Chapter 7 Notes

Chapter 7 NotesLesson 7.2 and 7.7 – Writing a Polynomial Function from a GraphObjectives:8. Identify the zeros of a function given a graph.9. Determine the lowest possible degree of a function given a graph.10.Write the equation of a polynomial function given a graph.The x-intercepts of a function are called the zeros (y=0) or the roots of the function.What are the zeros of the function? -3, -1, 2What is the y-intercept? -20; (0,-20)-4 -3 -2 -1 1 2 3 4 5y-intercept = -20How many turning points? 2Degree = #turning points + 1What is the lowest possible degree? 3Write the equation of the function in factored form:1. Plug-in the roots (zeros).y = a (x – r 1 )(x – r 2 )y = a (x + 3)(x + 1)(x – 2)2. Solve for a (the vertical scale factor).Plug-in the y-intercept or another point on the graph.Do not use any of the x-intercepts.-20 = a (0 + 3)(0 + 1) (0 – 2)-20 = a (3)(1)(-2)-20 = a (-6)-20/-6 = a10/3 = a3. Substitute a into the factored form.y = 10/3 (x + 3)(x + 1)(x – 2)_____________________________________________________________________________Practice)y-int = 10ANSWER: y = -2(x + 1)(x – 5)-4 -3 -2 -1 1 2 3 4 5Assignment: p.374 #9, p.408 #1-4

Chapter 7 NotesLesson 7.7 – Writing a Polynomial Function with Multiplicity from a GraphObjectives:11.Identify the double roots of a function from a graph or equation.12.Write the equation of a polynomial function with multiplicity given a graph.Compare the graphs for each function in #6 page 409 with their equations.What patterns do you see?Factor raised to an even power – the graph touches but does not cross the x-axisA factor squared is called a double root.Factor raised to an odd power – the graph crosses the x-axisIf a factor is cubed, the graph crosses the x-axis in a curved fashion.So, when you are writing the equation of a line and the graph touches the x-axis but does notcross, square the factor for that root. If it crosses the x-axis in a curve, cube the factor.Example)Write the equation of the function in factored form:1) Plug-in the roots (zeros).y = a(x – r 1 )(x – r 2 ). . .y = a(x + 2) 2 (x + 1)(x – 1)(x – 3) 2-4 -3 -2 -1 1 2 3 4 5y-intercept = -102) Solve for a (the vertical scale factor).-10 = a(0 + 2) 2 (0 + 1)(0 – 1)(0 – 3) 2-10 = a(-36)5= a183) Substitute a into the factored form.y = 185(x + 2) 2 (x + 1)(x – 1)(x – 3) 2Practice)y-intercept = 8ANSWER: y = -(x + 2) 2 (x – 2)-4 -3 -2 -1 1 2 3 4 5Assignment: p. 409 #7, 8ab

Chapter 7 NotesLesson 7.7 (part 2) – End Behavior and Graphing Polynomial FunctionsObjectives:13.Describe the end behavior of a function by its leading coefficient and degree.14.Determine the sign of the leading coefficient from the graph of a function.15.Graph polynomial functions in factored form.Describing End BehaviorIf the degree of the function is. . .EVENODD+ leading coefficient + leading coefficient– leading coefficient – leading coefficientNote: The leading coefficient is the coefficient of the term with the highest power if the functionis written in general form. For functions in vertex or factored form, the leading coefficient is thevertical scale factor, a.Practice) Describe the end behavior of each function using a sketch.1. y = -5x 4 2. y = x 2 + 1 3. y = 2x4. y = x 5 – 2x 2 5. y = –(x + 5)(x – 3)(x – 7)Determine the sign of the leading coefficient from the graph of a function.Practice) page 410 #11 part i*Discuss extreme values.Sketch graphs of polynomial functions in factored form:1. Plot the zeros of the function.2. Use the end behavior to sketch each side of the graph.3. Determine how the graph will behave at each x-intercept. (cross axis, turn, curve)4. Substitute ZERO for x to find the y-intercept. Add this point to the graph.5. Connect each intercept with smooth curves.Practice) y = –(x + 5)(x – 3)(x – 7)y = 2(x + 5) 2 (x + 2)(x – 1)Assignment: page 410 #11 ii – iv, BoardworkSketch the graph of each function.1. y = (x + 4)(x – 1) 2 (x – 3)2. y = –(x + 3)(x – 3) 23. y = -2(x + 1)(x – 4) 34. y = 2(x – 5) 2 (x – 2)(x + 3) 2Be sure to include all x- and y-intercepts.

Chapter 7 NotesGraphing Calculator ToolsObjectives:16.Calculate finite differences on a graphing calculator.17.Graph polynomial functions on a graphing calculator.Finite Differences1. Enter the y-values in a list: STAT EDIT Type each value. PressENTER2. Exit the List screen: 2nd [QUIT]3. [LIST] OPS 7:∆List (name of list) *For L1, press2nd 2nd 14.ENTER5. ∆List (ANS) 2nd [ENTRY], change the list name to 2nd [ANS]6. Repeat step 5 as needed until the differences are constant.Practice: p. 365 #3a,c Use the calculator to determine the degree of each function.Graphing Polynomial Functions with a CalculatorSee Calculator Notes Book pages 18 - 21Y=GRAPHWINDOWTBLSET and TABLEZOOMCALC: zero, minimum/maximumExamples: Sketch the graph of each function. Identify all x- and y-intercepts. Find the pointswhere the local maximums and minimums occur. Round to the nearest 100 th .1. y = 3x 3 – 15x 2 – 39x – 212. y = –x 4 – 3x 3 + x 2 – x + 5Assignment:See directions above.1. y = x 3 – 3x 2 + 22. y = x 4 – 4x 3 – x 2 + 12x – 213. y = 4x 4 + 2x 2Chapter 7 Quiz Objectives1. Identify polynomial expressions and polynomial degree.2. Use finite differences to find the degree of a polynomial function.3. Write a polynomial function that models a data set.4. Identify the general, vertex, and factored form of a quadratic function.5. Identify the zeros and the degree of a function given a graph.6. Write a polynomial function in factored form given a graph.7. Describe the end behavior of a function by its leading coefficient and degree.8. Graph polynomial functions manually and with a graphing calculator.Chapter 7 Homework Check Next Week4. y = x 5 – 6x 3 + 9xpage 374 #10

Calculator Notes, continuedFinding maximums, minimums and zeros:1. Open the CALCULATE menu (2 nd , TRACE).2. Choose 2:zero, 3:minimum, or 4:maximum.3. Use the Left/Right arrow keys and ENTER to choose a left bound, right bound and aguess.**Make sure that your window is small enough to see the details of the graph.