Published on: **Mar 4, 2016**

Published in:
Education

- 1. Smooth, Continuous Graphs Two important features of the graphs of polynomial functions are that they are smooth and continuous. By smooth, we mean that the graph contains only rounded curves with no y y sharp corners. By Smooth rounded Smooth rounded continuous, we mean corner corner that the graph has no breaks and can be drawn without lifting x x your pencil from the rectangular coordinate system. These ideas are Smooth Smooth rounded illustrated in the figure. corner rounded corner
- 2. Graphs of polynomials are smooth and continuous. No sharp corners or cusps No gaps or holes, can be drawn without lifting pencil from paper This IS the graph This IS NOT the graph of a polynomial of a polynomial
- 3. and LEFT RIGHT HAND BEHAVIOUR OF A GRAPH The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behaviour of a graph.
- 4. The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f (x) anxn an-1xn-1 an-2xn-2 … a1x a0 (an 0) eventually rises or falls. In particular, 1. For n odd: an 0 an 0 If the If the leading leading Rises coefficient is Rises coefficient is right negative, the left positive, the graph rises graph falls to to the left the left and and falls to rises to the the right. Falls right. Falls right left
- 5. The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f (x) anxn an-1xn-1 an-2xn-2 … a1x a0 (an 0) eventually rises or falls. In particular, 1. For n even: an 0 an 0 If the If the leading leading Rises coefficient is coefficient is right negative, the positive, the Rises graph falls to graph rises left the left and to the left to the right. and to the Falls right. left Falls right
- 6. Text Example Use the Leading Coefficient Test to determine the end behavior of the graph of Graph the quadratic function f(x) x3 3x2 x 3. Rises right y Solution Because the degree is odd (n 3) and the leading coefficient, 1, is positive, the graph falls to the left and rises to the right, as shown in the figure. x Falls left
- 7. Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term. left hand right hand behaviour: rises behaviour: rises
- 8. Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative. turning points in the middle left hand behaviour: falls right hand behaviour: falls
- 9. Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive. turning Points in the middle right hand behaviour: rises left hand behaviour: falls
- 10. Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative. turning points in the middle left hand behaviour: rises right hand behaviour: falls