Question

how to demonstrate a graph of a four degree polynomial function

Answer 1

Up until this point, we have limited our discussion to linear and quadratic functions. These are special cases of what are known as polynomials. Our objective is to learn more about polynomials and their graphs. We begin by defining a polynomial function.







Polynomial Function

A polynomial of degree n is a function of the form



f(x) = anxn + an 1xn 1 + … + a2x2 + a1x + a0



where each coefficient ak is a real number, an ≠ 0, and n is a non-negative integer. The leading coefficient is an and the degree is n.








Below are the graphs of some degree 3 and degree 4 polynomials. Notice that both are smooth (like a parabola), but have more curves. We shall examine how many curves a polynomial can have a little later on.





Figure 1: Graphs of degree 3 and degree 4 polynomials





Example 1:



What are the degree and leading coefficients of the following polynomials?



(i) 10x3 + 4x2 + 3x + 1 and (ii) 2x4 + 3x + 5



Solution:



For (i), we see that the degree is 3 and the leading coefficient is 10. For (ii), we see that the degree is 4 and the leading coefficient is 2.

At this point, it is useful to introduce two more concepts about functions that will help us to describe their graphs better.







Increasing and Decreasing Functions

Suppose that f(x) is a function defined over an interval I on the number line. If x1 and x2 are in I, then we say that



f(x) increases on I if, whenever x1 < x2, f(x1) < f(x2)



and



f(x) decreases on I if, whenever x1 < x2, f(x1) > f(x2)










Figure 2: Increasing and Decreasing Functions





Example 2:



Use the graph of f(x) = 12x x3 (shown below) to identify the intervals where f(x) is increasing and decreasing.





Figure 3: Graph of f(x) = 12x x3



Solution:



From the graph, we see that f(x) is increasing on the interval (-2, 2) and decreasing on the intervals (-∞, -2) (2, ∞).





Let us take a closer look at the graph above. The places where the graph changed from increasing to decreasing and from decreasing to increasing are of particular importance in calculus. We shall call those points turning points, since the graph changes direction. In calculus, one can show that a polynomial of degree n has at most n 1 turning points.



While we are considering graphs of polynomials, it is worth mentioning some additional concepts. In particular, notice that there were two types of turning points. One that appeared at “the top of the hill” and the other which appeared at “the bottom of the valley”. More formally, we call those points a local maximum and a local minimum of the graph.



We may also be interested in determining the largest or smallest value of the function overall. We call those points the absolute maximum and absolute minimum, respectively.



To recap, we have the following:







Absolute and Local Extrema

Suppose c is in the domain of f(x). Then



(i) f(c) is an absolute (global) maximum if f(c) ≥ f(x) for all x in the domain of f(x)

(ii) f(c) is an absolute (global) minimum if f(c) ≤ f(x) for all x in the domain of f(x)

(iii) f(c) is a local (relative) maximum if f(c) ≥ f(x) when x is near c

(iv) f(c) is a local (relative) minimum if f(c) ≤ f(x) when x is near c



Note: By “near c”, we mean that there is an open interval in the domain of f(x) containing c, where f(c) satisfies the stated inequality.








Example 3:



Below is the graph of f(x) on the interval [-3, 4]. Use the graph to identify the points where f(x) has a local maximum and local minimum. Also identify the absolute maximum and absolute minimum of f(x) on the interval.









Figure 4: Graph of f(x) on [-3, 4]



Solution:



We observe that there is a local maximum at the point (1, 37) (with a value of 37). There are two local minimums, occurring at (-2, -152) and (3, -27) (with values -152 and -27, respectively). On the interval [-3, 4], the maximum value is 64 (and occurs at (4, 64)) and the minimum value is -152 (and occurs at (-2, -152)).





Finally, we turn our attention to graphing polynomials. If we have a polynomial in factored form, we can make a rough sketch of the graph by considering the following three steps.


1.List all of the x-intercepts of the graph and (lightly) draw vertical lines at those intercepts.
2.Choose points to the left and to the right of each intercept and determine if the function is positive or negative. (Whenever possible, it is preferable to choose integers, as the math is usually simpler.) Plot the points on the graph. (You may wish to shade the area in between the x-intercepts that the chosen point does not pass through. These are known as the excluded regions.)
3.Finally, draw a smooth curve between the points you have drawn.





This is best illustrated through an example.





Example 4:



Sketch a graph of f(x) = (x + 4)(x 1)(x 4).



Solution:



We begin by plotting the x-intercepts of the graph (which occur at x = -4, 1, and 4) and draw in light vertical lines at those points.



Figure 5: Plot of x-intercepts



Next, we choose points in between the dashed lines. For the sake of simplicity, we shall choose x = -5, x = -2, x = 2, and x = 5. Notice that f(-5) = -54, f(-2) = 36, f(2) = -12, and f(5) = 36. Plotting those points on the graph, we have the following.





Figure 6: x-intercepts and some points



We place shaded rectangles to show the places where the graph does not pass through. Also, at this point, we can remove our vertical lines. Doing so, we have the following:





Figure 7: x-intercepts, some points and excluded regions



Finally, we draw a smooth curve that connects the points, making sure not to cross into any of the shaded regions. Doing this, we have:





Figure 8: Rough sketch of f(x)



And finally, we remove our shaded regions to obtain our final graph.





Figure 9: Final sketch of f(x)





In the example above, the polynomial had no repeated factors. That is, none of the factors is squared or cubed or so on. But what happens if there are repeated factors. Again, we shall follow the three steps listed above. But there is a fact from calculus that will aid us.







The Behavior of a Polynomial Function near an x-intercept

Let f(x) be a polynomial and suppose that (x a)n is a factor of f(x). Then, in the immediate vicinity of the x-intercept at a, the graph of y = f(x) closely resembles that of y = A(x a)n.










Drawn below are the graphs of y = (x + 1)(x 1)(x 2), y = (x + 1)2(x 1)(x 2), and y = (x + 1)3(x 1)(x 2).





Figure 10: Behavior changes with repeated factors





In the first graph, notice that near the x-intercept of -1, the function looks like a straight line. In the second graph, near the x-intercept of -1, the function looks like a parabola, and in the third graph, near the x-intercept of -1, the function looks like a cubic.