The types of function in math covered in the Canadian secondary curriculum — linear, quadratic, exponential, logarithmic, trigonometric, and more — are one of the most important foundations in secondary school mathematics. Each has a distinct shape, equation form, and set of properties. This guide covers every major type, from Grade 9 through Grade 12, with definitions, equations, graphs, key properties, and what to watch for on tests.
Does your child understand functions?
Our free assessment shows whether your child has mastered the foundations before they move on to more difficult topics.
→ Get your free feedback report
What Is a Function in Math?
Before covering the types of functions in math, it is worth being precise about what a function actually is.
A function is a relation in which every input value (x) produces exactly one output value (f(x) or y). No input can have more than one output.
Function: Every x gives exactly one y Not a function: Some x gives more than one y
The vertical line test determines whether a graph represents a function: if any vertical line crosses the graph more than once, the relation is not a function.
Key vocabulary:
- Domain — the set of all valid input values (x values)
- Range — the set of all possible output values (y or f(x) values)
- Function notation — f(x) is read “f of x” and represents the output when x is the input
For a full explanation of function notation, composition, and inverses — including practice problems — see our composite functions maths guide.
Types of Function in Math 1: Linear Functions
Equation form: f(x) = mx + b
Graph: A straight line
Key properties:
- Slope m determines the steepness and direction (positive = rising left to right, negative = falling)
- y-intercept b is where the line crosses the y-axis
- Constant rate of change — the slope is the same everywhere on the line
- Domain: all real numbers; Range: all real numbers
Example: f(x) = 2x − 3
| x | f(x) |
|---|---|
| 0 | −3 |
| 1 | −1 |
| 2 | 1 |
| 3 | 3 |
Where it appears in the Ontario curriculum: Linear functions are the focus of Grade 9 MTH1W — slope, y-intercept, equation of a line, and applications. They reappear in MCR3U as a reference point for comparing other function types. See our Ontario Grade 9 math curriculum guide.
What makes it linear: The highest power of x is 1. If you add a constant amount to x, the output increases by a constant amount (the slope). This constant rate of change is the defining feature.
Type 2: Quadratic Functions
Equation forms:
- Standard form: f(x) = ax² + bx + c
- Vertex form: f(x) = a(x − h)² + k
- Factored form: f(x) = a(x − r)(x − s)
Graph: A parabola — U-shaped (a > 0) or ∩-shaped (a < 0)
Key properties:
- Vertex at (h, k) — the maximum or minimum point
- Axis of symmetry: x = h
- Opens up if a > 0; opens down if a < 0
- x-intercepts (roots/zeros): where f(x) = 0
- Domain: all real numbers; Range: y ≥ k (if a > 0) or y ≤ k (if a < 0)
Example: f(x) = (x − 2)² − 9 (vertex form)
- Vertex: (2, −9)
- Axis of symmetry: x = 2
- Opens up (a = 1 > 0)
- x-intercepts: (x − 2)² = 9 → x − 2 = ±3 → x = 5 or x = −1
Where it appears: Quadratic functions are the central topic of Grade 10 MPM2D and are reviewed in MCR3U. The ability to move fluently between the three forms — standard, vertex, and factored — is essential. See our Grade 10 math Ontario guide.
What makes it quadratic: The highest power of x is 2. The rate of change is not constant — it changes linearly (which is why the graph curves).
Type 3: Polynomial Functions
Equation form: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
Graph: Smooth, continuous curves with at most n − 1 turning points (for degree n)
Key properties by degree:
| Degree | Name | Example | Shape |
|---|---|---|---|
| 0 | Constant | f(x) = 5 | Horizontal line |
| 1 | Linear | f(x) = 2x + 1 | Straight line |
| 2 | Quadratic | f(x) = x² − 4 | Parabola |
| 3 | Cubic | f(x) = x³ − 3x | S-shaped curve |
| 4 | Quartic | f(x) = x⁴ − 4x² | W or M shape |
End behaviour: Determined 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
- If n is odd and aₙ > 0: left end down, right end up
- If n is odd and aₙ < 0: left end up, right end down
Factor and Remainder Theorems:
- Remainder Theorem: When f(x) is divided by (x − a), the remainder equals f(a)
- Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0
Where it appears: Polynomial functions are covered in MCR3U Unit 2 and extended in MHF4U. The Factor Theorem and Remainder Theorem are MCR3U-specific content that many students find conceptually challenging.
Type 4: Rational Functions
Equation form: f(x) = P(x)/Q(x) where P and Q are polynomials and Q(x) ≠ 0
Graph: Curves with asymptotes — lines the graph approaches but never reaches
Key properties:
- Vertical asymptotes where Q(x) = 0 (and P(x) ≠ 0)
- Horizontal asymptotes determined by comparing degrees of P and Q:
- deg P < deg Q: y = 0
- deg P = deg Q: y = leading coefficient of P / leading coefficient of Q
- deg P > deg Q: no horizontal asymptote (oblique instead)
- Holes (removable discontinuities) where both P(x) = 0 and Q(x) = 0 at the same x value
Example: f(x) = (x + 2) / (x − 3)(x + 1)
- Vertical asymptotes: x = 3 and x = −1
- Horizontal asymptote: y = 0 (degree of numerator < degree of denominator)
- Domain: all real numbers except x = 3 and x = −1
Where it appears: Rational functions are introduced in MCR3U Unit 2 and covered more extensively in MHF4U. Sketching rational functions — identifying asymptotes, intercepts, and behaviour near discontinuities — is a common test and exam question type.
Type 5: Exponential Functions
Equation form: f(x) = a · bˣ where b > 0, b ≠ 1
Graph: A curve that grows (b > 1) or decays (0 < b < 1) rapidly, always remaining above the x-axis
Key properties:
- Always positive: f(x) > 0 for all x (when a > 0)
- Horizontal asymptote at y = 0
- y-intercept at (0, a)
- Domain: all real numbers; Range: y > 0 (when a > 0)
- Growth factor b: if b > 1, exponential growth; if 0 < b < 1, exponential decay
Common bases:
- Base 2: doubling (population, computer memory)
- Base 10: scientific measurements
- Base e ≈ 2.718: natural exponential (continuous growth, calculus)
Example: f(x) = 3 · 2ˣ
- y-intercept: (0, 3)
- Growth factor: 2 (doubles every unit increase in x)
- Horizontal asymptote: y = 0
- Domain: all real numbers; Range: y > 0
Applications: Compound interest, population growth, radioactive decay, cooling — exponential functions are the most practically applied of all function types in MCR3U.
Where it appears: MCR3U Unit 3. The relationship between exponential and logarithmic functions is one of the most conceptually important connections in the senior mathematics curriculum.
Type 6: Logarithmic — One of the Hardest Types of Function in Math
Equation form: f(x) = logᵦ(x) where b > 0, b ≠ 1
Graph: A curve that grows slowly, crossing the x-axis at (1, 0), with a vertical asymptote at x = 0
Key properties:
- The logarithmic function is the inverse of the exponential: if bʸ = x then logᵦ(x) = y
- Vertical asymptote at x = 0
- x-intercept at (1, 0)
- Domain: x > 0; Range: all real numbers
- Passes through (b, 1) — since logᵦ(b) = 1
Logarithm laws:
| Law | Formula |
|---|---|
| Product | logᵦ(MN) = logᵦ(M) + logᵦ(N) |
| Quotient | logᵦ(M/N) = logᵦ(M) − logᵦ(N) |
| Power | logᵦ(Mᵖ) = p · logᵦ(M) |
| Change of base | logᵦ(x) = log(x)/log(b) |
Example: f(x) = log₂(x)
- Domain: x > 0
- Vertical asymptote: x = 0
- x-intercept: (1, 0)
- f(2) = 1, f(4) = 2, f(8) = 3, f(1/2) = −1
Where it appears: MCR3U Unit 4 and MHF4U. Logarithmic functions are among the most conceptually challenging of the types of functions in math because they require a different mental model from any function type encountered before Grade 11. For more detail including worked examples and the trigonometric identities reference, see our MCR3U complete guide.
Type 7: Trigonometric — the Most Complex Types of Function in Math for Grade 11
Primary functions: sine, cosine, tangent (and their reciprocals: cosecant, secant, cotangent)
Equation forms:
- f(x) = a sin(b(x − c)) + d
- f(x) = a cos(b(x − c)) + d
Graph: Periodic wave — repeats at regular intervals
Key properties:
| Parameter | Effect |
|---|---|
| |a| | Amplitude (half the height of the wave) |
| b | Period = 2π/b |
| c | Phase shift (horizontal translation) |
| d | Vertical displacement (midline) |
Properties of sin and cos:
- Domain: all real numbers
- Range: −1 ≤ f(x) ≤ 1 (for basic sin and cos)
- Period: 2π radians (360°)
- sin(0) = 0, sin(π/2) = 1, sin(π) = 0, sin(3π/2) = −1
- cos(0) = 1, cos(π/2) = 0, cos(π) = −1, cos(3π/2) = 0
Special angle values:
| Angle (degrees) | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
Key identity: sin²(x) + cos²(x) = 1
Where it appears: MCR3U Unit 5 (the largest unit) and MHF4U (extended with identity proofs). Trigonometric functions are consistently the hardest of all types of functions in math for Grade 11 students because the shift from right-triangle trigonometry to the unit circle is the biggest conceptual leap in the course. See our special triangles in trigonometry guide and trigonometric identities sheet.
Type 8: Radical (Square Root) Functions
Equation form: f(x) = a√(x − h) + k
Graph: Half of a sideways parabola — starts at the vertex and curves to the right (for basic √x)
Key properties:
- Domain restriction: the expression under the radical must be ≥ 0
- For f(x) = √x: domain x ≥ 0, range y ≥ 0
- Vertex at (h, k)
- Not a full parabola — only the half where the output is non-negative
Example: f(x) = √(x − 3) + 1
- Domain: x − 3 ≥ 0 → x ≥ 3
- Range: y ≥ 1
- Vertex: (3, 1)
Where it appears: MHF4U (radical functions as one of the function types studied in depth). Domain restrictions for radical functions are a common source of errors on tests.
Type 9: Absolute Value Functions
Equation form: f(x) = a|x − h| + k
Graph: A V-shape (or ∧-shape if a < 0) with vertex at (h, k)
Key properties:
- The absolute value of a number is its distance from zero — always non-negative
- |x| = x if x ≥ 0; |x| = −x if x < 0
- Domain: all real numbers; Range: y ≥ k (if a > 0)
Example: f(x) = 2|x − 1| − 4
- Vertex: (1, −4)
- V-shape opens upward (a = 2 > 0)
- x-intercepts: 2|x − 1| = 4 → |x − 1| = 2 → x = 3 or x = −1
Where it appears: MHF4U. Absolute value equations and the graphical interpretation of absolute value functions are consistently tested topics.
How the Types of Functions in Math Connect to Each Other
Understanding how different function types relate to each other is more useful than memorising them in isolation.
Inverse relationships:
- Exponential and logarithmic functions are inverses of each other: f(x) = bˣ and f⁻¹(x) = logᵦ(x)
- This means their graphs are reflections of each other over the line y = x
Transformations apply to all types: The general transformation form y = af(b(x − c)) + d applies identically to every function type. Once a student understands transformations for one function type, the same logic applies to all others. This is why MCR3U Unit 1 (function transformations) matters so much — it is taught once and applied six times.
Composition connects types: Composite functions (see our composite functions maths guide) can combine different function types — f(g(x)) where f is exponential and g is linear, for example. Understanding each type individually is the prerequisite for working with them in combination.
The curriculum progression through types of function in math: Linear (Grade 9) → Quadratic (Grade 10) → Polynomial, Exponential, Logarithmic, Trigonometric (Grade 11 MCR3U) → Extended versions of all (Grade 12 MHF4U) → Calculus of all types (MCV4U)
Each type builds on what came before. A student who does not genuinely understand quadratic functions in Grade 10 will struggle with polynomial functions in Grade 11. A student who does not understand exponential functions in MCR3U will struggle with logarithms immediately after.
Not sure which types of functions your child has truly mastered?
Our free assessment identifies their strengths, knowledge gaps, and the areas they should focus on next. You’ll receive a personalised feedback report with clear recommendations for improving their Grade 11 math skills.
→ Find out where your child stands
Types of Function in Math: Summary Comparison
| Function Type | Equation Form | Graph Shape | Domain | Where Taught |
|---|---|---|---|---|
| Linear | mx + b | Straight line | All reals | Grade 9 |
| Quadratic | ax² + bx + c | Parabola | All reals | Grade 10 |
| Polynomial | aₙxⁿ + … | Smooth curve | All reals | MCR3U, MHF4U |
| Rational | P(x)/Q(x) | Curves with asymptotes | x: Q(x) ≠ 0 | MCR3U, MHF4U |
| Exponential | a · bˣ | Rapid growth/decay | All reals | MCR3U |
| Logarithmic | logᵦ(x) | Slow growth | x > 0 | MCR3U, MHF4U |
| Trigonometric | a sin(b(x−c))+d | Wave | All reals | MCR3U, MHF4U |
| Radical | a√(x−h)+k | Half-parabola | x: radicand ≥ 0 | MHF4U |
| Absolute Value | a|x−h|+k | V-shape | All reals | MHF4U |
Practice Problems
Set A — Identifying function types
Identify the type of function and state its key feature:
- f(x) = 4x − 7 → ___
- f(x) = −2x² + 3x + 1 → ___
- f(x) = 5 · 3ˣ → ___
- f(x) = log₅(x) → ___
- f(x) = 3 sin(2x − π) + 1 → ___
- f(x) = (x + 3)/(x − 2) → ___
- f(x) = √(2x + 6) → ___
- f(x) = −|x + 4| + 3 → ___
Set B — Domain and range
State the domain and range:
- f(x) = 2ˣ
- f(x) = log₃(x)
- f(x) = √(x − 5)
- f(x) = 1/(x + 3)
- f(x) = (x − 1)² + 2
- f(x) = 4 cos(x)
Set C — Key values
- For f(x) = 3 · 2ˣ, find x when f(x) = 24.
- For f(x) = log₂(x), find x when f(x) = 4.
- For f(x) = 2 sin(x) + 1, what is the maximum value?
- For f(x) = (x² − 9)/(x − 3), is there a vertical asymptote at x = 3? Explain.
Answers:
Set A: 1) Linear; slope 4 2) Quadratic; opens down 3) Exponential; growth factor 3 4) Logarithmic; base 5 5) Trigonometric; amplitude 3, period π 6) Rational; vertical asymptote x = 2 7) Radical; domain x ≥ −3 8) Absolute value; vertex (−4, 3), opens down
Set B: 9) Domain: all reals; Range: y > 0 10) Domain: x > 0; Range: all reals 11) Domain: x ≥ 5; Range: y ≥ 0 12) Domain: x ≠ −3; Range: y ≠ 0 13) Domain: all reals; Range: y ≥ 2 14) Domain: all reals; Range: −4 ≤ y ≤ 4
Set C: 15) 3 · 2ˣ = 24 → 2ˣ = 8 → x = 3 16) 2⁴ = 16 → x = 16 17) Maximum = 2(1) + 1 = 3 18) No — factor (x² − 9) = (x−3)(x+3), so (x−3) cancels; there is a hole at x = 3, not an asymptote
Frequently Asked Questions
What are the types of function in math? The main types of function in math covered in the Canadian secondary curriculum are: linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric, radical, and absolute value. Each has a distinct equation form, graph shape, and set of properties. They are introduced progressively from Grade 9 through Grade 12.
What is the difference between linear and quadratic functions? A linear function has the form f(x) = mx + b and produces a straight-line graph. A quadratic function has the form f(x) = ax² + bx + c and produces a parabola. The key difference is the highest power of x: 1 for linear, 2 for quadratic. Linear functions have a constant rate of change; quadratic functions have a rate of change that changes linearly.
Which types of functions in math are hardest? For most Ontario students, trigonometric functions (Grade 11, MCR3U Unit 5) are the most challenging — the conceptual shift from right-triangle trigonometry to the unit circle is the biggest leap in the curriculum. Logarithmic functions are the most conceptually unfamiliar when first encountered. Rational functions with asymptotes are the most graphically complex.
What is the easiest way to identify a function type? Look at the equation structure: straight x → linear; x squared → quadratic; bˣ (variable in exponent) → exponential; log → logarithmic; sin or cos → trigonometric; √x → radical; |x| → absolute value; P(x)/Q(x) → rational.
Do all types of function in math have the same domain? No. Linear, quadratic, polynomial, trigonometric, and absolute value functions generally have domain of all real numbers. Logarithmic functions require x > 0. Radical functions require the radicand ≥ 0. Rational functions exclude values where the denominator is zero. Domain restrictions are one of the most consistently tested topics across all function types.
Where do types of functions in math appear in EQAO? Function types first appear formally in EQAO Grade 9 (linear functions in the MTH1W data and linear relations strand). The full range of function types is tested throughout MCR3U and MHF4U assessments. See our EQAO Grade 9 complete guide for the Grade 9 context.
See our related guides: composite functions maths guide · MCR3U complete guide · Grade 11 Functions Ontario guide · MHF4U Advanced Functions guide · Grade 10 math Ontario guide · Ontario Grade 9 math curriculum guide · special triangles in trigonometry · trigonometric identities sheet · math order of operations guide · choosing high school math courses
Understanding every type of function in math is what makes Grade 11 and 12 manageable. Build that understanding now.
See how Think Academy teaches difficult concepts.
Experience a live lesson with one of our expert teachers and discover why thousands of Canadian families trust Think Academy.
