1、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 coefficienta0 is called the constant term,Example 1,Consider the functionDetermine: (i) The
2、 degree (ii) The leading coefficient (iii) The constant term,Example 1: Solution,The degree is 4The leading coefficient is -3The 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
3、zeroThe 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
4、#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 degreeFor 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-valueT
5、he maximum of a function is the greatest y-valueA local minimum is a point on a function that has the least y-value in some intervalA 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 m
6、inima,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 ofFind 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 M
7、inimum,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 valueIn Example 4 a polynomial
8、 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 descri
9、bes 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
10、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 Determ
11、ine 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
12、 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 coefficienta0 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),
