# x2 t04 01 curve sketching - basic features/ calculus

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- 1. (A) Features You ShouldNotice About A Graph

2. (A) Features You ShouldNotice About A Graph (1) Basic Curves 3. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; 4. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y x (both pronumerals are to the power of one) 5. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y x (both pronumerals are to the power of one)b) Parabolas: y x 2 (one pronumeral is to the power of one, the other the power of two) 6. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y x (both pronumerals are to the power of one)b) Parabolas: y x 2 (one pronumeral is to the power of one, the other the power of two)NOTE: general parabola is y ax 2 bx c 7. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y x (both pronumerals are to the power of one)b) Parabolas: y x 2 (one pronumeral is to the power of one, the other the power of two)NOTE: general parabola is y ax 2 bx c c) Cubics: y x 3 (one pronumeral is to the power of one, the other the power of three) 8. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y x (both pronumerals are to the power of one)b) Parabolas: y x 2 (one pronumeral is to the power of one, the other the power of two)NOTE: general parabola is y ax 2 bx c c) Cubics: y x 3 (one pronumeral is to the power of one, the other the power of three)NOTE: general cubic is y ax 3 bx 2 cx d 9. (A) Features You ShouldNotice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y x (both pronumerals are to the power of one)b) Parabolas: y x 2 (one pronumeral is to the power of one, the other the power of two)NOTE: general parabola is y ax 2 bx c c) Cubics: y x 3 (one pronumeral is to the power of one, the other the power of three)NOTE: general cubic is y ax 3 bx 2 cx dd) Polynomials in general 10. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) 11. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power) 12. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power)g) Circles: x 2 y 2 r 2 (both pronumerals are to the power of two,coefficients are the same) 13. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power)g) Circles: x 2 y 2 r 2 (both pronumerals are to the power of two,coefficients are the same) h) Ellipses: ax 2 by 2 k (both pronumerals are to the power of two, coefficients are NOT the same) 14. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power)g) Circles: x 2 y 2 r 2 (both pronumerals are to the power of two,coefficients are the same) h) Ellipses: ax 2 by 2 k (both pronumerals are to the power of two, coefficients are NOT the same)(NOTE: if signs are different then hyperbola) 15. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power)g) Circles: x 2 y 2 r 2 (both pronumerals are to the power of two,coefficients are the same) h) Ellipses: ax 2 by 2 k (both pronumerals are to the power of two, coefficients are NOT the same)(NOTE: if signs are different then hyperbola) i) Logarithmics: y log a x 16. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power)g) Circles: x 2 y 2 r 2 (both pronumerals are to the power of two,coefficients are the same) h) Ellipses: ax 2 by 2 k (both pronumerals are to the power of two, coefficients are NOT the same)(NOTE: if signs are different then hyperbola) i) Logarithmics: y log a x j) Trigonometric: y sin x, y cos x, y tan x 17. 1 e) Hyperbolas: y OR xy 1x(one pronomeral is on the bottom of the fraction, the other is not ORpronumerals are multiplied together) f) Exponentials: y a x (one pronumeral is in the power)g) Circles: x 2 y 2 r 2 (both pronumerals are to the power of two,coefficients are the same) h) Ellipses: ax 2 by 2 k (both pronumerals are to the power of two, coefficients are NOT the same)(NOTE: if signs are different then hyperbola) i) Logarithmics: y log a x j) Trigonometric: y sin x, y cos x, y tan x 1 1 1 k) Inverse Trigonmetric: y sin x, y cos x, y tan x 18. (2) Odd & Even Functions 19. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch 20. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) 21. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) 22. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x 23. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) 24. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) (4) Dominance 25. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) (4) DominanceAs x gets large, does a particular term dominate? 26. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) (4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates e.g. y x 4 3 x 3 2 x 2, x 4 dominates 27. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) (4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates e.g. y x 4 3 x 3 2 x 2, x 4 dominatesb) Exponentials: e x tends to dominate as it increases so rapidly 28. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) (4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates e.g. y x 4 3 x 3 2 x 2, x 4 dominatesb) Exponentials: e x tends to dominate as it increases so rapidlyc) In General: look for the term that increases the most rapidlyi.e. which is the steepest 29. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f x f x (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f x f x (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = xe.g. x 3 y 3 1 (in other words, the curve is its own inverse) (4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates e.g. y x 4 3 x 3 2 x 2, x 4 dominatesb) Exponentials: e x tends to dominate as it increases so rapidlyc) In General: look for the term that increases the most rapidlyi.e. which is the steepestNOTE: check by substituting large numbers e.g. 1000000 30. (5) Asymptotes 31. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero 32. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero 33. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero NOTE: if order of numerator order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check) 34. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero NOTE: if order of numerator order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check)(6) The Special Limit 35. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero NOTE: if order of numerator order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check)(6) The Special Limitsin x Remember the special limit seen in 2 Unit i.e. lim 1x0 x , it could come in handy when solving harder graphs. 36. (B) Using Calculus 37. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections. 38. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.(1) Critical Points 39. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.(1) Critical Points dy When dx is undefined the curve has a vertical tangent, these points are called critical points. 40. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.(1) Critical Points dy When dx is undefined the curve has a vertical tangent, these points are called critical points.(2) Stationary Points 41. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.(1) Critical Points dy When dx is undefined the curve has a vertical tangent, these points are called critical points.(2) Stationary PointsdyWhen dx 0 the curve is said to be stationary, these points may beminimum turning points, maximum turning points or points ofinflection. 42. (3) Minimum/Maximum Turning Points 43. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx 44. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx dyd2y b) When 0 and 2 0, the point is called a maximum turning point dxdx 45. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx dyd2y b) When 0 and 2 0, the point is called a maximum turning point dxdxdy NOTE: testing either side offor change can be quicker for harderdx functions 46. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx dy d2y b) When 0 and 2 0, the point is called a maximum turning point dx dxdy NOTE: testing either side offor change can be quicker for harderdx functions(4) Inflection Points 47. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx dy d2y b) When 0 and 2 0, the point is called a maximum turning point dx dxdy NOTE: testing either side offor change can be quicker for harderdx functions(4) Inflection Pointsd2y d3y a) When 2 0 and 3 0, the point is called an inflection pointdxdx 48. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx dy d2y b) When 0 and 2 0, the point is called a maximum turning point dx dxdy NOTE: testing either side offor change can be quicker for harderdx functions(4) Inflection Points d2yd3y a) When 2 0 and 3 0, the point is called an inflection point dxdx d2yNOTE: testing either side of2for change can be quicker for harder dxfunctions 49. (3) Minimum/Maximum Turning Points dyd2y a) When 0 and 2 0, the point is called a minimum turning point dxdx dy d2y b) When 0 and 2 0, the point is called a maximum turning point dx dxdy NOTE: testing either side offor change can be quicker for harderdx functions(4) Inflection Pointsd2yd3y a) When 2 0 and 3 0, the point is called an inflection pointdx dxd2yNOTE: testing either side of 2 for change can be quicker for harderdx functionsdyd2yd3y b) When 0, 2 0 and 3 0, the point is called a horizontaldxdx dx point of inflection 50. (5) Increasing/Decreasing Curves 51. (5) Increasing/Decreasing Curves dy a) When 0, the curve has a positive sloped tangent and is dxthus increasing 52. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing 53. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation 54. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions 55. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 56. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 Note: the curve has symmetry in y = x 57. y y=x x 58. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 Note: the curve has symmetry in y = x it passes through (1,0) and (0,1) 59. y y=x 1 1 x 60. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 Note: the curve has symmetry in y = x it passes through (1,0) and (0,1) it is asymptotic to the line y = -x 61. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 Note: the curve has symmetry in y = x it passes through (1,0) and (0,1) it is asymptotic to the line y = -x y 3 1 x3 i.e. y 3 x 3y x 62. y y=x 1 1x y=-x 63. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 On differentiating implicitly; Note: the curve has symmetry in y = x it passes through (1,0) and (0,1) it is asymptotic to the line y = -x y 3 1 x3 i.e. y 3 x 3y x 64. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1 On differentiating implicitly; 2 dy Note: the curve has symmetry in y = x 3x 3 y 20 it passes through (1,0) and (0,1) dx it is asymptotic to the line y = -x dy x 2 2 y 1 x 3 3 dx y i.e. y 3 x 3y x 65. (5) Increasing/Decreasing Curvesdy a) When 0, the curve has a positive sloped tangent and isdxthus increasingdy b) When 0, the curve has a negative sloped tangent and isdxthus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3 y 3 1On differentiating implicitly;2 dy Note: the curve has symmetry in y = x 3x 3 y 2 0 it passes through (1,0) and (0,1)dx it is asymptotic to the line y = -xdy x 2 2 y 1 x 3 3dx y dyi.e. y 3 x 3 This means that 0 for all x dx y xExcept at (1,0) : critical point & (0,1): horizontal point of inflection 66. yy=x 1 x3 y 3 11 xy=-x