Investigating Graphs of Polynomial Functions Section 6 7

Investigating Graphs of Polynomial Functions Section 6 -7 10/2/2020 8: 56 AM 6 -7 - Investigating Graphs of Polynomial Functions 1

Graphs of Polynomial Functions Graphs of Polynomial functions: • Are Continuous – No Breaks • Have smooth turns • With degrees of n, have at most n-1 turns. – In other words, if the polynomial has a degree of 3, there will be 2 turns. – If the polynomial has a degree of 4, there will be 3 turns, etc… 10/2/2020 8: 56 AM 6 -7 - Investigating Graphs of Polynomial Functions 2

Turns • If the first coefficient is POSITIVE, it will RISE to the right (positive x side). • If the first coefficient is NEGATIVE, it will FALL to the right. • If the top degree is an EVEN number, the graph will RISE to the left (negative x side). • If the top degree is an ODD number, the graph will FALL to the left. • Exception to the rule, if the degree has a negative coefficient, its OPPOSITE. >> FOLLOW THE ARROWS << 10/2/2020 8: 56 AM 6 -7 - Investigating Graphs of Polynomial Functions 3

Polynomials How many turns are there? Linear: 0 turns Quadratic: 1 turn 10/2/2020 8: 56 AM 6 -7 - Investigating Graphs of Polynomial Functions 4

Polynomials 2 turns 10/2/2020 8: 56 AM 3 turns 6 -7 - Investigating Graphs of Polynomial Functions 4 turns 5

END BEHAVIOR

END BEHAVIOR

END BEHAVIOR

Determining End Behavior • Degree • Sign of Leading Coefficient – Touchdown– Offsides– Disco 1– Disco 2 -

End Behavior 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. As x goes to negative infinity, P(x) goes to negative infinity 10/2/2020 8: 57 AM 6 -7 - Investigating Graphs of Polynomial Functions 10

Example 1 How many turns? 2 How would you describe the graph from negative x? 10/2/2020 8: 57 AM 6 -7 - Investigating Graphs of Polynomial Functions 11

Example 2 How many turns? 2 How would you describe the graph from positive x? 10/2/2020 8: 57 AM 6 -7 - Investigating Graphs of Polynomial Functions 12

Example 3 How many turns? 4 How would you describe the graph from negative x to positive x? 10/2/2020 8: 57 AM 6 -7 - Investigating Graphs of Polynomial Functions 13

Example 4 How many turns? 3 How would you describe the graph from negative x to positive x? 10/2/2020 8: 57 AM 6 -7 - Investigating Graphs of Polynomial Functions 14

Example 5 Identify the end behavior of P(x) = 2 x 5 + 3 x 2 – 4 x – 1 As x –∞, P(x) –∞. As x +∞, P(x) +∞. 10/2/2020 8: 57 AM 6 -7 - Investigating Graphs of Polynomial Functions 15

Example 6 Identify the end behavior of P(x) = 2 x 5 + 6 x 4 – x +4 As x –∞, P(x) –∞. As x +∞, P(x) +∞. 10/2/2020 8: 58 AM 6 -7 - Investigating Graphs of Polynomial Functions 16

Example 7 Identify the end behavior of P(x) = 2 x 5 + 6 x 4 – x +4 As x –∞, P(x) –∞. As x +∞, P(x) –∞. 10/2/2020 8: 58 AM 6 -7 - Investigating Graphs of Polynomial Functions 17

Graphing Polynomials 1. Plug the equation into the graphing calculator, Y= 2. Look at the table to plot points 3. Sketch the graph 4. Describe the end behaviors As x –∞, P(x) –∞. As x +∞, P(x) –∞. 10/2/2020 8: 58 AM 6 -7 - Investigating Graphs of Polynomial Functions 18

Example 8 Graph the function. f(x) = x 3 + 4 x 2 + x – 6. As x –∞, P(x) –∞. As x +∞, P(x) +∞. 10/2/2020 8: 58 AM 6 -7 - Investigating Graphs of Polynomial Functions 19

Assignment Page 457 15 -22, 32 -35, 36 -41 my. hrw. com mstarkey 42 panthermath 10/2/2020 8: 58 AM 6 -7 - Investigating Graphs of Polynomial Functions 20