Subject
College Algebra
Functions and their graphs, polynomial and rational behavior, exponentials and logarithms, systems, sequences, and series.
Begin at I if you’re new to College Algebra.
Function Domain
Find the values of x where a function is defined. The main rule for rational functions: denominators can't be zero.
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Parent Functions
The basic function shapes you'll meet again and again: linear, quadratic, cubic, square root, absolute value, reciprocal.
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Function Transformations
Shift, reflect, and stretch a parent function to build new functions. The rules become mechanical once you see the pattern.
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Piecewise Functions
Functions that use different rules on different intervals. To evaluate, decide which piece applies, then apply that rule.
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Even and Odd Functions
Classify a function as even (symmetric about y-axis), odd (symmetric about origin), or neither.
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Average Rate of Change
Slope between two points on any function: (f(b) - f(a)) / (b - a). Generalizes 'rise over run' beyond lines.
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Polynomial End Behavior
What a polynomial does at the far left and far right of its graph. Decided entirely by the leading term: degree even/odd, coefficient positive/negative.
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Zeros of Polynomials
Find all rational zeros of a polynomial using the Rational Root Theorem and synthetic division.
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Complex Zeros and the Fundamental Theorem
Every degree-n polynomial has n complex zeros. With real coefficients, complex zeros come in conjugate pairs.
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Rational Functions and Asymptotes
Find the vertical and horizontal asymptotes of f(x) = p(x)/q(x). VAs come from denominator zeros; HAs from comparing degrees.
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Polynomial and Rational Inequalities
Solve inequalities with a sign chart: factor, find boundaries, test intervals. Each problem here has a bounded solution — report its length.
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Partial Fractions
Split a rational expression into simpler pieces. The cover-up method finds each coefficient in one step.
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Exponential Functions
Evaluate functions of the form f(x) = a · b^x. The variable is in the exponent — apply exponent rules to compute.
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Logarithmic Functions
log_b(x) asks: what exponent on b gives x? Evaluate by rewriting the input as a power of the base.
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Change of Base
Rewrite log_b(x) as a ratio of logs in a friendlier base. Pick a base where both numbers are clean powers.
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Exponential Growth and Decay
Apply A = A₀ · b^(t/p) to populations, half-lives, and investments. Find the amount after a given time, or the time to reach a given amount.
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Nonlinear Systems
Substitute, solve the resulting quadratic, and check. Lines meet parabolas in 0–2 points; same for lines and circles.
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Arithmetic and Geometric Sequences
Two patterns that show up everywhere: arithmetic adds a constant each step, geometric multiplies by one. Use the formulas to find any term.
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Series and Sigma Notation
Sum the terms of a sequence. Use sigma notation to compress a long sum, and the arithmetic and geometric series formulas to skip the term-by-term work.
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Binomial Theorem
Expand (a + b)^n using binomial coefficients. Pick out individual terms without multiplying everything.
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