2016届湖北省武汉六中（国际部）高考数学一轮复习课件11_多项式函数定义

Unit 1: Polynomial Functions,Lesson 1: Properties of Polynomial Functions,What is a Polynomial Function?,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 coefficient (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 function 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 Example 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 function 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: Solution,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 & 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 degree 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-values 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, 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 behaviour,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 direction 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-interce