4.3 - End Behavior and Graphing Worksheet Key 1) 4x3 – 17x2 – 39x – 18 2) x4 – 9x3 + 39x2 – 225x + 350 3) x4 + 4x3 + 4x2 + 4x + 3 4) x5 – 4x4 - 2x3 + 8x2 - 24x + 96 5) { +3/2i, -1, 3 } 6) { + 𝒊 𝟑 , 1, -4 } 7) { 1 + i, -3, 4 } 8) { + 2i, -6 } 9) omit 1) x3 + 2x2 + 13x + 10 2) x3 + 4x2 + 3x 3) x3 - 10x2 + 29x - 20 4) x3 - 7x2 + 36 5) x3 - 12x2 + 44x - 48 6) x3 - 25x 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

Graphs of Polynomial Functions Section 4.3 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Graphing End Behavior Every graph is continuous. There are no gaps, jumps, holes, or sharp corners. End Behavior When a polynomial function has an ODD degree, one end of its graph shoots upward and the other end downwards. When a polynomial function has an EVEN degree, both ends will shoot upwards or downwards 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 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 1 How would you describe the graph from negative x to positive x? 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Your Turn How would you describe the graph from negative x to positive x? 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 2 Identify the end behavior of P(x) = 2x5 + 3x2 – 4x – 1 As x  –∞, P(x) –∞ As x  +∞, P(x) +∞ 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Your Turn Identify the end behavior of P(x) = –0.6x5 + 4x2 + x – 4 As x  –∞, P(x) +∞ As x  +∞, P(x) –∞ 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Multiplicity Intercepts The graph of any polynomial function of a degree n has only one y-intercept, which is equal to the constant term Has at most n x-intercepts Multiplicity is the amount of many times a particular number is a zero for a given polynomial Given (x – k)n If n is ODD, the zero CROSSES the x-axis If n is EVEN, the zero TOUCHES the x-axis 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 3 Given the graph below, state the zero and identify whether each function touches or crosses the x-axis crosses the x-axis and identify whether it is odd or even, y-intercept, and sketch the graph 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 4 Find all the zeros of f(x) = (x + 1)2(x + 2)(x – 3)3, state the multiplicity of each zero, and identify whether each function touches or crosses the x-axis and identify whether it is odd or even, y-intercept, and sketch the graph 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Your Turn Find all the zeros of f(x) = (x – 1)4(x + 1)(x + 3)5, state the multiplicity of each zero, and identify whether each function touches or crosses the x-axis and identify whether it is odd or even and sketch the graph 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Functions To determine the Highest Degree, it is the amount of “peaks” or “valleys” and ADD 1. Take the number. ODD graphs are symmetrical through the origin. EVEN graphs is always symmetrical about the vertical axis (that is, we have a mirror image through the y-axis) The LEADING COEFFICIENT is the right side of the graph. If it rises, it is positive. If it falls, it is negative. To determine the SMALLEST Degree, it is the amount of “peaks” or “valleys” and ADD 1. 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 5 Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function? 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 6 Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function? 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Example 7 Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function? 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Your Turn Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function? 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing

4.3 - End Behavior and Graphing Assignment Page 271 7-11 odd and include end behavior, 15 and 17 label multiplicity, zeros crosses or touches, and if multiplicity is odd/even, 19-24 all, 43, 44 2/22/2019 6:55 PM 4.3 - End Behavior and Graphing