Difficulty: Introductory | Prerequisites: Basic algebra (function notation, exponents, graphing in the coordinate plane)
This section of Calculus I is a catalogue of the function families you will use throughout the course. Before you can take limits, derivatives, or integrals, you need to recognise each family on sight, know its domain and range, and understand how its parameters change the graph. If you missed the precalculus review, start here: everything that follows in the course assumes you can identify these functions cold. The two families covered in this first set of notes are polynomials (including linear and quadratic as special cases) and power functions (including roots and the reciprocal).
Polynomials are sums of whole-number-power terms: the degree tells you the function's broad shape. Power functions generalise this by allowing the exponent to be any real number, which gives you roots (fractional exponents) and reciprocals (negative exponents). Know the general form, the degree, and the domain/range for each family, and you can read any graph the course throws at you.
Polynomial function
A function of the form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer and each aᵢ is a constant.
In simple terms, a polynomial is a sum of terms where x is raised only to whole-number powers (0, 1, 2, 3, ...) and multiplied by constants.
Degree (of a polynomial)
The highest power of x that appears with a non-zero coefficient. If aₙ ≠ 0, the degree is n.
Think of it as the number that tells you the polynomial's "complexity" and the broad shape of its graph.
Linear function
A polynomial of degree 1, written y = mx + b, where m is the slope and b is the y-intercept.
In simple terms, a straight line.
Quadratic function
A polynomial of degree 2, written y = ax² + bx + c. When a > 0 the parabola opens upwards; when a < 0 it opens downwards.
Think of it as the U-shaped (or upside-down U-shaped) curve.
Power function
A function of the form f(x) = xᵃ, where a can be any real number, not just a positive integer.
In simple terms, a single term with x raised to some power. Unlike a polynomial, the exponent does not have to be a whole number.
Root function
A power function where the exponent is a fraction of the form 1/n. For example, f(x) = x^(1/2) = √x (square root) or f(x) = x^(1/3) = ∛x (cube root).
Think of it as "undoing" an integer power.
Reciprocal function
The function f(x) = 1/x, which is x raised to the power -1.
In simple terms, dividing 1 by x. It is the simplest example of a rational function.
Rational function
A function that can be written as the ratio of two polynomials, P(x)/Q(x).
Think of it as a fraction where both the top and bottom are polynomials.
Linear functions (degree 1)
General form: y = f(x) = mx + b
m = slope (rate of change)
b = y-intercept (where the line crosses the y-axis)
Graph is always a straight line
Quadratic functions (degree 2)
General form: y = f(x) = ax² + bx + c
When a > 0, the parabola opens upwards (has a minimum)
When a < 0, the parabola opens downwards (has a maximum)
The sign of a is the single thing that determines the direction the parabola faces
General polynomial form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
The exponents run from n down to 0 (note that a₁x = a₁x¹ and a₀ = a₀x⁰)
n must be a non-negative integer: 0, 1, 2, 3, ...
The coefficients a₀ through aₙ are constants
The degree is n, provided the leading coefficient aₙ ≠ 0
Degree 1 gives you a linear function; degree 2 gives you a quadratic
Positive integer exponents
f(x) = xᵃ where a is a positive integer gives you a monomial
a = 1: f(x) = x (the identity line)
a = 2: f(x) = x² (the standard parabola)
a = 3: f(x) = x³ (the cubic S-curve)
Higher integer values of a produce increasingly "flat" graphs near the origin and increasingly steep graphs far from it
Fractional exponents (root functions)
When a = 1/n for a positive integer n, you get a root function
Square root: f(x) = x^(1/2) = √x
Domain: [0, ∞)
Range: [0, ∞)
Cube root: f(x) = x^(1/3) = ∛x
Domain: (-∞, ∞)
Range: (-∞, ∞)
The key difference: even roots (square root, fourth root, ...) require non-negative inputs. Odd roots (cube root, fifth root, ...) accept any real number.
Negative exponents
When a is negative, you get reciprocal-type functions
The simplest case: a = -1 gives f(x) = 1/x, the reciprocal function
1/x is a rational function because it can be written as P(x)/Q(x) with P(x) = 1 and Q(x) = x
Rational functions in general are any ratio of two polynomials
Why power functions matter separately from polynomials
A polynomial only allows non-negative integer exponents. A power function allows any real exponent (fractions, negatives, irrational numbers). This is a broader family, and many real-world models (square-root scaling, inverse-square laws) use non-integer powers.
P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0General polynomial. Degree = n when aₙ ≠ 0.
y = mx + bLinear function (degree 1). m = slope, b = y-intercept.
y = ax^2 + bx + cQuadratic function (degree 2). Sign of a controls direction of opening.
f(x) = x^aGeneral power function. a can be any real number.
f(x) = \frac{P(x)}{Q(x)}Rational function. Both P(x) and Q(x) are polynomials.
Linear functions model any situation with a constant rate of change: a phone plan that charges a flat fee plus a per-minute rate, or the distance a car travelling at constant speed covers over time.
Quadratic functions describe projectile motion (a ball thrown in the air follows a parabolic path) and are used in optimisation problems such as maximising the area of a fenced enclosure.
Power functions with fractional exponents turn up in physics and engineering. The period of a pendulum scales with the square root of its length. Inverse-square laws (gravity, light intensity) use negative-exponent power functions.
Students often think every power function is a polynomial. It is not. A polynomial requires non-negative integer exponents. f(x) = x^(1/2) is a power function but not a polynomial.
Students sometimes confuse the degree of a polynomial with its number of terms. A polynomial like 5x⁴ has degree 4, even though it is a single term.
The square root function √x is often assumed to accept negative inputs. It does not. The domain of √x is [0, ∞). Only odd roots like ∛x accept all real numbers.
Students often forget that the sign of the leading coefficient a in a quadratic determines whether the parabola opens up or down. They focus on the vertex instead, which does not control direction.
⚠️ You will be asked to identify whether a given function is a polynomial, and if so, state its degree. Know the definition cold.
⚠️ Expect questions that give you a power function with a fractional or negative exponent and ask for domain and range. The square root vs. cube root distinction is a common test item.
⚠️ Be able to write 1/x as a rational function P(x)/Q(x). This bridges the power-function material to the rational-function material that follows.
⚠️ Parabola direction (sign of a) is tested frequently, often embedded inside a word problem about projectile motion or optimisation.
True or False: f(x) = x^(2/3) is a polynomial.
Fill in the blank: The degree of 7x⁵ + 3x² - 1 is ___.
True or False: The domain of f(x) = √x is all real numbers.
Fill in the blank: A rational function is the ratio of two ___.
True or False: If a < 0 in y = ax² + bx + c, the parabola opens upwards.
Answers: 1. False, 2. 5, 3. False (domain is [0, ∞)), 4. polynomials, 5. False (it opens downwards)
Q: Is f(x) = 3x⁴ + 2x - 7 a polynomial? If so, what is its degree?
A: Yes. It is a polynomial of degree 4 because the highest power of x with a non-zero coefficient is 4.
Q: What is the domain and range of f(x) = √x?
A: Domain = [0, ∞). Range = [0, ∞). The square root function only accepts non-negative inputs and only produces non-negative outputs.
Q: Write f(x) = 1/x as a rational function P(x)/Q(x). What are P(x) and Q(x)?
A: P(x) = 1, Q(x) = x. Both are polynomials, so 1/x qualifies as a rational function.
Q: A ball is launched upwards and its height is modelled by h(t) = -16t² + 64t + 5. Does the parabola open upwards or downwards? What does that tell you physically?
A: The leading coefficient is -16, which is negative, so the parabola opens downwards. Physically, this means the ball reaches a maximum height and then falls back down.
Q: Explain why f(x) = x^(-2) is a power function but not a polynomial.
A: A power function is any function of the form xᵃ with a real. A polynomial requires all exponents to be non-negative integers. Since -2 is negative, x^(-2) is a power function but not a polynomial.
Polynomial and power functions are the first functions you will differentiate and integrate in this course. The power rule for derivatives (d/dx of xⁿ = nxⁿ⁻¹) applies directly to everything here, making this section essential groundwork.
Rational functions connect to the study of limits and asymptotes. You will revisit 1/x when learning about vertical and horizontal asymptotes, and again when studying improper integrals.
Root functions reappear in implicit differentiation and related rates problems, where you often need to differentiate expressions involving √x or x^(1/3).
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