When x >1, [Maple Math] . For -1 < x < 0, [Maple Math] and for x < -1 [Maple Math]

Example 3

Graph [Maple Math]

Solution .

The graph is the result of turning [Maple Math] upside down, moving it 2 units to the right, and moving it up 3 units.

[Maple Plot]

Example 4

Graph [Maple Math]

Solution .

[Maple Plot]

Summary

If n is positive and even, then [Maple Math] is symmetric about the y-axis and touches the x-axis at the x-intercept of (0, 0). As n gets larger, the graph is flatter near (0, 0) but steeper elsewhere.

If n is positive and odd, then [Maple Math] is symmetric about the origin and crosses the x-axis at the x-intercept of (0, 0). As n gets larger, the graph is flatter near (0, 0) but steeper elsewhere.