21 topics
Pre-calculus help, function by function
Pre-calculus is less a subject than a collection of function families, and the students who struggle are usually treating it as ten unrelated units instead of one question asked ten times: what does this family look like, what breaks it, and what do the parameters do. The course gets dramatically smaller once that pattern is visible.
Where students get stuck
Transformations — the horizontal ones go the wrong way
They genuinely are backwards, and it is not a trick. f(x − 3) shifts right because you now need an x that is 3 larger to feed the function the same input. Same reason f(2x) compresses rather than stretches. Vertical transformations act on the output and behave the way you expect; horizontal ones act on the input and invert. Two rules, not eight.
When is it a hole and when is it an asymptote?
Factor first, always. A factor that cancels between numerator and denominator is a hole; a factor that survives in the denominator is a vertical asymptote. Graphing before factoring is why holes get missed — a calculator will not show you a single missing point, and the marks for it are usually on the sketch.
Log rules feel arbitrary
They are exponent rules wearing a different hat. log(ab) = log a + log b is just x^m · x^n = x^(m+n) read backwards, because a logarithm is an exponent. If you can state what log_b(x) means in one sentence — the power b must be raised to in order to get x — every rule in the unit is derivable rather than memorised, including why log of a negative is undefined.
I can't tell arithmetic from geometric, or which formula to use
Subtract consecutive terms: constant difference means arithmetic. Divide consecutive terms: constant ratio means geometric. Then the only remaining choice is term versus sum — 'the 20th term' needs the general term formula, 'the first 20 terms' needs the series formula. Most lost marks here are using a sum formula for a term question, not the algebra.
What's covered
Pre-Calculus topics you can work through with a tutor, generate practice on, or turn into flashcards and a study plan.
Functions
- Domain, range, and function notation
- Even, odd, and piecewise functions
- Transformations of parent functions
- Composition and inverse functions
- Function operations and their domains
Polynomial and rational functions
- End behaviour and degree
- The factor and remainder theorems
- Roots, multiplicity, and sketching
- Vertical, horizontal, and oblique asymptotes
- Holes and removable discontinuities
- Polynomial and rational inequalities
Exponentials and logarithms
- Exponential growth and decay
- The definition of a logarithm
- Log laws and change of base
- Solving exponential and logarithmic equations
- Compound interest and half-life models
Sequences, series, and more
- Arithmetic and geometric sequences
- Finite and infinite geometric series
- Sigma notation
- The binomial theorem
- Conic sections
Pre-Calculus questions
Is this the same as Functions or Advanced Functions?
Largely, yes. Course names differ by region — Pre-Calculus, Functions, Advanced Functions, Algebra 2 — but the content above is the shared core. Tell it which course you are in and it will match the notation and conventions your class uses.
Will it help me get ready for calculus specifically?
Yes, and the honest answer is that most first-week calculus trouble is unfinished pre-calculus — factoring, log rules, and the unit circle. You can ask it to drill exactly those, which is a better use of a week than starting limits early.
Can I build a study plan across the whole course?
Yes. It can lay out a multi-day plan across the units above, generate practice for each, and track which topics you keep missing so the plan reweights toward them.
Stuck on pre-calculus right now?
Talk it through out loud, share your screen, and watch it worked out step by step on a whiteboard.
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