Quadratic Functions Review
Consolidate all methods for solving quadratic equations: factoring, completing the square, and the quadratic formula. Convert among standard, vertex, and factored forms and graph parabolas with full annotation.
A lesson of Algebra 2, in Review: Linear, Quadratic & Complex Numbers, about 10 minutes.
By the end of this lesson, you will be able to convert between standard, vertex, and factored forms of quadratics; identify vertex, axis, intercepts, and domain/range; and use transformations to graph any quadratic efficiently.
The Big Idea
Quadratics appear throughout Algebra 2 — as factors of polynomials, as models for projectiles and optimization, and as components of conic sections. This review consolidates everything from Algebra 1 into one unified picture: three forms, each revealing different information about the same function.
Three Forms of a Quadratic
| Form | Equation | Best reveals |
|---|---|---|
| Standard | $f(x) = ax^2 + bx + c$ | $y$-intercept ($c$), direction ($a$) |
| Vertex | $f(x) = a(x-h)^2 + k$ | Vertex $(h, k)$, axis $x = h$ |
| Factored | $f(x) = a(x - r_1)(x - r_2)$ | $x$-intercepts $r_1, r_2$ |
Converting Between Forms
Standard → Vertex: complete the square OR use $h = -\dfrac{b}{2a}$, $k = f(h)$
Standard → Factored: factor the trinomial (if factorable)
Vertex → Standard: expand $(x-h)^2$, distribute $a$, add $k$
Factored → Standard: FOIL, distribute $a$
Example 1 — Standard to Vertex
$f(x) = 2x^2 - 8x + 3$. Find the vertex.
$h = -\dfrac{-8}{2(2)} = 2$; $k = f(2) = 8 - 16 + 3 = -5$
Vertex form: $f(x) = 2(x-2)^2 - 5$
Example 2 — Vertex Form to Standard
$g(x) = -3(x+1)^2 + 7$. Expand to standard form.
$= -3(x^2 + 2x + 1) + 7 = -3x^2 - 6x - 3 + 7 = -3x^2 - 6x + 4$
Key Features from Each Form
Vertex: $(h, k)$ — maximum if $a < 0$, minimum if $a > 0$
Axis of symmetry: $x = h$
$y$-intercept: $(0, c)$ from standard form or $(0, a(0-h)^2+k)$
$x$-intercepts: set $f(x) = 0$ and solve (quadratic formula if needed)
Domain: all real numbers (always for quadratics)