Name LESSON 6-7 Date Class Reteach Investigating Graphs of Polynomial Functions Examine the sign and the exponent of the leading term (term of greatest degree) of a polynomial P! x " to determine the end behavior of the function. Even degree functions: Exponent of leading term is even. Read: As x approaches positive infinity, P! x " approaches negative infinity. Negative leading coefficient As x ! , P! x " ! . As x " , P! x " ! . Positive leading coefficient As x ! , P! x " " . As x " , P! x " " . Y Y X X Example: P! x " # 3x " 2x ! 5 Leading term: 3x 4 End behavior: As x ! , P! x " " . As x " , P! x " " . 4 3 Sign: positive Degree: 4, even Odd degree functions: Exponent of leading term is odd. Negative leading coefficient As x ! , P! x " " . As x " , P! x " ! . Positive leading coefficient As x " , P! x " " . As x ! , P! x " ! . Y Y X X Example: P! x " # !2x 5 ! 6x 2 " x Leading term: !2x 5 End behavior: As x ! , P! x " " . As x " , P! x " ! . Sign: negative Degree: 5, odd Identify the end behavior of each function. 1. P! x " # 4x 3 " 8x 2 ! 5 Leading term: 4x as x 6 3 as x !$, P!x" "$ , P ! x " Copyright © by Holt, Rinehart and Winston. All rights reserved. a207c06-7_rt.indd 54 3 !9x 6 Leading term: Positive, 3, odd Sign and degree: End behavior: 2. P! x " # !9x " 2x ! x " 7 Process Black Sign and degree: !$ End behavior: as x "$ 54 negative, 6, even as x !$, P!x" "$, P ! x " !$ !$ Holt Algebra 2 12/29/05 8:29:20 PM Name LESSON 6-7 Date Class Reteach Investigating Graphs of Polynomial Functions (continued) You can use the graph of a polynomial function to analyze the function. P ! x ": odd degree and positive leading coefficient if: Y Notice the graph increases, then decreases, and then increases again. As x as x ! , P ! x " ! and " , P !x " x " . X P ! x ": odd degree and negative leading coefficient if: Y Notice this graph is the reverse. It decreases, then increases, and then decreases again. As x as x ! , P !x " x " , P !x " x " and ! . X P ! x ": even degree and positive leading coefficient if: Y Look at the end behavior. The graph increases at both ends. As x as x ! , P !x " x " , P !x " x " and " . X P ! x ": even degree and negative leading coefficient if: Y This is the reverse. The graph decreases at both ends. As x as x ! , P !x " " , P !x " ! and ! . X Identify whether each function has an odd or even degree and a positive or negative leading coefficient. 3. Y 4. Y 5. X Y X X Even; positive Copyright © by Holt, Rinehart and Winston. All rights reserved. a207c06-7_rt.indd 55 Process Black Odd; negative 55 Odd positive Holt Algebra 2 12/29/05 8:29:24 PM