Unit 3 of Algebra 2: Quadratic Functions.
This unit covers vertex form, completing the square, discriminant and complex numbers — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers vertex form, completing the square, discriminant and complex numbers — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Vertex Form
Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. The value of a determines direction (up if positive, down if negative) and width. Students must convert between standard and vertex form and extract key features directly from vertex form.
Key Points
- Vertex is (h, k) — watch the sign: y = (x - 3)² + 1 has vertex (3, 1), not (-3, 1)
- Axis of symmetry is the vertical line x = h
- If a > 0, parabola opens up (minimum); if a < 0, opens down (maximum)
- The vertex gives the minimum or maximum value of the function
Write y = 2(x - 4)² - 5 in standard form, then identify the vertex and axis of symmetry.
Expand: 2(x - 4)² = 2(x² - 8x + 16) = 2x² - 16x + 32, so y = 2x² - 16x + 27. The vertex is read directly from vertex form as (4, -5). The axis of symmetry is x = 4.
2 Completing the Square
Completing the square converts y = ax² + bx + c into vertex form by creating a perfect square trinomial. This technique is tested both as an algebraic skill and as a method to find the vertex. When a ≠ 1, factor out a from the x-terms before completing the square.
Key Points
- Step: take half of b, square it, then add and subtract that value inside the expression
- If a ≠ 1, factor a out of the x² and x terms first before completing the square
- Whatever is added inside must be balanced by subtracting it (or adjusting outside)
- Used to derive the quadratic formula and to rewrite conics — expect it in both contexts
Convert y = x² - 6x + 11 to vertex form by completing the square.
Take half of -6, which is -3, then square it to get 9. Rewrite as y = (x² - 6x + 9) - 9 + 11, which simplifies to y = (x - 3)² + 2. The vertex is (3, 2).
3 Discriminant
The discriminant is b² - 4ac, the expression under the radical in the quadratic formula. It tells you the number and type of solutions without solving the equation. Exams frequently ask students to determine the nature of roots or find a missing value that produces a specific number of solutions.
Key Points
- b² - 4ac > 0: two distinct real solutions
- b² - 4ac = 0: exactly one real solution (repeated root); parabola touches x-axis at vertex
- b² - 4ac < 0: no real solutions; two complex conjugate solutions
- Can be used to find an unknown constant — set up an inequality or equation with b² - 4ac
Find the value of k so that 3x² - kx + 3 = 0 has exactly one real solution.
For exactly one real solution, the discriminant must equal zero: k² - 4(3)(3) = 0. This gives k² = 36, so k = ±6. Either value produces a double root.
4 Complex Numbers
Complex numbers have the form a + bi, where i = √(-1) and i² = -1. Students must simplify square roots of negative numbers, perform arithmetic with complex numbers, and recognize complex solutions to quadratics. Complex solutions always come in conjugate pairs (a + bi and a - bi).
Key Points
- √(-n) = i√n for any positive n; for example, √(-16) = 4i
- Add/subtract by combining real parts and imaginary parts separately
- Multiply using FOIL and replace i² with -1
- If a quadratic has no real roots (discriminant < 0), its complex solutions are conjugates
Solve x² + 4x + 13 = 0 using the quadratic formula.
The discriminant is 16 - 52 = -36. Applying the formula: x = (-4 ± √(-36)) / 2 = (-4 ± 6i) / 2. Simplifying gives x = -2 ± 3i, so the two complex conjugate solutions are x = -2 + 3i and x = -2 - 3i.
Questions, answered.
What is Quadratic Functions?
Quadratic Functions is Unit 3 of Algebra 2, covering vertex form, completing the square, discriminant and complex numbers.
How to study for Algebra 2 Unit 3?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 30+ review questions across 5 different game modes.