---
type: CKG Bundle
title: Pre-Calculus
tags: [STEM & Engineering]
timestamp: 2026-06-18T00:00:00Z
ckg:
  id: pre-calc
  nodes: 307
  license: CC BY 4.0
  source: Graphify.md ckg-mcp
---

# Pre-Calculus — Compressed Knowledge Graph

```csv
1,Real Number System,,FOUND
2,Number Line,1,FOUND
3,Order of Operations,1,FOUND
4,Variables and Expressions,1|3,FOUND
5,Coordinate Plane,2,FOUND
6,Ordered Pairs,5,FOUND
7,Input and Output,4,FOUND
8,Function Definition,7,FUNC
9,Function Notation,8,FUNC
10,Domain,8,FUNC
11,Range,8,FUNC
12,Independent Variable,7,FOUND
13,Dependent Variable,7|12,FOUND
14,Interval Notation,2,FOUND
15,Set-Builder Notation,4,FOUND
16,Vertical Line Test,8|5,FUNC
17,Function Evaluation,9,FUNC
18,Piecewise Functions,8|10|14,FUNC
19,Domain Restrictions,10,FUNC
20,Graphing Functions,5|9,FUNC
21,Intercepts,20,FUNC
22,X-Intercept,21,FUNC
23,Y-Intercept,21,FUNC
24,Increasing Functions,20|27,FUNC
25,Decreasing Functions,20|27,FUNC
26,Constant Functions,8,FUNC
27,Function Behavior,20,FUNC
28,Positive and Negative Regions,20|22,FUNC
29,Bounded Functions,11|20,FUNC
30,Even Functions,20|32,FUNC
31,Odd Functions,20|32,FUNC
32,Symmetry,20,FUNC
33,Change in Tandem,7|12|13,RATE
34,Average Rate of Change,33|9,RATE
35,Slope,34,RATE
36,Slope Formula,35|6,RATE
37,Rate of Change Units,34,RATE
38,Secant Line,36|20,RATE
39,Linear Rate of Change,35,RATE
40,Nonlinear Rate of Change,34|39,RATE
41,Concavity from Rates,40,RATE
42,Increasing Rate of Change,40|24,RATE
43,Decreasing Rate of Change,40|25,RATE
44,Linear Function Definition,8|35,LNQD
45,Slope-Intercept Form,44|23,LNQD
46,Point-Slope Form,44|36,LNQD
47,Standard Form of a Line,44,LNQD
48,Parallel Lines,35|44,LNQD
49,Perpendicular Lines,35|44,LNQD
50,Linear Regression,44|6,LNQD
51,Residuals,50,LNQD
52,Correlation,50,LNQD
53,Quadratic Function Definition,8|66,LNQD
54,Standard Form of Quadratic,53,LNQD
55,Vertex Form,53|60,LNQD
56,Factored Form,53,LNQD
57,Vertex,55,LNQD
58,Axis of Symmetry,57|32,LNQD
59,Parabola,53|20,LNQD
60,Completing the Square,54,LNQD
61,Quadratic Formula,54|60,LNQD
62,Discriminant,61,LNQD
63,Complex Numbers,1,LNQD
64,Imaginary Unit,63,LNQD
65,Complex Zeros,62|64,LNQD
66,Polynomial Definition,4,POLY
67,Degree of a Polynomial,66,POLY
68,Leading Coefficient,66,POLY
69,Monomial,66,POLY
70,Binomial,66,POLY
71,Trinomial,66,POLY
72,Polynomial Addition,66,POLY
73,Polynomial Multiplication,66|72,POLY
74,Factoring Polynomials,66|73,POLY
75,Factor Theorem,74|79,POLY
76,Remainder Theorem,75|77,POLY
77,Synthetic Division,66,POLY
78,Long Division of Polynomials,66,POLY
79,Zeros of Polynomials,66|22,POLY
80,Multiplicity of Zeros,79,POLY
81,Real Zeros,79,POLY
82,Complex Conjugate Zeros,79|65,POLY
83,Fundamental Theorem of Algebra,67|79|63,POLY
84,Polynomial End Behavior,67|68,POLY
85,Leading Term Test,84,POLY
86,Turning Points,87,POLY
87,Polynomial Graphs,79|80|84|20,POLY
88,Local Maximum,86,POLY
89,Local Minimum,86,POLY
90,Global Maximum,88,POLY
91,Global Minimum,89,POLY
92,Intermediate Value Theorem,87|10,POLY
93,Rational Function Definition,66|8,RATL
94,Rational Expressions,93,RATL
95,Simplifying Rational Expr,94|74,RATL
96,Vertical Asymptotes,93|19,RATL
97,Horizontal Asymptotes,93|84,RATL
98,Slant Asymptotes,97|78,RATL
99,Holes in Graphs,95|96,RATL
100,Rational Function Zeros,93|79,RATL
101,Rational Function Domain,93|19,RATL
102,End Behavior of Rationals,97|98,RATL
103,Rational Function Graphs,96|97|99|100|20,RATL
104,Rational Inequalities,103|28,RATL
105,Parent Functions,20|44|53|93,TMOD
106,Vertical Translation,105,TMOD
107,Horizontal Translation,105,TMOD
108,Vertical Stretch,105,TMOD
109,Vertical Compression,108,TMOD
110,Horizontal Stretch,105,TMOD
111,Horizontal Compression,110,TMOD
112,Reflection Over X-Axis,105,TMOD
113,Reflection Over Y-Axis,105,TMOD
114,Combined Transformations,106|107|108|110|112|113,TMOD
115,Transformation Order,114,TMOD
116,Writing Transformed Eqns,114|9,TMOD
117,Function Model Selection,105|34,TMOD
118,Data Analysis for Models,117|6,TMOD
119,Regression Models,50|118,TMOD
120,Model Assumptions,117,TMOD
121,Model Validation,119|51,TMOD
122,Extrapolation,119,TMOD
123,Interpolation,119,TMOD
124,Residual Analysis,51|121,TMOD
125,Goodness of Fit,124|52,TMOD
126,Competing Models,125,TMOD
127,Sequence Definition,8,SEQX
128,Arithmetic Sequence,127|35,SEQX
129,Common Difference,128,SEQX
130,Arithmetic Sequence Formula,128|129,SEQX
131,Geometric Sequence,127,SEQX
132,Common Ratio,131,SEQX
133,Geometric Sequence Formula,131|132,SEQX
134,Recursive Formulas,127,SEQX
135,Explicit Formulas,127|130|133,SEQX
136,Sequence Convergence,131|11,SEQX
137,Sequence Divergence,131|136,SEQX
138,Exponential Function Def,8|131,SEQX
139,Base of Exponential,138,SEQX
140,Growth Factor,138|132,SEQX
141,Decay Factor,138|132,SEQX
142,Exponential Growth,138|140,SEQX
143,Exponential Decay,138|141,SEQX
144,Natural Base e,138,SEQX
145,Continuous Growth,144|142,SEQX
146,Compound Interest,142|145,SEQX
147,Exponential Equations,138|3,SEQX
148,Exponential Graphs,138|20,SEQX
149,Exponential Transformations,148|114,SEQX
150,Exponential Modeling,142|143|118,SEQX
151,Half-Life,143,SEQX
152,Doubling Time,142,SEQX
153,Percent Rate of Change,140|141,SEQX
154,Logarithm Definition,138,LOGR
155,Common Logarithm,154,LOGR
156,Natural Logarithm,154|144,LOGR
157,Logarithmic Form,154,LOGR
158,Exponential-Log Conversion,154|147,LOGR
159,Properties of Logarithms,154,LOGR
160,Product Rule for Logs,159,LOGR
161,Quotient Rule for Logs,159,LOGR
162,Power Rule for Logs,159,LOGR
163,Change of Base Formula,159|155|156,LOGR
164,Logarithmic Equations,158|159,LOGR
165,Logarithmic Graphs,154|20,LOGR
166,Logarithmic Transformations,165|114,LOGR
167,Logarithmic Modeling,165|150,LOGR
168,Logarithmic Scale,154|2,LOGR
169,Semi-Log Plot,168|148,LOGR
170,Linearizing Exp Data,169|50,LOGR
171,Log-Log Plots,169,LOGR
172,Exponential Regression,170|119,LOGR
173,Power Regression,171|119,LOGR
174,Composition of Functions,8|9|17,FUNC
175,Domain of Compositions,174|10|19,FUNC
176,Decomposing Functions,174,FUNC
177,Inverse Function Definition,8|174,FUNC
178,One-to-One Functions,177,FUNC
179,Horizontal Line Test,178|16,FUNC
180,Finding Inverses Algebraically,177|9,FUNC
181,Inverse Function Graphs,177|20|113,FUNC
182,Restricting Domains,178|10,FUNC
183,Angle Definition,,TRIG
184,Degree Measure,183,TRIG
185,Radian Measure,183|144,TRIG
186,Degree-Radian Conversion,184|185,TRIG
187,Arc Length,185,TRIG
188,Sector Area,185|187,TRIG
189,Standard Position,183|5,TRIG
190,Terminal Side,189,TRIG
191,Reference Angle,190|184,TRIG
192,Coterminal Angles,190|184,TRIG
193,Unit Circle Definition,5|185,TRIG
194,Unit Circle Coordinates,193|6,TRIG
195,Special Angles,194|191,TRIG
196,Quadrantal Angles,194,TRIG
197,Signs by Quadrant,194,TRIG
198,Sine Function,193|194,TRIG
199,Cosine Function,193|194,TRIG
200,Tangent Function,198|199,TRIG
201,Reciprocal Trig Functions,198|199|200,TRIG
202,Secant Function,201|199,TRIG
203,Cosecant Function,201|198,TRIG
204,Cotangent Function,201|200,TRIG
205,Trig Function Evaluation,198|199|200|195,TRIG
206,Exact Trig Values,205|195,TRIG
207,Right Triangle Trig,183|198|199|200,TRIG
208,SOH-CAH-TOA,207,TRIG
209,Sine Graph,198|20,TRIG
210,Cosine Graph,199|20,TRIG
211,Amplitude,209|210,TRIG
212,Period,209|210|185,TRIG
213,Midline,209|210,TRIG
214,Frequency,212,TRIG
215,Phase Shift,209|210|107,TRIG
216,Vertical Shift of Trig,209|210|106,TRIG
217,Sinusoidal Functions,211|212|213|215|216,TRIG
218,Tangent Graph,200|20,TRIG
219,Period of Tangent,218|212,TRIG
220,Asymptotes of Tangent,218|96,TRIG
221,Secant Graph,202|20,TRIG
222,Cosecant Graph,203|20,TRIG
223,Cotangent Graph,204|20,TRIG
224,Graphing Trig Transformations,217|114,TRIG
225,Sinusoidal Modeling,217|118,TRIG
226,Periodic Data Analysis,225,TRIG
227,Tidal Models,225,TRIG
228,Temperature Models,225,TRIG
229,Daylight Hour Models,225,TRIG
230,Sinusoidal Regression,225|119,TRIG
231,Period from Data,226|212,TRIG
232,Amplitude from Data,226|211,TRIG
233,Pythagorean Identity,198|199,TRIG
234,Reciprocal Identities,201,TRIG
235,Quotient Identities,200|198|199,TRIG
236,Cofunction Identities,198|199|191,TRIG
237,Even-Odd Identities,198|199|30|31,TRIG
238,Sum and Difference Formulas,233|198|199,TRIG
239,Double Angle Formulas,238,TRIG
240,Verifying Identities,233|234|235|236,TRIG
241,Simplifying Trig Expr,240,TRIG
242,Arcsine,198|177|182,TRIG
243,Arccosine,199|177|182,TRIG
244,Arctangent,200|177|182,TRIG
245,Inverse Trig Domains,242|243|244|10,TRIG
246,Inverse Trig Ranges,242|243|244|11,TRIG
247,Inverse Trig Graphs,242|243|244|181,TRIG
248,Composing Trig Inverses,242|243|244|174,TRIG
249,Solving Trig Equations,205|242|243|244,TRIG
250,General Solutions,249|192,TRIG
251,Solutions in an Interval,249|14,TRIG
252,Trig Inequalities,249|28,TRIG
253,Multiple Angle Equations,249|239,TRIG
254,Polar Coordinate System,5|185,POLR
255,Polar Point Plotting,254,POLR
256,Rectangular to Polar,254|198|199,POLR
257,Polar to Rectangular,254|198|199,POLR
258,Multiple Polar Represent,254|192,POLR
259,Polar Function Definition,254|8,POLR
260,Polar Graphs,259|255,POLR
261,Circles in Polar Form,260,POLR
262,Cardioids,260,POLR
263,Limaçons,260,POLR
264,Rose Curves,260,POLR
265,Petal Count of Roses,264,POLR
266,Symmetry of Polar Graphs,260|32,POLR
267,Polar Rate of Change,260|34,POLR
268,Polar Graph Intersections,260,POLR
269,Parametric Equations,5|8,POLR
270,Parameter,269,POLR
271,Parametric Graphs,269|20,POLR
272,Eliminating the Parameter,269|9,POLR
273,Parametric Motion,271|270,POLR
274,Position Functions,273,POLR
275,Velocity from Parametrics,273|34,POLR
276,Direction of Motion,275,POLR
277,Parametric Modeling,273|118,POLR
278,Projectile Motion Model,277|53,POLR
279,Vector Definition,,VECM
280,Vector Notation,279,VECM
281,Magnitude of a Vector,279|1,VECM
282,Direction of a Vector,279|183,VECM
283,Component Form,279|5,VECM
284,Unit Vectors,281|283,VECM
285,Vector Addition,283,VECM
286,Vector Subtraction,283|285,VECM
287,Scalar Multiplication,279|283,VECM
288,Dot Product,283|285,VECM
289,Vector Applications,285|286|287,VECM
290,Matrix Definition,,VECM
291,Matrix Dimensions,290,VECM
292,Matrix Addition,290|291,VECM
293,Matrix Subtraction,290|292,VECM
294,Scalar Mult of Matrices,290,VECM
295,Matrix Multiplication,290|291,VECM
296,Identity Matrix,290|295,VECM
297,Determinant,290|295,VECM
298,Inverse of a Matrix,296|297,VECM
299,Linear Transformations,295|283,VECM
300,Rotation Matrix,299|185,VECM
301,Reflection Matrix,299,VECM
302,Matrix Applications,295|298|299,VECM
303,Calculator Strategies,17|147|164|249,VECM
304,Free Response Strategy,303,VECM
305,Multiple Choice Strategy,303,VECM
306,Time Management on Exams,303,VECM
307,Showing Work on Free Resp,304,VECM
```