Essential Questions

1. Essential Questions • How do we use properties of end behavior to analyze, describe, and graph polynomial functions? • How do we identify and use maxima and minima of polynomial functions to solve problems?

2. Polynomial functions are classified by their degree. The graphs of polynomial functions are classified by the degree of the polynomial. Each graph, based on the degree, has a distinctive shape and characteristics.

3. End behavior is a description of the values of the function as x approaches infinity (x +∞) or negative infinity (x –∞). The degree and leading coefficient of a polynomial function determine its end behavior. It is helpful when you are graphing a polynomial function to know about the end behavior of the function.

5. Example 1: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. Q(x) = –x4+ 6x3 – x + 9 The leading coefficient is –1, which is negative. The degree is 4, which is even.

6. Example 2: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. P(x) = 2x5+ 6x4 – x + 4 The leading coefficient is 2, which is positive. The degree is 5, which is odd.

7. Example 3: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. P(x) = 2x4+ 3x2 – 4x –1 The leading coefficient is 2, which is positive. The degree is 4, which is even.

8. Example 4: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. S(x) = –3x3+ x+ 1 The leading coefficient is –3, which is negative. The degree is 3, which is odd.

9. Example 5: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. and an odd degree. P(x) has a negative leading coefficient

10. Example 6: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. and an even degree. P(x) has a positive leading coefficient

11. Example 7: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. and an odd degree. P(x) has a negative leading coefficient

12. Example 8: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. and an even degree. P(x) has a positive leading coefficient

13. Now that you have studied factoring, solving polynomial equations, and end behavior, you can graph a polynomial function.

14. Example 9: Graphing Polynomial Functions Graph the function f(x) = x3 + 4x2 + x – 6. Step 1Find the roots of the equations. The y-intercept is -6. These are the x-intercepts.

15. Example 9: Graphing Polynomial Functions Graph the function f(x) = x3 + 4x2 + x – 6. Step 2Determine end behavior of the graph. The leading coefficient is 1, which is positive. The degree is 3, which is odd. Because all of the zeros are multiplicity of 1, they go straight through the x-axis

16. Example 10: Graphing Polynomial Functions Graph the function f(x) = x3 – 2x2 – 5x + 6. Step 1Find the roots of the equations. The y-intercept is 6. These are the x-intercepts.

17. Example 10: Graphing Polynomial Functions Graph the function f(x) = x3 – 2x2 – 5x + 6. Step 2Determine end behavior of the graph. The leading coefficient is 1, which is positive. The degree is 3, which is odd. Because all of the zeros are multiplicity of 1, they go straight through the x-axis

18. Example 11: Graphing Polynomial Functions Graph the function f(x) = – 2x2 – x + 6. Step 1Find the roots of the equations. These are the x-intercepts. The y-intercept is 6.

19. Example 11: Graphing Polynomial Functions Graph the function f(x) = – 2x2 – x + 6. Step 2Determine end behavior of the graph. The leading coefficient is -2, which is negative. The degree is 2, which is even. Because all of the zeros are multiplicity of 1, they go straight through the x-axis

20. A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing. A turning point corresponds to a local maximum or minimum.

21. Lesson 4.2 Practice A