Unit 2 · 11th Grade Mathematics

Quadratic
Functions.

Features, forms, complex numbers, and solving quadratic equations — all in one guide.

Start studying Topic B Topic C

Unit 2 — Lesson Video

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Topic A: Features of Quadratic Functions

Lessons 1–4 · Vertex, axis of symmetry, zeros, and forms of quadratics.

01 — Lesson 1

Key Features of Parabolas

A quadratic function creates a U-shaped parabola with distinct features.

  • Vertex: the highest or lowest point — (h, k).
  • Axis of symmetry: vertical line x = h through the vertex.
  • Opens up (a > 0): minimum at vertex. Opens down (a < 0): maximum.
  • Zeros/roots: x-intercepts where f(x) = 0.
  • Y-intercept: f(0) = the constant term c in ax² + bx + c.
f(x) = −(x−3)² + 9 → vertex (3, 9) is a maximum; axis: x = 3; opens down.
Practice — Lesson 1
Q1 · Multiple Choice

For f(x) = 2(x−1)² − 4, what is the vertex?

Q2 · Multiple Choice

f(x) = −3x² + 6x − 1. Does this parabola have a maximum or minimum?

02–03 — Lessons 2–3

Forms of Quadratic Functions

Quadratics can be written in three equivalent forms, each revealing different features.

  • Standard form: f(x) = ax² + bx + c → y-intercept = c.
  • Vertex form: f(x) = a(x−h)² + k → vertex = (h, k).
  • Factored/Intercept form: f(x) = a(x−p)(x−q) → zeros at x = p and x = q.
  • Convert between forms by expanding or completing the square.
f(x) = (x−2)(x−6) → zeros: x = 2, x = 6; axis: x = 4; vertex: (4, −4)
Practice — Lessons 2–3
Q3 · Multiple Choice

Which form of a quadratic most easily reveals its x-intercepts?

Q4 · Short Answer

What is the y-intercept of f(x) = 3x² − 5x + 7?

04 — Lesson 4

Transformations of Quadratics

Transformations shift, stretch, or flip the parent function f(x) = x².

  • f(x) = x² + k: shifts up k units (down if k < 0).
  • f(x) = (x − h)²: shifts right h units (left if h < 0).
  • f(x) = ax²: a > 1 narrows; 0 < a < 1 widens; a < 0 flips.
f(x) = −2(x+1)² + 5 → flipped (a=−2), shifted left 1, up 5. Vertex: (−1, 5).
Practice — Lesson 4
Q5 · Multiple Choice

Compared to f(x) = x², how is g(x) = (x−4)² + 3 transformed?

Q6 · Multiple Choice

Which quadratic is narrower than f(x) = x²?

Topic B: Complex Numbers

Lessons 5–7 · Imaginary numbers, complex number operations, and complex solutions.

05 — Lesson 5

Imaginary & Complex Numbers

When a quadratic has no real zeros, its solutions involve the imaginary unit i.

  • i = √(−1), so i² = −1.
  • Complex number: a + bi, where a is real and bi is imaginary.
  • Pure imaginary: no real part, e.g. 3i.
  • √(−9) = √9 · √(−1) = 3i
√(−25) = 5i  ·  (2 + 3i) + (1 − i) = 3 + 2i
Practice — Lesson 5
Q7 · Short Answer

Simplify √(−16). Write in the form bi.

Q8 · Multiple Choice

What is i²?

06–07 — Lessons 6–7

Operating with Complex Numbers

Add, subtract, multiply, and divide complex numbers — treat i like a variable, but replace i² with −1.

  • Addition/Subtraction: combine real parts and imaginary parts separately.
  • Multiplication: FOIL, then replace i² = −1.
  • Conjugate of a + bi: a − bi. Used for division.
  • Division: multiply numerator and denominator by the conjugate.
(2 + 3i)(1 − 2i) = 2 − 4i + 3i − 6i² = 2 − i + 6 = 8 − i
Practice — Lessons 6–7
Q9 · Short Answer

Multiply: (3 + i)(3 − i). (Hint: it's a difference of squares.)

Q10 · Multiple Choice

What is the complex conjugate of 5 − 2i?

Topic C: Solving Quadratic Equations

Lessons 8–11 · Factoring, completing the square, and the quadratic formula.

08–09 — Lessons 8–9

Solving by Factoring

Set the quadratic equal to zero, factor it, then use the zero product property.

  • Zero product property: if AB = 0, then A = 0 or B = 0.
  • Factor: find two binomials whose product equals the quadratic.
  • GCF first: always factor out the greatest common factor first.
  • Works cleanly when solutions are rational numbers.
x² − 5x + 6 = 0 → (x−2)(x−3) = 0 → x = 2 or x = 3
Practice — Lessons 8–9
Q11 · Short Answer

Solve: x² + 7x + 12 = 0. Give the smaller solution.

Q12 · Multiple Choice

The zero product property says: if (x−5)(x+2) = 0, then...

10–11 — Lessons 10–11

Quadratic Formula & Discriminant

When factoring is difficult, use the quadratic formula to find exact solutions.

  • Formula: x = [−b ± √(b²−4ac)] / 2a
  • Discriminant (b²−4ac): determines the number of solutions.
  • > 0: two real solutions. = 0: one real solution. < 0: two complex solutions.
x² − 4x + 1 = 0 → discriminant = 16−4 = 12 > 0 → two real solutions:
x = (4 ± √12)/2 = 2 ± √3
Practice — Lessons 10–11
Q13 · Multiple Choice

For 2x² + 3x + 5 = 0, the discriminant is 9 − 40 = −31. What does this tell us?

Q14 · Multiple Choice

Using the quadratic formula on x² − 6x + 9 = 0, what is x?