Definitions:

• Polynomials are expressions involving x raised to a whole number power (exponent).

  Some examples are:

  
  
• Degree of a polynomial: The highest power (exponent) of x.
  
  
• Leading coefficient: The coefficient of the highest power of x.

  In the case of it is +3.

• Constant term: The number not associated with any power of x.

  In the case of it is +7.

We shall refer to the degree, leading coefficient, and the constant term frequently in discussing the graphs of polynomials.

Graphing polynomials accurately:

We will refer to ways that a calculator can assist in graphing as well as which important points to graph accurately. Important points 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. Locate the y-intercept by letting x = 0 (the y-intercept is the constant term) and locate the x-intercept(s) by setting the polynomial equal to 0 and solving for x or by using the TI-83 calculator under and the 2.zero function. The intercepts provide accurate points to help in sketching the graphs.

Locate the maximum or minimum points by using the TI-83 calculator under and the 3.minimum or 4.maximum functions.

Graphing polynomials of degree 2:

1. is a parabola and its graph opens upward from the vertex .
   The graph is shown below using the WINDOW (-5, 5) X (-2, 16). Note that the leading coefficient is positive and that is why the parabola opens upward. Notice that the constant term 7 is the y-intercept. The y-intercept is the point where x = 0.
   
   It is a good practice to plot points near the minimum point which in this case has approximate coordinates (0.17, 6.92).

If
If .

The points (-1, 11) and (2, 17) can be plotted to sketch a more accurate graph.



2. is a parabola and its graph opens downward from the vertex (1, 3).
   
   The graph is shown below using the WINDOW (-5, 5) (-8, 8). Note that the leading coefficient is negative and that is why the parabola opens down. Notice that the constant term 1 is the y-intercept. The maximum point is located at (1, 3). Points near this are (-1, -5) and (2, 1) which can help in sketching the graph.

   

In general, a parabola (polynomial of degree 2) is given by . If a is positive, the parabola opens up and if a is negative, the parabola opens down.

Polynomials of even degree greater than 2:

Polynomials of even degree open up or down depending on whether the leading coefficient is positive or negative. The end behavior is the same for both the left and right sides of the graph.

1. 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 that the graph opens up both on the left and right sides of the graph. The degree of this polynomial is 4 and is even. Thus both the left and right sides of the graph show the same end behavior, opening up.
   
   The y-intercept is 4 and the x-intercepts are -2.82 and -1.34 approximately. The minimum is approximately (-2.25, -4.54). Other points in this window which assist in graphing accurately are (-2, -4), (-1, 2), and (1, 8). It might be necessary to plot points for x values of -0.5 and 0.5 for an accurate sketch. These points are (-0.5, 3.6875) and (0.5, 4.4375).



2. Let . We expect the behavior of the graph to be the same on both the left and right sides of the graph because the degree of is even. Because the leading coefficient is positive, the graph will be going upward. The graph is shown below.
   
   The y-intercept is 0 and the x-intercepts are -1.5, 0, and 1. There are two minimum points on the graph at (0.70, -0.65) and (-1.07, -2.04). There is a maximum at (0, 0). Other points on the graph which can help sketch an accurate graph are (-2, 12), (-1, -2), and (1.5, 6.75).



3. Let We expect the end behavior of the graph to be the same on both the left and right sides of the graph. Because the leading coefficient is negative, the graph will be going downward. The graph is shown below.
   
   The y-intercept is 1 and the x-intercepts are approximately -1.42 and 0.8. There are two maximum points at (-1.11, 2.12) and (0.33, 1.22). There is a minimum at (-0.34, 0.78). Other points on the graph are (-1, 2), and (1, -2).

   

Polynomials of odd degree greater than 2:

Polynomials of odd degree open up or down depending on whether the leading coefficient is positive or negative. However, the end behavior is different for the left and right sides of the graph. If the right side of the graph goes up, the leading coefficient is positive and the left side will go down. If the right side goes down, the leading coefficient is negative and the left side goes up.

1. 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 graph opens up to the right and down to the left. This is because the leading coefficient is positive.
   
   The y-intercept is 4 and is also a minimum point. A maximum is found at (-2, 8). The x-intercept is approximately -3.36. Other points which can assist in sketching this graph accurately are (-3, 4), (-1, 6), and (1, 8).



2. If , the graph opens down to the right and up to the left because the leading coefficient is negative.
   
   The y-intercept is 0. The x-intercepts are located at 0, 0.5, and 3. Other points on the graph are (-0.5, 3.5), (1, 2), and (1.5, 4.5). There is a minimum at approximately (0.24, -0.34) and a maximum at approximately (2.09, 6.05).



3. 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).