Higher Order Polynomials |
Contents: This page corresponds to § 3.2 (p. 255) of the text.
Suggested Problems from Text:
p. 263 #1-8, 11, 14, 16, 18, 19, 21, 23, 24, 30, 33, 37, 38, 75
Graphs
Leading Coefficient Test
The graphs of all polynomial functions have two important properties.
1. Polynomial functions are continuous, which means that their graphs do not have breaks or jumps. The graph of a continuous function can be drawn without lifting your pencil from the paper.
Continuous
Discontinuous
2. Graphs of polynomial functions are smooth, which means that they have no sharp corners.
Smooth
Not Smooth
In the last section we saw how to get an accurate sketch of the graph of any quadratic polynomial function. Quadratic functions are easy to graph because their graphs are always parabolas, so they have the same basic shape. For higher degree polynomials the situation is more complicated.
The applets Cubic and Quartic below generate graphs of degree 3 and degree 4 polynomials respectively. These applets use the fact that 4 points determine a degree 3 polynomial function and 5 points determine a degree 4 polynomial function. As you drag the points indicated in the graphs, the function and graph are updated.
As you drag points in the Quartic applet, you see that degree 4 polynomial graphs can have a variety of shapes.
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Cubic (degree 3) |
Quartic (degree 4) |
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Monomials of the form xn have graphs that can be sketched easily even when n is larger than 2. To begin with, all such graphs go through the origin (0, 0) and the point (1, 1). [Why do you think is is true?]
When n is even, the graph of xn is symmetric with respect to the y-axis and contains the point (-1, 1).
When n is odd, the graph of xn is symmetric with respect to the origin and contains the point (-1, -1).
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n even |
n odd |
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The graphs of xn when n is even all have a similar shape, the difference is in steepness. For larger n, the graph is flatter at (0,0) and steeper at (1,1) and (-1,1). When n is odd the graphs still have similar shapes and for larger n the graphs are flatter at (0,0) and steeper at (-1,-1) and (1,1).
Using these general characteristics we can make good sketches of graphs of xn as well as graphs obtained from these by reflecting, stretching/shrinking, or shifting.
Example 1.
Sketch the graphs of y = x3 and y = (x - 2)3 - 4.
The second graph is obtained from the first by shifting 2 units to the right and 4 units down.
Example 2.
Sketch the graphs of y = x4 and y = 3 - x4.
The second graph is obtained from the first by reflecting through the x-axis, then shifting 3 units up. This is more obvious if you rewrite the second equation as y = -x4 + 3.
Sketch the graphs of y = x5 and y = (x + 2)5 - 1 without using a graphing utility. Use a graphing utility to check your answer.
Look again at the examples of polynomial graphs that we have drawn and at the Cubic and Quartic applets. Notice that as x approaches infinity or - infinity, the function values also become unbounded. So, the points on the graph either continue upward or continue downward as we move to the right or left.
So, the graph below could not be the graph of a polynomial, because the points on the graph approach the dotted line y = 1 as x goes to infinity.
Graphs of polynomials either "rise to the right" or they "fall to the right", and they either "rise to the left" or they "fall to the left."
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rise to left and fall to right |
rise to left and rise to right |
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Note: When we think of the graphs above as "rising to the left," we are thinking of moving along the from right to left. In both cases the graphs are decreasing on interval of the form (-inf, a). Remember that when we are checking to see if a graph is increasing or decreasing, we always move from left to right.
The behavior of a polynomial graph as x goes to infinity or negative infinity is determined by the leading coefficient, which is the coefficient of the highest degree term.
Leading Coefficient Test:
Let f be the polynomial function f(x) = anxn + ... + a1x + a0, so an is the leading coefficient.
1. When n is odd:
If an is positive, then the graph of f falls to the left and rises to the right.
You can remember this by using the example x3.
If an is negative, then the graph of f rises to the left and falls to the right.
You can remember this by using the example -x3.
So, when n is odd, the graph goes in a different direction when x goes left than when x goes right.
2. When n is even:
If an is positive, then the graph of f rises to the left and rises to the right.
You can remember this by using the example x2.
If an is negative, then the graph of f falls to the left and falls to the right.
You can remember this by using the example -x2.
So, when n is even, the graph goes in the same direction when x goes left and right.
(a) Go to the Cubic applet above and drag points until you have a graph that rises to the left and falls to the right. Look at the leading coefficient of the function formula in the text box below the graph and note that it is negative.
(b) Go to the Quartic applet above and drag points until you have a graph that rises to the left and rises to the right. Look at the leading coefficient and note that it is positive.
