and Translations
In this section we will examine the graphs of
for various values of n.
Example 1
Graph
and
on the same coordinate system. Note any similarities and differences.
Solution .
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.
From the graphs we also note that for -1 < x < 1,
. When
,
.
Example 2
Graph
and
on the same coordinate system. Note any similarities and differences.
Solution .
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,
. When x >1,
. For -1 < x < 0,
and for x < -1
Example 3
Graph
Solution .
The graph is the result of turning
upside down, moving it 2 units to the right, and moving it up 3 units.
Example 4
Graph
Solution .
Summary
If n is positive and even, then
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
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.