Graph Sketching. Review: 1) Interval Notation. Set Notation Interval Notation Set Notation Interval Notation. 2) Solving Inequalities

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1 Lesson. Graph Sketching Review: ) Interval Notation Set Notation Interval Notation Set Notation Interval Notation a) { R / < < 5} b) I (, 3) ( 3, ) c){ R} d) I (, ] (0, ) e){ R / > 5} f) I [ 3,5) ) Solving Inequalities a) ( + )( 4)( + 5) < 0 b)

2 3) Summary of Graphing Functions a) Polynomial Functions: Basic Shapes: y y 3 y 4 y n i) y ( )( + 4) ii) f() ( ) ( + ) ( 6) Steps: i) find roots (-intercepts) ii) find y-intercept iii) interval chart iv) approimate ma/min v) graph vi) end behaviour v) Domain & Range b) Rational Functions: i) y ( 3)( + 3) Steps: i) find Vertical Asymptotes ii) find Horizontal Asymptotes roots (-intercepts) ii) find y-intercept iii) interval chart iv) approimate ma/min v) graph HW: Pg.9# ab,3acghi, 4abc, 5acg, Pg. #,, Pg. 30 # 6,7,8

3 Lesson. Increasing and Decreasing Functions Definition: A function is said to be on an interval, I, if as over the interval, f() or y. A function is said to be on an interval, I, if as over the interval, f() or y. Consider the graph On which intervals is the function, f() increasing? decreasing? Slope of tangents (A, B) f is (B, C) f is (C, D) f is How do the slopes of the tangents compare? What do you notice about the ma/min points? Conclusion: find the intervals on which the following functions are increasing and decreasing. (Hint: Find turning points and check intervals on either side). f() f() HW: Pg. 37 # -3, 6, 8, 0

4 Lesson.3 Absolute Maimum and Minimum Values of a Function Definition: ) Absolute ma/min: the highest or lowest the function (y value) can ever reach ) Relative (Local) ma/min: a turning point (bump), place where the graph changes direction but not the highest or lowest the function (y value) can ever reach Fermat s Theorem: States that if f() has any local ma or min values at c then either f (c) 0 or f (c) DNE. i) Find the critical numbers for each function. ii) Determine the Absolute ma/min values on the given interval. iii) Identify any local ma/min values.. f() on I [-4, 4]. f() on I [-, ] 3. f() + on I [-½, 4] First ask yourself what is this function? What could it look like? HW: Pg. 37 #, (not k,l), 5, 7,

5 Lesson.4 First Derivative Test What do we know how to do? Consider the graph: First Derivative Test : (Putting it all together) If c is a critical number of the function f() then: a) b) c) d) i) Find the critical numbers for each function. ii) Determine the Absolute local ma/min values on the given interval. iii) Sketch.. f() f() HW: Pg. 37 # 3(not j), 4, 6, 9,

6 Lesson.5 Concavity and Points of Inflection Definitions: a) Concave Upward: b) Concave Downward: Consider the graph:. Concave upwards? B F G. Concave downwards? A C E 3. At which points does the concavity change? D c) Points of Inflection: Consider Quadratics:. f() f() 4 6

7 Test for Concavity: a) b) c) Determine where the function is concave upwards and downwards, whether there are any points of inflection and then sketch the graph.. f() f() HW: Pg. 338 # 3, 5, 7, 8, 0

8 Lesson.6 The Second Derivative Test A) First Derivative Test: Critical points occur when f '(c) 0 or f '(c) DNE. a) if f '(c) 0 and f '() changes from increasing to decreasing at c, then there will be a local ma. b) if f '(c) 0 and f '() changes from decreasing to increasing at c, then there will be a local min. B) Concavity: a) The graph f() is concave upwards if f ''() > 0. b) The graph f() is concave downwards if f ''() < 0. c) A point of inflection occurs when the concavity changes. Combining A & B leads to the Second Derivative Test Second Derivative Test: a) b) c) Use the second derivative test to find local ma/min and points of inflection and then sketch. (You may still use other tools as well. E. or y intercepts etc). f() f() + 3 HW: Pg. 338 # 6, 8 (not g or h), a)

9 Lesson.7 Vertical Asymptotes Asymptote: A line a graph approaches but never touches or crosses. Remember limits? lim lim + lim Take a closer look at What about? + 3. Find any vertical asymptotes if they eist. a) b) 4. Try these limits. + a) lim b) + lim c) lim 3 3 d) lim e) 3 3 lim ( ) 3 3 HW: Pg. 347 #, 3ace, 4-6

10 Lesson.8 Horizontal Asymptotes Back to What happens as? In other words: lim Test some cases: i) ii) iii) iv) 3 n Remember trick from Advanced Functions: Find the horizontal asymptotes.. 5. f 3 ) ( Find all asymptotes, & y intercepts and sketch a) b) c) HW: Pg. 359 #, 3-5, 7

11 Lesson.9 Oblique Asymptotes Review:. Graph the following lines : i) y 3 ii) y -/ + iii) y -/3 /. Find the Vertical and Horizontal Asymptotes for the following: i) 5 ii) 7 iii) ( )( + 3) 8 So what do we do when there is no horizontal asymptote? Note: Oblique asymptotes happen when Find the Oblique asymptotes: HW: Pg. 359 #, 6, 8, 9, 4a)

12 Lesson.0 Curve Sketching (Putting it All Together!!) Tools: ) Vertical Asymptotes ) Horizontal or Oblique Asymptotes 3) & y intercepts 4) Critical Points ( st derivative) 5) Concavity & Ma/Min (Local or Absolute) ( nd derivative) 6) Points of Inflection ( nd derivative) 7) Basic shapes (direction of opening) 8) Domain & Range Type : Polynomial Functions f 5 3 ( ) 6 0 Tools: ) & y intercepts ) Critical Points ( st derivative) 3) Concavity & Ma/Min (Local or Absolute) ( nd derivative) 4) Points of Inflection ( nd derivative) 5) Basic shapes (direction of opening) 6) Domain & Range Type : Rational Functions 3 4 Tools: ) Vertical Asymptotes ) Horizontal or Oblique Asymptotes 3) & y intercepts 4) Critical Points ( st derivative) 5) Concavity & Ma/Min (Local or Absolute) ( nd derivative) 6) Points of Inflection ( nd derivative) 7) Basic shapes (direction of opening) 8) Domain & Range HW: Pg. 370 # 3-5

MA 220 Lesson 28 Notes Section 3.3 (p. 191, 2 nd half of text)

MA 220 Lesson 28 Notes Section 3.3 (p. 191, 2 nd half of tet) The property of the graph of a function curving upward or downward is defined as the concavity of the graph of a function. Concavity if how