Polynomial Functions of Higher Degree Section 2.2 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions End Behavior Every graph is continuous. There are no gaps, jumps, holes, or sharp corners. End Behavior The leading coefficient focuses on the RIGHT side. If it is POSITIVE, the graph will RISE TO THE RIGHT If it is NEGATIVE, the graph will FALL TO THE RIGHT The highest degree focuses on the LEFT side. EVEN: The left behavior is the SAME as the right behavior; EQUAL  EVEN ODD: The left behavior is the OPPOSITE of the right behavior; OPPOSITE  ODD 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 1 Describe the right-hand and left-hand behavior of the graph below. 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 2 Identify the end behavior of 𝒇 𝒙 =−𝟐 𝒙 𝟓 +𝟑 𝒙 𝟐 −𝟒𝒙−𝟏 and describe increasing, decreasing, or constant. 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 3 Identify the end behavior of 𝒇 𝒙 = 𝒙 𝟒 −𝟓 𝒙 𝟐 +𝟒 and describe increasing, decreasing, or constant. 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Your Turn Identify the end behavior of 𝒇 𝒙 = −𝒙 𝟑 +𝟒𝒙 and describe increasing, decreasing, or constant. 12/7/2018 8:59 AM §2.2: Polynomial Functions

Establishing Real Zeros & Turning Points Make the equation and solve for the zeros Establish all multiple roots through the power of the exponent To determine the Highest Degree, it is the amount of “peaks” or “valleys” and subtract 1. ODD graphs are symmetrical through the origin. EVEN graphs are always symmetrical about the vertical axis (that is, we have a mirror image through the 𝒚-axis) 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 4 Find all zeros and maximum turns of 𝒇 𝒙 =−𝟐 𝒙 𝟒 +𝟐 𝒙 𝟐 . 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 5 Find all zeros and maximum turns of 𝒇 𝒙 = 𝒙+𝟏 𝟐 𝒙−𝟐 𝒙−𝟑 𝟑 . 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Your Turn Find all zeros and maximum turns of 𝒇 𝒙 = 𝒙 𝟑 −𝟏𝟐 𝒙 𝟐 +𝟑𝟔𝒙. 12/7/2018 8:59 AM §2.2: Polynomial Functions

Sketching the Polynomial Graph Apply the end behavior rules Determine the real zeros Plot additional points by making a table and testing points 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 6 Sketch the graph 𝒇 𝒙 =𝟑 𝒙 𝟒 −𝟒 𝒙 𝟑 −∞,𝟎 𝟎, 𝟒 𝟑 𝟒 𝟑 ,∞ 𝒙=−𝟏 𝒚=𝟕 𝒙=𝟏 𝒚=−𝟏 𝒙=𝟐 𝒚=𝟏𝟔 −∞,0 0, 4 3 4 3 ,∞ 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Example 7 Sketch the graph 𝒇 𝒙 =− 𝒙 𝟒 +𝟑 𝒙 𝟑 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Your Turn Sketch the graph 𝒇 𝒙 =𝟐 𝒙 𝟑 −𝟔 𝒙 𝟐 12/7/2018 8:59 AM §2.2: Polynomial Functions

§2.2: Polynomial Functions Assignment Page 133 9-14 all, 19-29 odd, 35-39 odd (A to C and then sketch the graph), 43, 47 12/7/2018 8:59 AM §2.2: Polynomial Functions