If you have an arbitrary polynomial P(x) of the n’th degree in x, and you graph the equation y = P(x), describe the overall shape of this graph. What happens when x gets large in the positive and negative directions, and how many peaks and valleys does the graph have? How many times does the graph cross the x-axis? (To simplify, let the sign of the highest power term in P(x) be positive.) Hint: See chapter 11 of the text.
I recommend that you read chapter 11 and take a stab at this yourself.
In general, what happens at very large values of x and -x depends upon the sign of the highest order term, x^n, and whether n is evn or odd. The higher the value of n, the more times the function goes up and down, but the number of maxima, minima and zeroes will depend upon the coefficients of all terms. An nth order polynomial can have n roots, but some may be equal and some may be complex numbers. Complex roots come in conjuagate pairs.
When graphing the equation y = P(x), where P(x) is an arbitrary polynomial of degree n, the overall shape of the graph depends on the degree of the polynomial and the leading coefficient (the coefficient of the highest power term).
1. Degree of the Polynomial:
- If the degree of the polynomial (n) is even, the graph will have a similar overall shape to a parabola, either concave up or concave down.
- If the degree of the polynomial (n) is odd, the graph will not have the same symmetry as a parabola and may have multiple peaks and valleys.
2. Leading Coefficient:
- If the leading coefficient (the coefficient of the highest power term) is positive, the graph will open upward if n is even or have a positive slope at the ends if n is odd.
- If the leading coefficient is negative, the graph will open downward if n is even or have a negative slope at the ends if n is odd.
As x gets large in the positive direction:
- If the degree of the polynomial (n) is even, the values of y will also get large in the positive direction if the leading coefficient is positive, or get large in the negative direction if the leading coefficient is negative.
- If the degree of the polynomial (n) is odd, there is no fixed behavior for y as x gets large in the positive direction. The graph may have multiple peaks and valleys depending on the specific polynomial.
As x gets large in the negative direction:
- The behavior of the graph as x gets large in the negative direction is the same as when x gets large in the positive direction. It depends on the degree and leading coefficient of the polynomial.
The number of peaks and valleys:
- The number of peaks and valleys in the graph corresponds to the number of local maximum and minimum points on the graph.
- For a polynomial of degree n, there can be at most n - 1 peaks and valleys. This means that if the degree is even, the graph may have 0, 1, 2, ..., (n-1)/2 peaks and valleys. If the degree is odd, it may have 0, 1, 2, ..., (n-1)/2 peaks and (n-1)/2 valleys.
The number of times the graph crosses the x-axis:
- The number of times the graph crosses the x-axis is equal to the number of real roots of the polynomial.
- The Fundamental Theorem of Algebra states that a polynomial of degree n has at most n distinct complex roots, so it may have up to n real roots.
- The actual number of crossings of the x-axis will depend on the specific polynomial.
To fully analyze the graph of an arbitrary polynomial, it is necessary to determine the degree, leading coefficient, and the roots of the polynomial. This can involve factoring, using the Rational Root Theorem, or using numerical methods like graphing calculators or computer programs. The chapter 11 of your text should provide more detailed instructions and examples on how to analyze polynomial graphs.