PreCalculus 113.2 Investigating Quadratic Functions in Standard Form (p.163)

Quadratic Function in Standard Form:

f(x) = ax2 + bx + c a determines the shape and opening of the graph

b determines the position of the graph

c determines the y-intercept of the graph

Key Concepts:

Identifying characteristics of a Quadratic function in Standard form (from the graph):

a) Direction of opening: _____________

b) Vertex: _____________

c) Maximum or Minimum value _____________

d) Axis of Symmetry _____________

e) x-intercepts: ______________y-intercept: _____________

f) Domain: _____________ Range: _________________

Chapter 3.2 Quadratic Function in Vertex Form

PreCalculus 11Converting the function from Vertex Form to Standard form

- just expand and remove the brackets

E.g. f(x) = 2(x – 3)2 + 4f(x) = 2(x – 3)(x – 3) + 4f(x) = 2(x2 – 3x -3x + 9) + 4f(x) = 2(x2 – 6x + 9) + 4f(x) = 2x2 – 12x + 18 + 4f(x) = 2x2 – 12x + 22

Practice:Convert the following quadratic function into standard form:

a) f ( x )=−3 (x+1 )2−5 b) f ( x )=12

( x−2 )2+3

Converting the function from Standard form to Vertex form (Using a formula):Development of the formula to find p and q:Vertex form: Standard form:

f(x) = a(x - p)2 + qf(x) = a(x – p)(x – p) + qf(x) = a (x2 – px – px + p2) + qf(x) = a(x2 – 2px + p2) + qf(x) = ax2 – 2apx + ap 2 + q f(x) = ax2 + bx + c

Chapter 3.2 Quadratic Function in Vertex Form

p=−b2aq=c−a p2

PreCalculus 11Example: f(x) = -x2 – 2x + 3

p = -(-2)2(−1)

= 2−2

=−1 q=(3 )— (−1)(−1 )2=3+1=4

Vertex = (-1, 4), a = -1

f ( x )=−( x+1 )2+4

Practice:Covert the following quadratic function from standard to vertex form using the formula:

1a) f ( x )=3 x2−6 x+4 b) f ( x )=−12x2+4 x + 2

2) Complete the table of the following using information from both vertex and standard forms:

a) f(x) = x2 + 5x + 4 b) f(x) = 2(x – 1)2 - 8

a) Direction of opening: _____________ ________________

b) Vertex: _____________ _______________

c) Maximum or Minimum value _____________ _______________

d) Axis of Symmetry _____________ _______________

e) y-intercept: _____________ _______________

f) x-intercepts: _____________ _______________(factoring if possible)

Homework: WB: Pg 123 #1ac, 2ab,3bc,4, 6, 9TB: Pg.175 #12, 15

Chapter 3.2 Quadratic Function in Vertex Form