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Learn · Algebra 2

Quadratic Functions Mastery: Vertex Form, Completing the Square, and the Discriminant

Algebra 1 introduced parabolas. Algebra 2 makes you fluent — switch between forms, complete the square in your head, and read the discriminant to know exactly how many real roots before solving.

10 minTEKS 4A,4B,4C,4D,4FAlgebra 2

From "knowing the formula" to "fluent across forms"

Algebra 1 gave you the quadratic formula and one form. Algebra 2 expects fluency: vertex form for graphing, standard form for the formula, factored form for roots — and the ability to switch between them in seconds.

Anatomy of a parabola: y = 0.5(x − 1)² − 2xy−2−112341−1−2axis of symmetryx = 1vertex (1, −2)root (−1, 0)root (3, 0)y-int (0, −1.5)Vertex form y = a(x−h)² + k shows vertex (h, k) directly · Roots come from setting y = 0 · Y-intercept at x = 0

The three forms

Standard: f(x) = ax2 + bx + cVertex: f(x) = a(x − h)2 + kFactored: f(x) = a(x − r1)(x − r2)Same parabola, three lenses. Use the form that matches what you're being asked.
Which form when?

Need the vertex? → vertex form. Need the roots? → factored form. Need to plug into the quadratic formula? → standard form.

Reading the vertex from vertex form

f(x) = 2(x − 3)2 + 5Vertex form a(x − h)2 + k → vertex (h, k)Vertex: (3, 5)
Read the vertex
Find the vertex of f(x) = 2(x − 3)² + 5.
Quadratic formula: works when nothing else doesx =−b ± √(b² − 4ac)2adiscriminant = b² − 4acΔ > 0: 2 real rootsΔ = 0: 1 double rootΔ < 0: 2 complex roots

Completing the square

Goal: convert standard form to vertex form. The trick is to take half of b, square it, and add (and subtract!) it inside.

x2 + 6x + 5 = 0x2 + 6x = −5 (move c to RHS)x2 + 6x + 9 = −5 + 9 (add (b/2)² = 9 to both sides)(x + 3)2 = 4x + 3 = ±2x = −1 or x = −5
Complete the square
Solve by completing the square: x² + 6x + 5 = 0.

The discriminant tells you how many roots

Discriminant = b2 − 4acPositive: two distinct real roots (parabola crosses x-axis twice)Zero: one repeated real root (parabola just touches x-axis)Negative: no real roots — two complex roots (parabola misses x-axis)
Compute the discriminant
The discriminant of x² + 4x + 5 = 0 is:
Three forms of a quadratic — each reveals different information at a glanceStandardy = ax² + bx + creveals: y-intercept (c),used for quadratic formulaVertexy = a(x − h)² + kreveals: vertex (h, k),easy to graphFactoredy = a(x − r₁)(x − r₂)reveals: roots r₁, r₂,x-intercepts

Real-world: projectile motion

A ball is thrown upward with the height equation h(t) = −16t² + 64t + 5 (height in feet, t in seconds). The maximum height is at the vertex.

tmax = −b / (2a) = −64 / (−32) = 2 secondsh(2) = −64 + 128 + 5 = 69 feet
Maximum height
A ball is thrown so that h(t) = −16t² + 64t + 5, where h is height in feet and t is seconds. What is the maximum height?

3-second recap

  • Three forms: standard, vertex, factored. Pick the one that matches the question.
  • Vertex from standard: x = −b / (2a).
  • Completing the square: add (b/2)2 to both sides.
  • Discriminant b2−4ac: + → 2 real, 0 → 1 repeated, − → 2 complex.

Check yourself

Quick check #1
Convert y = x² + 6x + 5 to vertex form.
Quick check #2
How many real solutions does x² + 4x + 5 = 0 have?