Common Mistakes
Pre-calculus · AP · Polynomial and Rational Functions
Students lose points on Change in Tandem when interval language, function rules, or input/output roles get fuzzy. Use this guide to catch and repair those mistakes early.
Frequent mistakes
### 1. “Increasing” without an interval
What it looks/sounds like: “The function is increasing” while the graph rises then falls.
Why it happens: Eyes track the overall trend instead of every pair of inputs on a stated interval.
How to fix: Require the phrase “increasing on ___.” Split the domain at turning points.
### 2. Wrong inequality direction for decreasing
What it looks/sounds like: Writes \(a < b \Rightarrow f(a) < f(b)\) while claiming decreasing.
Why it happens: The increasing definition is memorized and reused unchanged.
How to fix: Say: larger input → smaller output, so \(f(a) > f(b)\) when \(a < b\).
### 3. Mixing up input and output
What it looks/sounds like: Treats the dependent variable as the input in a verbal story.
Why it happens: The story leads with the interesting quantity (temperature, price) instead of the independent variable.
How to fix: Underline “as ___ changes” to identify the input; define \(x\) and \(f(x)\) explicitly.
### 4. Matching the wrong representation
What it looks/sounds like: Picks a table that shares some values but violates the rule for other inputs.
Why it happens: Checking only one or two pairs instead of the full rule.
How to fix: Test every given input; for graphs, verify several sample points against the equation or table.
### 5. Calling a non-function a function
What it looks/sounds like: Accepts a relation with two outputs for one input because “it looks continuous.”
Why it happens: Vertical-line / unique-output test is skipped.
How to fix: Ask: “Can this input produce two different outputs?” If yes, it is not a function.
### 6. Using later Unit 1 tools too early
What it looks/sounds like: Talks about holes, asymptotes, or end behavior on a pure tandem item.
Why it happens: Unit 1 vocabulary bleeds across topics.
How to fix: Stick to representations and increasing/decreasing definitions unless the stem asks for more.
How to avoid them
1. Always name the interval in the same sentence as increasing/decreasing. 2. Pick concrete \(a < b\) and compare outputs before writing a conclusion. 3. Label input/output in every table and story. 4. For matching items, verify all given pairs, not just the first. 5. Separate “is it a function?” from “is it increasing?”
Quick self-check
- [ ] My claim includes an interval (or explains why the whole domain works)
- [ ] My inequality direction matches increasing vs decreasing
- [ ] Input and output roles are labeled
- [ ] I did not invent asymptotes/holes on a tandem-only prompt
- [ ] I can redo one missed item with a full written justification
Related materials
- Exam Strategy — timed MC/FRQ moves for 1.1
- Study Guide — practice pacing and vocabulary
- Family Overview — home support questions
- Practice Problems and Review under topic 1.1