That shift is the point of the course, and the tool below turns your own notes into questions about it.
WeSolve+ reads the whole document and writes the questions for you
Upload your PDF, photograph your notebook, or point the camera. WeSolve+ writes questions from that material, explains why each answer is right, reads the chapter back to you as a podcast, and remembers every item you missed until you own it.
The tool below is a small browser-only tool and it is not WeSolve+: paste a few lines and text rules turn them into cards on the spot. The real app, the one that uses AI, is behind the link above.
Function behaviour, from your notes
This is a browser-only tool, and that is all it isIt splits the text you paste by rule, and nothing else. WeSolve+ is a different thing entirely: it reads your whole PDF with AI, writes the reasoning behind every question, speaks the chapter back to you, and remembers what you missed so it can return it. Try the real app now, free!
This tool splits the text you paste by rule and returns cards. It cannot factor, graph or solve anything, so it works on the rules you already wrote down.
Degree is the first thing to read off a polynomial
It caps how many real zeros and turning points are possible and it decides the end behaviour together with the leading coefficient. Two numbers, taken from the expression without any algebra, tell you the overall shape before you plot a single point. Students who start by factoring are working harder than the question requires.
Multiplicity is the detail that separates grades
A zero of even multiplicity touches the axis and turns back; odd multiplicity crosses. That is why two polynomials with identical zeros can look completely different, and why a sketch that ignores multiplicity is marked down even when every intercept is right. It is one rule and it pays every time a graph is involved.
Rational functions break in three distinguishable ways
A vertical asymptote where the denominator vanishes and the numerator does not. A hole where a factor cancels from both. And an end behaviour set by comparing degrees. Rational functions are examined on telling these apart, so factoring first and looking second is the reliable order.
Rate of change is what the course is actually about
Average rate of change over an interval is the slope of the line joining the endpoints, and this unit introduces it deliberately early. Every later topic, and the whole of calculus after it, is a refinement of that one idea. Treating it as a formula rather than as the subject is the most common way to find the course harder than it is.
Where marks are lost
On describing rather than justifying. Saying a graph rises on an interval is an observation; saying it rises because the leading coefficient is positive and the degree is odd is a reason. Questions in this unit almost always want the reason, and the reason is usually one clause longer than the observation.
What to photograph for Polynomial and Rational Functions
Graphs you sketched with your reasoning written beside them. Related: Calculus AB Unit 1, photo to quiz and pricing.
Sources used on this page
- College Board, AP Precalculus
- Polynomial
- Rational function
- Function (mathematics)
- Trigonometric functions
- Active recall
- Spaced repetition
- Testing effect
- Forgetting curve
- Generation effect
- Judgment of learning
- Metacognition
- Desirable difficulty
- Distributed practice
- Formative assessment
- Flashcard
- Cloze test
- Multiple choice
- Test (assessment)
- Educational assessment
- Advanced Placement
- Curriculum
- Study skills
- Study guide
- Note-taking
- Overlearning
- Instructional scaffolding
- Item analysis
- Mastery learning
| Feature | Where it comes from | What it tells you |
|---|---|---|
| Degree | Highest exponent | Maximum zeros and turning points |
| Leading coefficient | Sign of the first term | Which way the ends go |
| Multiplicity | Repeated factors | Touch or cross at each zero |
| Vertical asymptote | Denominator zero, numerator not | Where the function breaks |
| Hole | Factor cancels in both | A single missing point |
| Horizontal asymptote | Comparing degrees | Behaviour far from the origin |
What is AP Precalculus Unit 1?
Polynomial and Rational Functions: reading a function's behaviour, including zeros, multiplicity, asymptotes, end behaviour and average rate of change.
What should I read off a polynomial first?
Degree and leading coefficient. Those two tell you the maximum number of zeros and turning points and which way the ends go, before any factoring.
Why does multiplicity matter?
Because an even multiplicity touches the axis and turns back while an odd one crosses. Two polynomials with the same zeros can look completely different.
How do I tell an asymptote from a hole?
A hole appears where a factor cancels from both numerator and denominator. A vertical asymptote appears where only the denominator vanishes.
Why introduce rate of change so early?
Because it is what the course is about. Every later topic refines the idea, and calculus is its continuation.
What loses marks here?
Describing without justifying. Say that the graph rises because the degree is odd and the leading coefficient positive, not simply that it rises.
Last updated: 2026-08-15