Polynomials are expressions involving x raised to a whole number power (exponent).
Degree of a polynomial: The highest power (exponent) of x.
Relative maximum: The point(s) on the graph which have maximum y values or second coordinates “relative” to the points close to them on the graph.
Relative minimum: The point(s) on the graph which have minimum y values or second coordinates “relative” to the points close to them on the graph.
Absolute maximum: The point on the graph which has the largest value of y (the second coordinate of the point).
Absolute minimum: The point on the graph which has the smallest value of y (the second coordinate of the point).
We shall refer to the degree and maximum and minimum points frequently in discussing the graphs of polynomials in this lesson.Finding minimum and maximum values of a polynomials accurately:
Important points on a graph of a polynomial include the x- and y-intercepts, coordinates of maximum and minimum points, and other points plotted using specific values of x and the associated value of the polynomial. This lesson will focus on the maximum and minimum points. These points are sometimes referred to as max, min, extreme values, or extrema. Locate the maximum or minimum points by using the TI-83 calculator under and the “3.minimum” or “4.maximum” functions. Let's Practice:
Some of the examples below are also discussed in the Graphing Polynomials lesson. It is highly recommended that the reader review that lesson to have a greater understanding of the graphs in these examples. Max/min of polynomials of degree 2:
Polynomials of degree greater than 2:
- is a parabola and its graph opens upward from the vertex . The vertex can be found at . In this case, a = 3 and b = -1 which gives . The minimum value of the polynomial is . The graph is shown at right using the WINDOW (-5, 5) X (-2, 16). Of course this vertex could also be found using the calculator.
- is a parabola and its graph opens downward from the vertex (1, 3) since . The graph is shown at right using the WINDOW (-5, 5) X (-8, 8). The maximum point is found at x = 1 and the maximum value of P(x) is 3. The coordinates of this point could also be found using the calculator.
Polynomials of degree greater than 2 can have more than one max or min value. The largest possible number of minimum or maximum points is one less than the degree of the polynomial. The following examples illustrate several possibilities. In each case, the accompanying graph is shown under the discussion.
- A polynomial of degree higher than 2 may open up or down, but may contain more “curves” in the graph. Notice in the case of . The minimum is located at x = -2.25 and the minimum value is approximately -4.54. There is only one minimum and no maximum point. The value -4.54 is the absolute minimum since no other point on the graph is lower.
- Let There are two minimum points on the graph at (0.70, -0.65) and (-1.07, -2.04). The minimum value of -0.65 at x = 0.70 is called a relative minimum because it is not the minimum or absolute, smallest value of the function. It is a minimum value “relative” to the points that are close to it on the graph. The minimum value of -2.04 at x = -1.07 is called an absolute minimum because it is the smallest value of P(x). There is a maximum at (0, 0). This maximum is called a relative maximum because it is not the maximum or absolute, largest value of the function. It is a maximum value “relative” to the points that are close to it on the graph.
- Let There are two maximum points at (-1.11, 2.12) and (0.33, 1.22). There is a minimum at (-0.34, 0.78). The maximum value of 2.12 at x = -1.11 is the absolute maximum. The maximum value of 1.22 at x = 0.33 is a relative maximum, being the largest value “relative” to points close to this on the graph. The minimum value of 0.78 at x = -0.34 is a relative minimum, being the smallest value “relative” to points close to this on the graph.
- Let . There is a relative minimum value of -0.34 at approximately x = 0.24 and a relative maximum value of 6.05 at approximately x = 2.09.
- If , the graph opens up to the right and down to the left because the leading coefficient is positive. The y-intercept is -2, the constant term. The x-intercepts are located at -1, -0.5, 0.5, 1, and 2. Other points which can assist in making an accurate sketch are located at (-0.25, -1.58), (0.25, -1.23), and (1.5, -5).
Notice that there are two relative maxima and two relative minima. These four points can occur because P(x) is a polynomial of degree 5. The maximum points are located at x = 0.77 and -0.80. The maximum values at these points are 0.69 and 1.57 respectively. The minimum points are located at x = -0.05 and 1.68. The minimum values are -2.02 and -6.00 respectively.