1. 6.2: E VALUATING AND GRAPHING POLYNOMIAL FUNCTIONS Objectives: Students will be able to identify, evaluate and graph a polynomial function.

2. P OLYNOMIAL F UNCTIONS exponents are whole numbers (positive) coefficients are real numbers (Notice exponents are in descending order: STANDARD FORM) -2: Leading coefficient 4: Degree -7: Constant term

3. D EGREE OF A POLYNOMIAL Exponent of the leading term when in standard form. SUMMARY: (after 4, say “degree 5, degree 6, etc…) Degree:Type:Example: 0Constant f(x) = 2 1Linear f(x) = x + 3 2Quadratic f(x) = 2x 2 +3x+7 3Cubic f(x) = x 3 +x 2 +x+5 4Quartic f(x) = x 4 +3x 3 +2x+x+5

4. D ECIDE WHETHER THE FUNCTION IS A POLYNOMIAL. I F SO, WRITE IN STANDARD FORM AND STATE ITS DEGREE, TYPE AND LEADING COEFFICIENT.

5. Y OU CAN EVALUATE A POLYNOMIAL FUNCTION USING : Direct substitution Synthetic substitution Direct Substitution:  replace each x with the given value and evaluate (PEMDAS!!) Evaluate f(x) = 2x 4 – 8x 2 + 5x -7 for x = 3. (same as saying find f(3)) f(3) = 2(3) 4 – 8(3) 2 + 5(3) -7 =

6. S YNTHETIC S UBSTITUTION : Write the polynomial in standard form (include all degree terms) Write coefficients ONLY (include a 0 if polynomial skips a degree) Use the given x value and follow the process on next slide.

7. E VALUATE F ( X ) = 2 X 4 – 8 X 2 + 5 X -7 FOR X = 3. USING SYNTHETIC SUBSTITUTION. 3 20 -8 5 -7 6 18 30 105 2 6 10 35 98 1.) Multiply the 2 by 3 and write in next column 2.) Add the column; repeat. 3.) The last number in bottom right corner is the value of f(3). f(3) = 98

8. U SE SYNTHETIC SUBSTITUTION TO EVALUATE : 1. f(x)=3x 5 -x 4 -5x+10 when x = -2 2. Find f(1) if f(x) = 7x 4 +3x 3 +2x 2 +x-7.

9. G RAPHING POLYNOMIALS !!! END BEHAVIOR: What happens to your function’s value as your x values get large and positive and large and negative (as x ∞ and as x - ∞) What the graph does at the ends

10. F OR THE FOLLOWING FUNCTIONS : Write in standard form. Indicate degree and leading coefficient. Graph on your calculator. Describe what is happening at x ∞ and as x - ∞

11. E ND BEHAVIOR : TO DETERMINE LOOK AT THE DEGREE AND LEADING COEFFICIENT Degree: Even Leading Coefficient : Positive End behavior: Same Example: f ( x ) = x 2

12. Degree: Even Leading Coefficient : Negative End behavior: Same Example: f ( x ) = –x 2

13. Degree: Odd Leading Coefficient : Positive End behavior: Alternates Example: f ( x ) = x 3

14. Degree: Odd Leading Coefficient : Negative End behavior: Alternates Example: f ( x ) = –x 3

15. T O GRAPH A POLYNOMIAL : Determine end behavior Make a table of values Plot points and draw a smooth curve. Things to help you graph… Degree:Max # zerosMax # turning points 0O or infinity0 110 221 332 443 n (odd)nn -1 n (even)nn -1

16. G RAPH : F ( X ) = X 3 + 2 X 2 – X + 3

17. G RAPH : F ( X ) = X 4 + 6 X 2 -5

18. The number of new words students in a language course were asked to learn each week is modeled by y =.0003x3 + 50, where x is the number of weeks since the course began. Graph the model. Use the graph to estimate the number of words the students must learn in week 32.