AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS
Unit 1A — REVIEW
Notes — Six Big Ideas in One Place
💡 What this review covers This document consolidates the six core ideas of Unit 1A:
Use it as a cumulative reference. Each section restates the essential ideas in compact form, then highlights the key skills and the warnings most students need to remember on exam day. |
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The six concepts of Unit 1A at a glance — each will resurface throughout the rest of the course.
1. Change in Tandem (Topic 1.1)
A function pairs every input with exactly one output. Two related quantities are said to vary in tandem when they change together according to such a function.
- Increasing on (a, b): outputs grow as inputs grow.
- Decreasing on (a, b): outputs shrink as inputs grow.
- Concave up: graph smiles; the rate of change is increasing.
- Concave down: graph frowns; the rate of change is decreasing.
- Inflection point: where concavity changes.
- Zeros: the inputs where f(x) = 0; x-intercepts on the graph.
2. Rates of Change (Topic 1.2)
💡 Two flavors of rate • Average rate of change (AROC) on [a, b] = ( f(b) − f(a) ) / ( b − a ) = slope of the secant line. • Instantaneous rate of change (IROC) at x = c is approximated by AROC on tiny intervals containing c. AROC has units of (output units)/(input units). The sign of AROC tells you whether the quantity grew (+), shrank (−), or returned to its starting value (0) over the interval. |
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3. Rates of Change in Linear and Quadratic Functions (Topic 1.3)
Function type | AROC behavior | Difference table signature |
|---|---|---|
Linear (degree 1) | Constant on every interval | 1st differences constant |
Quadratic (degree 2) | AROCs themselves are linear | 2nd differences constant |
Cubic (degree 3) | 2nd differences are linear | 3rd differences constant |
Polynomial of degree n | n − 1 layers of structure | n-th differences constant |
Concavity and AROCs:
- AROCs increasing across consecutive equal-length intervals → graph is concave up.
- AROCs decreasing across consecutive equal-length intervals → graph is concave down.
4. Polynomial Functions and Rates of Change (Topic 1.4)
A polynomial of degree n has form aₙxⁿ + … + a₀ with aₙ ≠ 0.
- Local maximum/minimum: a peak/valley relative to nearby inputs.
- Global maximum/minimum: the largest/smallest value of f on its entire domain.
- Between any two distinct real zeros, there must be a local extremum.
- Even-degree polynomials always have a global max or global min (based on sign of leading coefficient).
- Inflection points are where concavity changes — equivalently, where the rate of change shifts from increasing to decreasing or vice versa.
5. Polynomial Functions and Complex Zeros (Topic 1.5A)
💡 The Fundamental Theorem of Algebra A polynomial of degree n has exactly n complex zeros, counting multiplicity. Some may be real; non-real complex zeros of real-coefficient polynomials always come in conjugate pairs (if a + bi is a zero, so is a − bi). | ||
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Multiplicity at a real zero | Graph behaviour | Sign change? |
1 (or any odd m) | Crosses through the x-axis | Yes |
2 (or any even m) | Touches the x-axis and bounces | No |
3, 5, 7, … (odd, ≥ 3) | Crosses with flattening | Yes |
4, 6, 8, … (even, ≥ 4) | Bounces with stronger flattening | No |
To determine the degree of a polynomial that fits a table of values at equally spaced inputs: take repeated differences. The order at which they first become constant is the degree.
6. Even and Odd Polynomials (Topic 1.5B)
- Even function: f(−x) = f(x); graph is symmetric across the y-axis.
- Odd function: f(−x) = −f(x); graph is symmetric about the origin.
- A polynomial is even ⇔ all of its terms have even exponents.
- A polynomial is odd ⇔ all of its terms have odd exponents.
- Mixed exponents ⇒ neither even nor odd.
⚠️ Don't conflate ‘even degree’ with ‘even function’ Degree refers to the highest exponent. Even/odd function refers to symmetry. p(x) = x² + x has even degree but is neither even nor odd. |
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7. Polynomial Functions and End Behavior (Topic 1.6)
💡 Leading-term dominance For large |x|, p(x) behaves like its leading term aₙxⁿ. The end behavior is fully determined by:
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Degree | Leading coeff | x → ∞ | x → −∞ | Memory cue |
Even | Positive | +∞ | +∞ | Both ends UP |
Even | Negative | −∞ | −∞ | Both ends DOWN |
Odd | Positive | +∞ | −∞ | DOWN-left, UP-right |
Odd | Negative | −∞ | +∞ | UP-left, DOWN-right |
8. The Big Picture — A Diagnostic Routine
If you are handed a polynomial graph or formula on the AP exam, walk through this checklist:
- Step 1: Identify the degree (count zeros + multiplicities, or read from the formula) and the sign of the leading coefficient.
- Step 2: Predict end behavior using the four-cases table.
- Step 3: Locate all real zeros — note their multiplicities (touch vs cross).
- Step 4: Identify intervals of increase/decrease and any local extrema.
- Step 5: Identify intervals of concave up/concave down and any inflection points.
- Step 6: Check for symmetry (even/odd) — if you spot it, use it to halve your work.
9. Vocabulary Quick-Reference
Term | Meaning |
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Zero / root | An input where p(x) = 0 |
Multiplicity | How many times (x − a) appears as a factor |
Local extremum | A peak/valley relative to nearby inputs |
Global extremum | Largest/smallest value over the entire domain |
Inflection point | A point where the graph changes concavity |
AROC | Average rate of change = slope of secant line |
IROC | Instantaneous rate of change at one point |
Even function | f(−x) = f(x); y-axis symmetry |
Odd function | f(−x) = −f(x); origin symmetry |
End behavior | What p does as x → ±∞ |
Leading term | aₙxⁿ — the term with the highest exponent |
Conjugate root | Pair (a + bi, a − bi) of non-real zeros |
10. What Comes Next
Unit 1A focuses on polynomial functions. The next half of Unit 1 (1B) extends these ideas to rational functions — quotients of polynomials — bringing in vertical asymptotes, horizontal asymptotes, holes, and the connection between zeros of numerator and denominator. Most of the techniques you have learned (sign analysis, end behavior, multiplicity) carry directly into Unit 1B.