Dr. Carol JVF Burns: homepage

PRECALCULUS

Free Online Precalculus Course
with Randomly-Generated Practice Problems and Worksheets
in Every Section


1.   prerequisites: language of mathematics essentials
2.   prerequisites: function review; difference quotients
3.   solving linear inequalities in one variable
4.   solving nonlinear inequalities in one variable (introduction)
5.   the test point method for sentences like ‘$f(x) \gt 0$’
6.   the test point method for sentences like ‘$f(x) \gt g(x)$’
7.   the test point method ‘in a nutshell’ and additional practice
8.   absolute value as distance from zero
9.   absolute value as distance between two numbers
10.   circles and the ‘completing the square’ technique
11.   review of lines and slopes of lines
12.   distance between points; the midpoint formula
13.   sketching regions in the coordinate plane
14.   testing equations for symmetry
15.   more on difference quotients
16.   [stub] finding the domain and range of a function
17.   [stub] working with linear functions: finding a new point, given a point and a slope
18.   [stub] reading information from the graph of a function
19.   [stub] increasing and decreasing functions
20.   [stub] basic function models you must know
21.   [stub] piecewise-defined functions
22.   [stub] direct and inverse variation
23.   [stub] proportionality problems
24.   What is the graph of $\,y = f(x)\,$?
25.   shifting graphs up/down/left/right
26.   horizontal and vertical stretching/shrinking
27.   reflecting about axes, and the absolute value transformation
28.   multi-step practice with all the graphical transformations
29.   even and odd functions
30.   extreme values of functions (max/min)
31.   review of quadratic functions: vertex form, max/min, intercepts, more
32.   max/min problems resulting in quadratic functions
33.   combining functions to get new functions
34.   composition of functions
35.   writing a function as a composition
36.   using a function box ‘backwards’
37.   one-to-one functions
38.   ‘undoing’ a one-to-one function; inverse functions
39.   properties of inverse functions
40.   finding inverse functions (when there's only one $x$ in the formula)
41.   finding inverse functions (switch input/output names method)
42.   the graph of an inverse function
43.   introduction to polynomials
44.   relationship between the zeros (roots) and factors of polynomials
45.   end behavior of polynomials
46.   turning points of polynomials
47.   long division of polynomials
48.   the division algorithm
49.   synthetic division
50.   the remainder theorem
51.   ‘trapping’ the roots of a polynomial; an interval guaranteed to contain all real roots
52.   introduction to complex numbers
53.   arithmetic with complex numbers
54.   the square root of a negative number
55.   the complex conjugate
56.   the quadratic formula revisited; the discriminant
57.   multiplicity of zeros and graphical consequences
58.   discussion: graphing polynomials with technology (optional; no exercises)
59.   Solve an equation? Find a Zero? Your Choice!
60.   the fundamental theorem of algebra
61.   polynomials with real number coefficients
62.   solving polynomial equations
63.   introduction to rational functions
64.   introduction to asymptotes
65.   introduction to puncture points (holes)
66.   finding vertical asymptotes
67.   finding ‘puncture points’ of graphs
68.   finding horizontal asymptotes
69.   finding slant asymptotes
70.   exponential functions: review and additional properties
71.   linear versus exponential functions
72.   recognizing linear and exponential behavior from tables of data
73.   the natural exponential function
74.   introduction to average rate of change
75.   introduction to instantaneous rate of change and tangent lines
76.   a special property of the natural exponential function
  1. simple versus compound interest
  2. the compound interest formula
  3. continuous compounding
  4. introduction to logarithmic functions
  5. properties of logarithmic functions
  6. exponential growth problems
  7. laws of logarithms
  8. change of base formula for logarithms
  9. solving exponential equations
  10. solving logarithmic equations
  11. doubling time; half-life
  12. introduction to trigonometry
  13. the right triangle approach to trigonometry
  14. the unit circle approach to trigonometry
  15. terminal points
  16. special triangles
  17. reference numbers
  18. radian measure
  19. the trigonometric functions
  20. signs of trigonometric functions
  21. finding trigonometric values of special angles
  22. more on finding trigonometric values
  23. fundamental identities: the Pythagorean Identity; cosine is even; sine is odd
  24. periodic functions; the period of a periodic function
  25. graphs of sine and cosine
  26. graphing $y = a\sin(k(x-b)$ and $y = a\cos k(x-b)$
  27. amplitude, period, and phase shift
  28. graphs of tangent and secant
  29. length of a circular arc
  30. Why the name ‘radian measure’?
  31. area of a circular sector
  32. right triangle trigonometry: SOHCAHTOA
  33. area of a triangle
  34. the Law of Sines
  35. given two sides and a non-included angle, how many triangles?
  36. the Law of Cosines
  37. solving triangles; simple problems
  38. the arccosine function
  39. using the Law of Cosines in the ‘SSS’ case
  40. solving triangles; more advanced problems
  41. verifying trigonometric identities
  42. more trigonometric identities
  43. additional and subtraction formulas for sine and cosine
  44. double-angle formulas for sine and cosine
  45. trying to ‘undo’ trigonometric functions
  46. the arcsine function
  47. the arccosine function
  48. the arctangent function
  49. solving trigonometric equations
  50. solving pseudo-quadratic equations
  51. introduction to vectors
  52. working with the arrow representation for vectors
  53. working with the analytic representation for vectors
  54. unit vectors
  55. formula for the length of a vector
  56. horizontal and vertical components of a vector
  57. direction and magnitude of a vector
  58. bearing and velocity vectors; finding true speed and direction
  59. forces acting on an object
  60. partial fraction decomposition/expansion (PFE)
  61. PFE: distinct linear factors
  62. PFE: repeated linear factors
  63. PFE: irreducible quadratic factors
  64. introduction to conic sections
  65. identifying conics by the discriminant
  66. parabolas
  67. parabolas as reflectors/collectors
  68. equations of parabolas
  69. finding the equation of a parabola
  70. ellipses
  71. the reflecting property of ellipses
  72. equations of ellipses
  73. finding the equation of an ellipse
  74. hyperbolas
  75. graphing a hyperbola
  76. shifted conics