Scotch Egg Home
Unit 6: Quadratic and Polynomial Functions

Topic 1

Forms of Quadratic Equations

A quadratic equation is any equation that can be written in the form:
ax2+bx+c=0,a≠0.ax^2 + bx + c = 0, \quad a \ne 0.
Quadratics describe parabolic relationships and appear throughout the SAT in both algebraic and graphical contexts.
There are three main forms of quadratic equations, each revealing different information:
1. Standard Form: y=ax2+bx+cy = ax^2 + bx + c — easiest for identifying a,b,ca, b, c and using the quadratic formula.
2. Factored Form: y=a(x−r1)(x−r2)y = a(x - r_1)(x - r_2) — shows the zeros (x-intercepts) directly: x=r1,r2.x = r_1, r_2.
3. Vertex Form: y=a(x−h)2+ky = a(x - h)^2 + k — shows the vertex (h,k)(h, k) and direction of opening (up if a>0a > 0, down if a<0a < 0).
The three forms are algebraically equivalent, but each highlights different features of the same parabola. Converting between forms is a key SAT skill.

Core Skills

  • Identify aa, bb, and cc in the standard form.
  • Factor quadratics to find zeros (convert to factored form).
  • Complete the square to convert to vertex form.
  • Recognize how aa affects the parabola’s width and direction.
  • Interpret graphs and equations interchangeably between forms.

Example 1: Recognizing Forms

Given y=2x2−8x+6y = 2x^2 - 8x + 6:
  • Standard form: y=2x2−8x+6y = 2x^2 - 8x + 6
  • Factored form: y=2(x−1)(x−3)y = 2(x - 1)(x - 3)
  • Vertex form: y=2(x−2)2−2y = 2(x - 2)^2 - 2
Each form reveals something different: • Zeros: x=1,3x = 1, 3• Vertex: (2,−2)(2, -2)• Opens upward since a=2>0a = 2 > 0.

Example 2: Converting to Vertex Form

Convert y=x2−6x+8y = x^2 - 6x + 8 to vertex form.
y=(x2−6x)+8y = (x^2 - 6x) + 8
Complete the square:
(x2−6x+9)−9+8=(x−3)2−1(x^2 - 6x + 9) - 9 + 8 = (x - 3)^2 - 1
So:
y=(x−3)2−1y = (x - 3)^2 - 1
Vertex: (3,−1)(3, -1), opens upward.

Example 3: Converting to Factored Form

Convert y=x2−5x+6y = x^2 - 5x + 6 to factored form.
x2−5x+6=(x−2)(x−3)x^2 - 5x + 6 = (x - 2)(x - 3)
Zeros: x=2,3.x = 2, 3. Factored form: y=(x−2)(x−3)\boxed{y = (x - 2)(x - 3)}

Example 4: Interpreting a Factored Equation

Given y=−2(x+1)(x−4)y = -2(x + 1)(x - 4):
  • Zeros: x=−1,4x = -1, 4
  • Opens downward (a=−2a = -2)
  • Axis of symmetry halfway between zeros: x=−1+42=1.5x = \dfrac{-1 + 4}{2} = 1.5
Vertex: occurs at x=1.5x = 1.5.

Example 5: Understanding Coefficient Effects

Compare y=x2y = x^2, y=2x2y = 2x^2, and y=12x2y = \frac{1}{2}x^2:
  • a>1a > 1: narrower parabola.
  • 0<a<10 < a < 1: wider parabola.
  • a<0a < 0: opens downward.
Key insight: aa controls vertical stretch and direction.

Key Takeaways

  • Standard form gives coefficients for the quadratic formula.
  • Factored form gives zeros quickly.
  • Vertex form gives vertex and direction directly.
  • All forms describe the same parabola, just from different perspectives.
  • Practice converting among forms to move fluidly between algebra and graphs.