Polynomial, Rational, and Radical Functions

Outcome 1: Polynomial, Rational, and Radical Functions · Blueprint Pillar 3 · PGCC MAT-1350 (interim) · Download .docx

Objectives

Identify the degree, leading coefficient, and end behavior of a polynomial function.
Find zeros of a polynomial and interpret their multiplicity from the graph.
Identify vertical and horizontal asymptotes of a rational function.
State the domain of a radical function and explain domain restrictions.
Sketch a rough graph of a polynomial or rational function using key features.

Key terms

polynomial function: A sum of terms of the form axⁿ where n is a non-negative integer; degree is the highest power.
rational function: A quotient p(x)/q(x) where p and q are polynomials and q(x) ≠ 0.
zero: An x-value where f(x) = 0; appears as an x-intercept on the graph.
multiplicity: The number of times a root factor repeats; odd multiplicity = graph crosses; even = graph touches and turns.
vertical asymptote: x = a where q(a) = 0 and p(a) ≠ 0; the graph grows without bound near this line.
horizontal asymptote: A horizontal line y = L that the rational function approaches as |x| → ∞.
end behavior: Whether f(x) → +∞ or −∞ as x → +∞ and x → −∞; determined by degree and leading coefficient sign.
radical function: A function containing a root; f(x) = √(x) requires x ≥ 0 for real output.

The concept

Functions are the central object of precalculus. A function maps inputs to outputs — for each x in the domain, exactly one y value results. Understanding a function means being able to describe its shape, its zeros, its behavior at the extremes, and any values where it is undefined.

Polynomial functions are the most familiar. A polynomial of degree n looks like aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀. The degree controls how many turns the graph can make (at most n−1) and how many zeros it can have (at most n). End behavior is controlled entirely by the leading term aₙxⁿ: if n is even and aₙ > 0, both ends go up; if n is even and aₙ < 0, both ends go down; odd degree gives opposite ends (one up, one down).

Zeros are where the function equals zero — the x-intercepts. Factor the polynomial completely and set each factor to zero. Multiplicity matters. If (x − r)² appears (even exponent), the graph touches the x-axis at x = r and turns back — it does not cross. If (x − r)³ appears (odd exponent), the graph crosses with an S-curve at x = r. These behaviors are visible and testable.

Rational functions are more complex because they can be undefined. A vertical asymptote occurs wherever the denominator is zero and the numerator is nonzero. The function blows up — it goes to ±∞. A hole occurs if the numerator and denominator share a common factor — the graph has a missing point rather than an asymptote. Horizontal asymptotes describe what happens far out on the x-axis: if the degree of the numerator is less than the denominator's, y → 0; if equal, y → ratio of leading coefficients; if numerator degree is higher, there is no horizontal asymptote (oblique instead).

Radical functions involve roots. For even-index roots (square root, fourth root), the radicand must be non-negative — this restricts the domain. f(x) = √(x − 3) has domain x ≥ 3. For odd-index roots (cube root), there is no restriction — every real number has a real cube root.

Blueprint Pillar 3 — Technology and Society: Polynomial and rational models describe real phenomena — parabolic trajectories, population dynamics, signal attenuation over distance. Engineers use rational functions to model electrical circuits. Biologists use polynomial regression to fit population data. The ability to identify key features from an equation — without graphing technology — is a critical analytical skill in every STEM field.

Worked examples

Example 1: End behavior and zeros: f(x) = −2x³ + 5x. Leading term: −2x³. As x → +∞, f → −∞. As x → −∞, f → +∞. Factor: f(x) = x(−2x² + 5) = x(√(5/2) − x)(√(5/2) + x). Zeros at x = 0, x = ±√(5/2) ≈ ±1.58. Each zero has multiplicity 1 (odd) so the graph crosses at each.

Example 2: Horizontal asymptote: g(x) = (3x² + 1)/(x² − 4). Degrees equal (both 2). Horizontal asymptote: y = 3/1 = 3. Vertical asymptotes: x² − 4 = 0 → x = ±2. At x = 2 and x = −2, numerator = 3(4)+1 = 13 ≠ 0 → true vertical asymptotes, not holes.

Common mistakes

Confusing zeros and y-intercept. The y-intercept is f(0) — the output when x = 0. The zeros are the x-values when the output is 0. These are different questions with different answers.
Missing holes in rational functions. If (x − r) cancels from numerator and denominator, there is a hole at x = r, not a vertical asymptote. Factor fully before identifying asymptotes.
Incorrect domain for radical functions. √(x − 3) requires x − 3 ≥ 0, so domain is [3, ∞). Students often write just 'x ≥ 0' forgetting to shift for the inside expression.

Self-check

Try each one before you look. A miss here costs nothing and tells you exactly what to reread.

1. A polynomial has zeros at x = 1 (multiplicity 2) and x = −4 (multiplicity 1). At x = 1, the graph:

2. For g(x) = (x + 3)/(x − 2), the vertical asymptote is:

3. What is the domain of f(x) = √(x + 5)?

4. The end behavior of f(x) = −3x⁴ + x as x → +∞ is:

5. For rational function h(x) = (x²)/(x + 1), which statement is true?

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