1. 7/6/20169-1 & 9-2 9-1& 9-2 Polynomial Models & Their Graphs

2. 7/6/20169-1 & 9-2Definitions Polynomial – An expression the form: Coefficients – The constant numbers that are multiplying the variables. (a n, a n-1, etc) Leading Coefficient – The coefficient multiplying the variable with the largest exponent. Standard Form – A polynomial with the exponents written in descending order. Polynomial Function – A function whose rule can be written as a polynomial.

3. 7/6/20169-1 & 9-2Definitions Monomial – Polynomial with 1 term Binomial – Polynomial with 2 terms Trinomial – Polynomial with 3 terms Degree of a Monomial – The sum of the exponents on the variables Degree of a Polynomial – The highest degree of any monomial in the polynomial

4. 7/6/20169-1 & 9-2 Finding Degrees Find the degree of the following monomials: Monomial:Degree:

5. 7/6/20169-1 & 9-2 Finding Degrees Find the degree of the following polynomials: PolynomialDegree PolynomialDegree

6. 7/6/20169-1 & 9-2 Extrema of Functions Maximum – The largest value of a function Minimum – The smallest value of a function Extreme Values or Extrema – Min or Max Relative Extrema – The highest or lowest point on a section of the graph

7. 7/6/20169-1 & 9-2 Extrema of Functions

8. 7/6/20169-1 & 9-2 Zeros of Poly Functions Zeros or Roots – All values of x in a polynomial function that cause the function to equal 0 (same as x-intercepts, p(x)=0) To Find Zeros: (for now) Graph the function in your calculatorGraph the function in your calculator Select 2 nd TraceSelect 2 nd Trace Select “Zero”Select “Zero” Set Left and Right BoundSet Left and Right Bound Select GuessSelect Guess

9. 7/6/20169-1 & 9-2 Finding Relative Extrema To Find Relative Extrema: Graph function in your calculatorGraph function in your calculator Select 2 nd TraceSelect 2 nd Trace Select “Maximum” or “Minimum”Select “Maximum” or “Minimum” Set Left and Right BoundSet Left and Right Bound Select GuessSelect Guess Repeat for each extreme valueRepeat for each extreme value

10. Example 7/6/20169-1 & 9-2 Using your calculator, find the extrema and the zeros of the function.

11. 7/6/20169-1 & 9-2 Increasing vs. Decreasing To determine if an interval is increasing or decreasing: Read the graph from left to right.Read the graph from left to right. If the Rate of Change of any part of the graph is positive, the function is increasing on that interval (written as an inequality in terms of the x-coordinates)If the Rate of Change of any part of the graph is positive, the function is increasing on that interval (written as an inequality in terms of the x-coordinates) If the Rate of Change of any part of the graph is negative, the function is decreasing.If the Rate of Change of any part of the graph is negative, the function is decreasing. Q: What part(s) of the function determine WHERE these intervals exist and possibly change? Let’s revisit our function….

12. 7/6/20169-1 & 9-2 Positive vs. Negative To determine if an interval is Positive or Negative: Read the graph from left to right.Read the graph from left to right. If the graph is above the x-axis, the function is positive on that interval (written as an inequality in terms of the x-coordinates)If the graph is above the x-axis, the function is positive on that interval (written as an inequality in terms of the x-coordinates) If the graph is below the x-axis, the function is negative.If the graph is below the x-axis, the function is negative. Q: What part(s) of the function determine WHERE these intervals exist and possibly change? Let’s revisit our function again….

13. 7/6/20169-1 & 9-2Example: State the intervals of: a) Increase vs. Decrease Decreasing x < a Increasing a < x < b Decreasing b < x < c Increasing c < x < d Decreasing d < x < e Increasing x > e F G H I J K

14. 7/6/20169-1 & 9-2Example: State the intervals of: b) Positive vs. Negative Positive x < f Negative f < x < g Positive g < x < h Negative h < x < i Positive i < x < j Negative j k Positive x > k F G H I J K

15. Note: What vs. Where: If the question asks “what” something is, they are asking you for the value of the function. That is the y-coordinates. If the question asks “where” something is, they are asking you for the intervals of the function. That is the x-coordinates, like we just did. 7/6/20169-1 & 9-2

16. Example7/6/2016 Using your calculator, the intervals where the function is increasing and decreasing. Also, find the intervals where the function is negative and positive.