1. Math 30-11

2. PolynomialsTrig Functions Exponential Functions Log FunctionsLogistic Functions Most functions that are encountered in the real world are continuous and smooth Absolute Value CONTINUOUS FUNCTION graphs are connected A function is continuous if “it can be drawn without lifting your pencil from your paper.” (formal definition involves calculus topics) 2

3. Are these graphs continuous? Math 30-13

4. Discontinuous Function graphs are not connected A location on the graph where you must lift your pencil is called a discontinuity. The location is described using the value of x. Many rational functions are discontinuous functions, meaning their graphs contain one or more jumps, breaks, or holes. This occurs at an excluded value, or non-permissible value of the variable. Breaks in continuity can take 2 forms: A point discontinuity (a “hole” in the curve of the function, represented with an open circle) A vertical asymptote (line which the function approaches but does not intersect) Math 30-14

5. A point of discontinuity occurs in a graph where the graph is not continuous, sometimes referred to as a “hole in the graph”. Continuous at x = 0 Discontinuity at x = 1 Discontinuity at x = 0 Vertical asymptotes also describe discontinuity. Math 30-1 5

6. For rational functions, discontinuities can be found by determining where the denominator is equal to zero, the non-permissible values for the variable. The rational function has a point of discontinuity or a vertical asymptote for each real zero of Q(x). Discontinuity from the Function Equation If P(x) and Q(x) has no common zeros, then the graph has a vertical asymptote for each zero of Q(x). If P(x) and Q(x) have a common zero, then there is a hole at each of the common zeros. Examine the factors of the numerator and denominator to determine locations of asymptotes and zeros. Math 30-16

7. Finding Breaks in Continuity Factor the numerator and denominator, then determine the excluded value from the domain If the excluded value comes from a factor that is in BOTH the numerator and denominator, this will cause a point discontinuity (hole) If the excluded value comes from a factor that is ONLY in the denominator, there will be a vertical asymptote at this excluded value Hole at x = c Vertical Asymptote at x = c Math 30-17

8. Asymptotes Property Vertical Asymptote Words If the rational expression of a function is written in simplest form and the function is undefined for x = a, then x = a is a vertical asymptote Example The rational expression of the function is in simplest form. x = 3 is a zero of Q(x). Therefore, x = 3 is a vertical asymptote. Graph 8 Horizontal Asymptote HA at y = 1 The degrees match

9. Point Discontinuity Property Point Discontinuity Words If the original function is undefined for x = a but the rational expression of the function in simplest form is defined for x = a, then there is a hole in the graph at x = a. Example The original function is undefined for x = –2. However, the simplified function is defined for x = –2. Therefore there is a hole at x = –2. Model Math 30-19 The height of the hole at x = –2 can be found by subbing into the simplified function.

10. Rational Functions A point of discontinuity: A vertical asymptote : Graph g(x) = x 2 - 6 x - 2 x 2 - 4 x + 2 Graph f(x) = The graph has a hole at x = –2. (-2, -4) The graph has a vertical asymptote at x = 2, Math 30-110

11. Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of Factor the numerator and denominator of the rational expression. The function is undefined for x = 1 and x = 5. 1 1 x = 5 is a vertical asymptote, x = 1 is a hole in the graph (1, -1/2) HA at y = 1 Rational Functions x – 1 is a factor that is common to both the numerator and the denominator. Math 30-111

12. Rational Functions What are the coordinates of the point of discontinuity? What does (x + 1) represent? Hole x = 1 Vertical Asymptote x = 5 x-int at -1 Math 30-112 degrees match HA at y = 1

13. Determine the equations of any vertical asymptotes, the coordinates for any holes in each graph, and the x- intercept. Analysing Rational Functions from the Equation Vertical Asymptotes at x = 1 and at x= –3 HA at y = 0 Vertical Asymptote at x = –3 Horizontal Asymptote at y = 0 Vertical Asymptote at x = –5 x- intercept at x = 2 Hole at x = 2 (2, 1/5) Hole at x = –3 (-3, -3) x-intercept at x = 3 Math 30-113

14. Match the equation of each rational function with the most appropriate graph. Rational Functions Graph 1Graph 2Graph 3Graph 4 Math 30-114

15. 1. If function f is defined by, then at x = 3 A.The graph of f has a vertical asymptote B.The graph of f has a hole on the x-axis C.f(x) = 0 D.The graph of f has a hole at the point (3,2) Rational Functions (Multiple Choice) 2. Which of the following functions has a hole at (1, 4)? Math 30-115

16. 3. Determine the function f whose graph is given. Math 30-116

17. Page 451 1, 3, 4, 5, 6, 7a, 8 12, 13, model a situation 13 Math 30-117