Overview
This lecture explains how to transform a quadratic function from vertex form to general form, using step-by-step examples and two expansion methods.
Forms of Quadratic Functions
- Vertex form: ( y = a(x-h)^2 + k )
- General form: ( y = ax^2 + bx + c )
- Both forms use ( y ) or ( f(x) ) interchangeably.
Expanding Binomials (Methods)
- To expand ( (x + d)^2 ), multiply the binomial by itself: ( (x + d)(x + d) ).
- FOIL method: First, Outer, Inner, Last (e.g., ( (x+3)^2 = x^2 + 6x + 9 )).
- Square of binomial rule: ( (x + d)^2 = x^2 + 2dx + d^2 ).
Example 1: ( y = (x + 3)^2 )
- Expand: ( (x + 3)^2 = x^2 + 6x + 9 ) (both FOIL and binomial square yield this).
- General form: ( y = x^2 + 6x + 9 ).
- Coefficients: ( a = 1 ), ( b = 6 ), ( c = 9 ).
Example 2: ( y = 3(x - 2)^2 + 4 )
- Expand ( (x - 2)^2 = x^2 - 4x + 4 ).
- Distribute 3: ( 3(x^2 - 4x + 4) = 3x^2 - 12x + 12 ).
- Add 4: ( y = 3x^2 - 12x + 16 ).
- Coefficients: ( a = 3 ), ( b = -12 ), ( c = 16 ).
Example 3: ( y = -2(x + 5)^2 + 7 )
- Expand ( (x + 5)^2 = x^2 + 10x + 25 ).
- Distribute -2: ( -2(x^2 + 10x + 25) = -2x^2 - 20x - 50 ).
- Add 7: ( y = -2x^2 - 20x - 43 ).
- Coefficients: ( a = -2 ), ( b = -20 ), ( c = -43 ).
Key Terms & Definitions
- Vertex Form — ( y = a(x-h)^2 + k ), shows vertex at ((h, k)).
- General Form — ( y = ax^2 + bx + c ), standard quadratic equation.
- FOIL Method — Used to expand two binomials: First, Outer, Inner, Last.
- Square of Binomial — Shortcut: ( (x+d)^2 = x^2 + 2dx + d^2 ).
- Distributive Property — Multiply each term inside the parentheses by the outside coefficient.
Action Items / Next Steps
- Assignment: Convert ( y = -5(x - 3)^2 + 3 ) to general form, and find ( a, b, c ).
- Post your answer in the comment section for review.