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Chapter 3

Chapter 3. The Nature of Graphs. Section 3.1 Section 3.2 Section 3.3 Section 3.4 Section 3.5 Section 3.6 Section 3.7 Section 3.8. Section 3.1. Symmetry and Coordinate Graphs. Symmetry.

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Chapter 3

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  1. Chapter 3 The Nature of Graphs

  2. Section 3.1 Section 3.2 Section 3.3 Section 3.4 Section 3.5 Section 3.6 Section 3.7 Section 3.8

  3. Section 3.1 Symmetry and Coordinate Graphs

  4. Symmetry Determine if the following is symmetric across the orgin, x-axis, y-axis, the line y=x or the line y=-x. Step 1: Pick a point that makes the equation true (Avoid zero and duplicating numbers, Such as (1,1) (2,2) etc) Step 2: Use the rules to see which one is also true! So it is symmetric across the origin, the line y=x and the line y=-x.

  5. Even and Odd Functions Even functions – Symmetric with respect to the y-axis (a,b)=(-a,b) Odd functions – Symmetric with respect to the origin (a,b)=(-a,-b)

  6. Homework: Pg 134: #14-26 E, 32-34 A, 42

  7. Answer to Homework Problems: 14. Yes 16. No 18. Yes 20. Yes. 22. x-axis 24. y-axis 26. x- and y- axis 32. y-axis 33. x-axis 34. Both 42. No. If an odd function has a y- intercept, then it must be the origin. If it were not, say it were (0,1), then the graph would have to contain (-1,0). This would cause the relationship to fail the vertical line test and would therefore not be a function. Not all functions have a y-intercept. Consider the graph y= 1/x

  8. Section 3.2 Families of Graphs

  9. Parent Graphs (e) &(f). Polynomial Function

  10. Rules for shifting a graph Change to the parent Function Change to the Parent Graph Visual Example Reflections: Is reflected over the x-axis Is reflected over the y a-axis Translations: Translates the graph c units up Translates the graph c units down Translates the graph c units left Translates the graph c units right Dilations: Expands the graph vertically Compresses the graph vertically Compresses the graph horizontally Expands the graph horizontally

  11. Examples Observe the graph of each function. State the parent graph of the function. Then describe how the graphs shift. 1. Write the equations for the graph g(x) if 2. Use the graph of a given parent function to describe the graph of the function. a. b. c. Translates up 3, anything below the x-axis reflects above x-axis. Translated left 1 unit, compressed vertically. Reflected over the x-axis, compressed horizontally 3. Use the graph of a given parent function to describe the graph of the function. a. b. c. Expands horizontally Translates right 5 units and down 2 units Expands vertically, translates up 6 units.

  12. Homework: Pg 143: #20-23 A, 26, 27, 35, 37, 39, 42part”a”

  13. Pg 143 Answers: 20a. Reflected over the x-axis, compressed horizontally 20b. Translated right 3 units, expanded vertically 20c. Compressed vertically, translated down 5 units 21a. Expanded horizontally 21b. Expanded vertically, translated down .4 units 21c. Reflected across x-axis, translated left 1 unit, expanded vertically 22a. Translated left 2 units and down 5 units 22b. Expanded horizontally, reflected over the x-axis 22c. Compressed horizontally, translated up 2 units 23a. Translated left 2 units, compressed vertically 23b. Reflected over y axis, translated down 7 units 23c. Expanded vertically, translated right 3 units and up 4 units 27. 35a. 35b. 35c. 37a. 37b. 37c. 39. The x-intercepts will be 42a. (1) (2) (3) (4)

  14. Section 3.3 Graphs of Nonlinear Inequalities

  15. Examples Determine whether the points are solutions for the inequality Graph 3. Graph

  16. 5. Graph 4. Solve

  17. Homework: Pg. 149: #14 – 18 E, 19, 20 – 23 A, 29 – 31 A, 34 – 38 E, 41, 44

  18. Pg 14 Answers: 14. No 16. No 18. Yes 20. 21. 22. 23. 29. 30. 31. 34. ( or 36.[ 38. 41. (5.5,10) 44. a. ( b. None c. d. 4 e.

  19. Secton 3-4 Inverse Functions and Relations

  20. Inverse Functions Inverse Functions – Two relations are inverse relations if and only if one relation contains the element (a,b) when the other relation contains the element (b,a) Example: Graph Then graph

  21. Definitions: Vertical Line Test – Test to see if is a function Horizontal Line Test – Test to see if is function. Example: Consider Is a function? b. Is a function? c. Graph on your paper. d. Graph using a graphing calculator.

  22. Examples: Determine if is a function. Then find 1. 2. 3.

  23. Homework: Pg 156: #16 – 40 E, 31, 42, 43, 45

  24. Pg 149 Answer Key: 16. 18. 20. 22.

  25. Pg 156 Answers 26. It is a function 28. It is NOT a function 30. It is NOT a function 32. It is a function 34. 36. 38. 40. .

  26. 41a. • 41b. No, the graph fails the horizontal line test. • 41c. gives the numbers that are 4 units from x on the number line. There are always two such numbers, so pairs two y values for one x value. • 42a. • 42b. Yes, the pump can propel water to a height of about 88ft. • 43a. • 43b. Yes, because the line y=x is the axis of symmetry and the line of reflection. • It must be translated up 6 units and 5 units to the left;

  27. Section 3.5 Continuity and End Behavior

  28. Definition Continuous Functions: Functions that continue from beginning to end. (You can trace the function with you pencil) Name the parent functions that are CONTINUOUS. • Polynomial Functions • 3. or • Absolute Value Functions • 4. • 3. Constant Functions • 3. • 4. Identity Functions • 4. • 5. Square Root Function • 5. Discontinuous Functions: Function that is not continuous from beginning to end. (You cannot trace the function with your pencil) Name the parent functions that are DISCONTINUOUS. • 1. Reciprocal Functions • 2. Greatest Integer Functions • 2. [[x]]

  29. Types of Discontinuous Functions Infinite Discontinuity – Means that becomes greater and greater as the graph approaches a given x- value. Jump Discontinuity – Indicates that the graph stops at a given value of the domain and then begins again at a different range value for the same value of the domain. Point Discontinuity – When there is a value in the domain for which the function is undefined, but the pieces of the graph match up.

  30. Continuity Test A function is continuous at x = c if it satisfies the following conditions: 1. The function is defined at c; in other words, exists. 2. The function approaches the same y-value on the left and the right side of x=c, the “y” value as the “x” value approaches some ‘c’ starting on the far left. the “y” value as the “x” value approaches some ‘c’ starting on the far right.

  31. Example Determine whether each function is continuous at the given x-value. a. Test 2: Does it approach the same y-value on the left and the right? Yes! So it is continuous! . Test 1: Does Yes! Go to Test 2. b. Test 1: Does exist? So no. It is not continuous. c. Test 2: No! So it is not continuous! Test 1: Does f(1) exist? Yes! Go to test 2.

  32. Homework: Pg 165 #5, 6, 12 – 19 A

  33. Pg 165 Answers No No Yes No Yes Yes No Yes

  34. End Behavior End behavior - describes what the y-values do as “x” approaches positive or negative infinity How to write end behavior: Example

  35. Increasing, Decreasing and Constant Functions Increasing – as x increases, y increases (Up the hill) Decreasing – as x increases, y decreases (Down the hill) Constant – as x increases, y remains constant (Flat Land) Examples: Graph each function. Determine the interval(s) on which the function is increasing and the interval(s) the function is decreasing. a. b. b. Decreasing c. c.

  36. Homework: Pg. 166 #12 – 18 E, 19, 20 – 30 E, 33, 34, 39

  37. Homework: Pg. 166 #12 – 18 E, 19, 20 – 30 E, 33, 34, 39 Page 166 Answer Key: 20. 22. 24. 26. Increasing: Decreasing: 28. Decreasing: 30. Increasing: Decreasing:

  38. Homework: Pg. 166 #12 – 18 E, 19, 20 – 30 E, 33, 34, 39 33a. Since f is even, its graph must be symmetric with respect to the y-axis. Therefore, f is decreasing for (-2,0) and increasing for . F must have a jump discontinuity when x=-3 and 33b. Since f is odd, its graph must be symmetric with respect to the origin. There for f is increasing for (-2,0) and decreasing for . F must have a jump discontinuity when x=-3 and 34a. Polynomial 34b. 34c. (

  39. 39. a=4 b=-2

  40. Section 3.6 Critical Points and Extrema

  41. Definitions Critical Point -- points on a graph at which a line drawn tangent to the curve is horizontal or vertical. Types of critical points: Maximum – When a graph goes from increasing to decreasing (Top of the hill) 2. Minimum – When a graph goes from decreasing to increasing (Bottom of hill) 3. Point of inflection – When a graph changes in curvature. Extrema– General Term for Max or Min

  42. Definitions Absolute Maximum – The greatest value that a function assumes over its domain. Relative Maximum – May not be the greatest value that a function assumes, but it is the greatest on some domain. Absolute Minimum – The least value that a function assumes over its domain. Relative Minimum - May not be the least value that a function assumes, but it is the least on some domain.

  43. Use a graphing calculator to graph each function and to determine and classify its extrema Relative Max: (0,0) Relative Min: (2, -16) 2. Relative Max: (-1,1) Relative Min: (1, -11)

  44. The function , has critical points at Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection. You can use a graphing calculator to determine it, but I am going to want to see work…. So use the graphing calculator to check your work. To do this by hand: Pick two points besides it You want to choose a small number, because sometimes local min and max are only one unit away. 2.769 3 2.829 Maximum -0.010 .009 0 POI -187.308 .-189 -187.087 Minimum

  45. Homework: Pg 177: #14 – 32 E, 38, 40, 51

  46. Homework: Pg 177: #14 – 32 E, 38, 40, 51 Page 177 Answers Key: 14. Absolute Max: (-1,3) Relative Min: (0.5, 0.5) Relative Max: (1.5,2) 16. Relative Max: (-6,4) Absolute Min: (-2,-3) 18. No extrema Relative Max: (-1.53, 1.13) Relative Min (1.53, -13.13) Absolute Min (-1.41, -6) Relative Max (0,-2) No Extrema

  47. 40. If a cubic has one critical point, then it must be a point of inflection. If it were a relative max or min, then the end behavior for a cubic would not be satisfied. If a cubic has three critical points, then one must be a max, another a min, and a third a point of inflection. 51. D Point of inflection Minimum Minimum Maximum Same answers: or

  48. Section 3.7 Graphs of Rational Functions

  49. Definitions Rational Functions – A quotient of two polynomial functions. Ex: The parent graph for rational functions is the reciprocal graph: Vertical Asymptotes – The Line x = a is a vertical asymptote for a function if or as from either the left or the right. Horizontal Asymptotes – The line y = b is a horizontal asymptote for a function if as Example Determine the asymptotes for the graph of Horizontal Asymptotes: Find the inverse of the function … Vertical Asymptotes: So the vertical Asymptote is The horizontal Asymptote is

  50. On your own: Use the parent graph . Describe the transformation(s) that take place. Then Identify the location of the asymptote.

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