One Mathematical Cat, Please!

This is a complete, sequenced, free course.

It has unlimited, randomly-generated exercises
and worksheets in every lesson.

It can be used for a year-long high school course
or a one-semester college course.
Or, just enjoy it yourself, at your own pace!

lessons 1–87

lessons 88–160
get a print version
of these lessons

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. finding the domain and range of a function
17. working with linear functions: finding a new point, given a point and a slope
18. increasing and decreasing functions
19. reading information from the graph of a function
20. basic function models you must know
21. piecewise-defined functions
22. direct and inverse variation
23. 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
77. (optional) a surprising appearance of the irrational number  e ; diluting a toxic liquid
78. simple versus compound interest
79. continuous compounding
80. logarithmic functions: review and additional properties
81. logarithm summary: properties, formulas, laws
82. exponential growth and decay: introduction
83. exponential growth and decay: relative growth rate
84. solving exponential equations
85. solving logarithmic equations
86. solving exponential growth and decay problems
87. doubling time, half-life
88. introduction to trigonometry
89. the right triangle approach to trigonometry
90. the unit circle approach to trigonometry
91. compatibility of the right triangle and unit circle approaches
92. special triangles and common trigonometric values
93. reference angles
94. radian measure: associating real numbers with points on the unit circle
95. the trigonometric functions
96. signs of all the trigonometric functions
97. trigonometric values of special angles
98. fundamental trigonometric identities
99. periodic functions
100. the period of a periodic function
101. graphs of sine and cosine
102. graphing generalized sine and cosines, like $y = a\sin k(x\pm b)$ and $y = a\cos (kx\pm B)$
103. amplitude, period, and phase shift
104. given amplitude, period, and phase shift, write an equation
105. graph of the tangent function
106. graph of the secant function
107. review of circles (and related concepts)
108. length of a circular arc (and related concepts)
109. (optional) the angle subtended by a geometric object at an external point
110. area of a circular sector
111. sine and cosine as scaling factors of hypotenuse
112. area of a triangle
113. law of sines
114. Given two sides and a non-included angle, how many triangles?
115. law of cosines
116. (optional) revisiting ‘equal’ versus ‘approximately equal’
117. using the Law of Cosines in the SSS case (& introduction to the arccosine function)
118. summary: solving triangles (all types)
119. verifying trigonometric identities
120. addition and subtraction formulas for sine and cosine
121. double-angle formulas for sine and cosine
122. trying to ‘undo’ trigonometric functions
123. inverse trigonometric function: arcsine
124. inverse trigonometric function: arccosine
125. inverse trigonometric function: arctangent
126. inverse trigonometric function problems: all mixed up
127. solving simple trigonometric equations
128. solving pseudo-quadratic equations
129. introduction to vectors
130. working with the arrow representation for vectors
131. working with the analytic representation for vectors
132. unit vectors
133. formula for the length of a vector
134. vectors: from direction/magnitude to horizontal/vertical components
135. vectors: from horizontal/vertical components to direction/magnitude
136. vector application: finding true speed and direction
137. vector application: forces acting an an object in equilibrium
138. introduction to partial fraction expansion/decomposition (PFE)
139. PFE: linear factors
140. PFE: irreducible quadratic factors
141. introduction to conic sections
142. identifying conics by the discriminant
143. (optional) intersecting an infinite double cone and a plane: looking at the equations
144. (optional) How is it that the conic discriminant tells us the type of conic?
145. parabolas: definition, reflectors/collectors, derivation of equations
146. finding the equation of a parabola
147. definition of an ellipse
148. reflecting property of an ellipse
149. equations of ellipses in standard form: foci on the $\,x$-axis
150. (optional) getting an ellipse by stretching/shrinking a circle
151. (optional) two useful transforms: reflection about the line $\,y = x\,$ and counterclockwise rotation by $\,90^\circ$
152. equations of ellipses in standard form: foci on the $\,y$-axis
153. summary: equations of ellipses in standard form
154. definition of a hyperbola
155. equations of hyperbolas in standard form
156. graphing hyperbolas
157. reflecting property of a hyperbola
158. shifting conics (shifting any equation in two variables)
159. polar coordinates
160. parametric equations