Using Vertex Form EX #7: Graph 𝑓𝑓(𝑥𝑥) = 2(𝑥𝑥 + 2)2 − 3 a = ______________

QUADRATIC FUNCTIONS and TRANSFORMATIONS y

Vocabulary

h = ______________ k = ______________

axis of symmetry: __________________

Parabola – the graph of a quadratic function x

vertex: __________________ point: ___________________

reflected point : ____________

domain: ____________________ range: ____________________ minimum/maximum: _________________________________

Describe and explain the transformation of the parent graph 𝑦𝑦 = 𝑥𝑥 2 to 𝑓𝑓(𝑥𝑥) = 2(𝑥𝑥 + 2)2 − 3.

Quadratic function – any function that can be written in the form 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐, where 𝑎𝑎 ≠ 0. Vertex form – any quadratic function written as 𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 − ℎ)2 + 𝑘𝑘, where 𝑎𝑎 ≠ 0.

Axis of symmetry –a line that divides the parabola into two mirror images, the equation for an axis of symmetry is 𝑥𝑥 = ℎ.

Vertex of the parabola –the intersection of the parabola and its axis of symmetry is (ℎ, 𝑘𝑘). 5

y

4

3

2

1

x −4

−3

−2

−1

1 −1

𝑦𝑦 = 𝑥𝑥 2

2

3

4

Graphing a Function of the form f(x) = ax2 EX #1: Graph

𝒇𝒇(𝒙𝒙) = 𝒂𝒂(𝒙𝒙 − 𝒉𝒉)𝟐𝟐 + 𝒌𝒌

y

𝑦𝑦 = 2𝑥𝑥 2

If 𝒂𝒂 > 0, then _________________________________

Vertex: _______________ Axis of Symmetry: ____________________

x

Point: __________________

Vertex: _______________ Axis of Symmetry: ____________________

Point: __________________

Reflected Point:_____________

If 𝒂𝒂 < 0, then _________________________________ (𝒉𝒉, 𝒌𝒌) is _______________________________________

𝒙𝒙 = 𝒉𝒉 is _______________________________________

Reflected Point:_____________

EX #2: Graph 1 𝑦𝑦 = 𝑥𝑥 2 2

VERTEX FORM OF A PARABOLA

(𝒙𝒙 − 𝒉𝒉) _________________ (𝒙𝒙 + 𝒉𝒉)_______________

y

EXPLORATION: Use a graphing calculator to discover the rules for vertex form. Then analyze graph characteristics. 1. y = (x – 2)2 + 3 x

2. y = (x + 2)2 – 3 3. y = −2(x – 1)2 + 2 4. y = −0.5(x + 1)2 + 2

Graphing a Function of f(x) = −ax2 EX #3: Graph

y

𝑦𝑦 = −𝑥𝑥 2

Vertex: _______________

x

Axis of Symmetry: ____________________

Point: __________________

Reflected Point:_____________

EX #4: Graph 𝑦𝑦 = −2𝑥𝑥 2

Vertex: _______________ Axis of Symmetry: ____________________ © 2014 Jean Adams Flamingo Math.com All rights reserved

Point: __________________

Reflected Point:_____________

y

x

Graphing Translations of f(x) =x2 EX #5: Graph

Writing a Quadratic Function

y

EX #8: Write a quadratic function to model the graph.

𝑦𝑦 = 𝑥𝑥 2 − 2

1. Find the following:

Vertex: _______________ Axis of Symmetry: ____________________

x

Reflected Point:_____________

Describe the translation from 𝑦𝑦 = 𝑥𝑥 2

___________________________________________________

EX #6: Graph

Point: __________________

y

6

Vertex: ________________

2. Substitute values into the vertex form and solve for the a value.

Point: __________________

8

4

2 x −6

f ( x ) = a ( x – h )2 + k

−4

−2

2 −2

−4

−6

−8

y

𝑦𝑦 = (𝑥𝑥 + 1)2

Vertex: _______________ Axis of Symmetry: ____________________

Point: __________________

Reflected Point:_____________

Describe the translation from 𝑦𝑦 = 𝑥𝑥 2

___________________________________________________

x

3. Write the quadratic function: 4. Name the domain, range and minimum value:

4