《湖北省高三（国际部）数学复习课件11__多项式函数定义 》由会员分享，可在线阅读，更多相关《湖北省高三（国际部）数学复习课件11__多项式函数定义 （24页珍藏版）》请在金锄头文库上搜索。
1、Unit 1: Polynomial Functions,Lesson 1: Properties of Polynomial Functions,Any function of the form: The degree of the function is n (the largest exponent) an is called the leading coefficient a0 is called the constant term,Example 1,Consider the function Determine: (i) The degree (ii) The leading co
2、efficient (iii) The constant term,Example 1: Solution,The degree is 4 The leading coefficient is -3 The constant term is 1,Common Examples,Intercepts of Polynomial Functions,The point where a function crosses the x-axis is known as the x-intercept Sometimes referred to as a zero The point where a fu
3、nction crosses the y-axis is known as the y-intercept The value of f (0) So, for a polynomial function, the y-intercept is the constant term a0 :,Example 2,Consider the graph of How many zeroes are there?,Example 2: Solution,Zero #3,This function has three zeros,Zero #2,Zero #1,Example 2: Notes,In E
4、xample 2 a polynomial function with degree 3 had 3 zeroes. The maximum number of x-intercepts of any polynomial function is its degree For example, a quadratic function (degree 2) can have 0, 1, or 2 x-intercepts.,Minima & Maxima,The minimum of a function is the least y-value The maximum of a functi
5、on is the greatest y-value A local minimum is a point on a function that has the least y-value in some interval A local maximum is a point on a function that has the greatest y-value in some interval,Example 3,Consider the graph of Determine the location of the local maxima and minima,Example 3: Sol
6、ution,Local Maximum at (-1,3),Local Minimum at (1,-3),This function has no minimum or maximum value,Example 4,Consider the graph of Find the local maxima and minima,Example 4: Solution,Local Maximum,This function has a minimum value of -5 but no maximum value,Local Minimum,Local Minimum,Example 3 &
7、4: Notes,In Example 3 a polynomial function with a degree of 3 had a total of 2 local maxima/minima For a polynomial function with an odd degree The maximum number of local maxima/minima is one less than its degree There will be no maximum or minimum value In Example 4 a polynomial function with a d
8、egree of 4 had a total of 3 local maxima/minima and 1 minimum For a polynomial function with an even degree The maximum number of local maxima/minima is one less than its degree There will be at least one maximum or minimum value,End Behaviour,The end behaviour of a function describes what the y-val
9、ues do as x approaches + i.e. as x gets very large Denoted by x + as x approaches - i.e. as x gets very large and negative Denoted by x -,Example 5,Describe the end behaviour of,Example 5: Solution,As x +, y +,As x -, y -,As x gets very large and positive, so does y,As x gets very large and negative
10、, so does y,In words:,Example 5: Notes,In this example you can see that for a polynomial function with an odd degree, the two ends of the function go in opposite directions All polynomial functions with an odd degree exhibit this end behaviour,Example 6,Consider the graph of Determine the end behavi
11、our,Example 6,As x +, y +,As x -, y +,As x gets very large and positive, so does y,As x gets very large and negative, y gets large and positive,In words:,Example 6: Notes,In this example you can see that for a polynomial function with an even degree, the two ends of the function go in the same direc
12、tion All even-degree polynomial functions with an even degree exhibit this end behaviour,Summary,Functions of the form: The degree of the function is n (the largest exponent) an is called the leading coefficient a0 is called the constant term (the value of the y-intercept),Practice Problems,P. 11-12 #1-3 Notes: For #3, dont do part (iv) P. 26-29 #1-3, 5, 6, 15, 16 Notes: For #2, dont do part (c) For #3, dont do parts (ii) & (iii) For #6, dont do part (d),
