Algebra 2 Unit 2 - Quadratics Packet - Dam

UNIT 2:   
Quadratic

Functions

SUCCESS CRITERIA 

 

I can…  factor quadratic expressions and solve quadratic equations. 

 

I can explain…  how to use any quadratic form to graph a parabola and label its key features. 

 

I can justify…  the number and type of roots of a quadratic function. 

 

I can model… a quadratic scenario using one of the three forms. 

 

 

  When you rewrite (x + 2)2 as x2 + 4 …

(x + 2)2 = ______________________ 

 
 
 
 

2 

  TABLE OF CONTENTS  PG # 
  4-8 
TITLE  9-14 
LESSON  15-16 
2.1  Three Forms of Quadratic Functions  17-19 
2.2  20-22 
2.3  Graphing Quadratic Functions  23-24 
2.4  25 
2.5  Rewriting Quadratic Equations into Standard Form  26-29 
2.6 
2.7  Rewriting Quadratic Equations into Factored Form 
2.8 
Quadratic Formula + Imaginary/Complex Numbers 
 
Identifying the Nature of Roots from the Discriminant 

Finding Roots from the Vertex Form 

Writing Quadratic Equations from a Scenario 

 
 
 
 
 
 
 
 
 
 
 
 

 
3 

Lesson 2.1 - Three Forms of Quadratic Functions 
 
 

  

S​ tandard Form    f (x) =  
  
F​ actored Form     f (x) =  
  
​Vertex Form        f (x) =  

 

Sorting Activity: Identify each equation as ​Standard​, F​ actored​, or ​Vertex​ form.

 

 
 
 
 
 
 
 

 
4 

Given a = 1 , write an equation for each graph using ​Vertex form​. (Hint: Use transformations)

5 

Vertex Form: f (x) = a (x − h)2 + k a =/ 0

 

 
6 

Factored Form: f (x) = a (x − r1)(x − r2) a =/ 0

 
 
 

 
7 

Standard Form: f (x) = ax2 + bx + c a =/ 0

 
 
 

 

8 

Lesson 2.2 - Graphing Quadratic Functions  a =/ 0
 
 

Vertex Form: f (x) = a (x − h)2 + k

Example: f (x) = − 2(x − 2)2 + 1

 
9 

You Try: h(x) = 2(x + 2)2 − 6

   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Practice 1: y = − (x + 3)2 − 4 Practice 2: y = − 1 (x − 2)2 − 1
2

 

 
10 

Factored Form: f (x) = a (x − r1)(x − r2) a =/ 0

Example: f (x) = − (x − 4)(x + 2)

 
11 

You Try: h(x) = 1 (x + 3)(x − 5)
2

Practice 1: y = 2x(x − 4) Practice 2: y = − 1 (x + 2)(x − 8)
5

12 

Standard Form: f (x) = ax2 + bx + c a =/ 0

Example: f (x) = x2 − x − 6

 
13 

You Try: h(x) = − 3x2 + 6x + 5

   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Practice 1: y = − x2 + 4x − 3 Practice 2: y = 3x2 + 6x − 2

 

 

 
14 

Lesson 2.3 - Rewriting Quadratic Equations into Standard Form 
 
 
Vertex Form → Standard Form

1) y = − 3(x + 2)2 + 1 3) y = − 2(x + 1)2 − 5

2) y = − 2(x − 3)2 − 3 4) y= 1 (x + 2)2 − 3
2

 
15 

Factored Form → Standard Form 3) y = 2(x + 1)2

1) y = − 3(x + 2)(x − 5)

2) y = − 1 (x − 2)(x + 4) 4) y= 1 (x + 4)(x − 4)
2 4

16 

Lesson 2.4 - Rewriting Quadratic Equations into Factored Form 
 
 
Standard Form → Factored Form 

Factoring Thinking Process: 

1. GCF 

2. Binomials​ (N​ ote: Sum of squares cannot be factored unless there is a GCF)​  

a. Difference of squares: a2 − b2 =  

b. Difference of cubes: a3 − b3 =  

c. Sum of cubes: a3 + b3 =  

3. Trinomials 

a. Perfect square trinomial: a2 + 2ab + b2 = a2 − 2ab + b2 =  

b. x2 + bx + c (a = 1)  

c. ax2 + bx + c (a =/ 1)  

4. Four terms →​ grouping 

GCF Practice: 3) 3m4 − 24m2n2
1) 10a3 − 18a2
4) 5u4 − 20u2v2 + 15v4  
2) 2u4 + 6u2v2 − 80v4

  5) − 25b2 + 16a2
6) 25x2 − 9y2
Difference of Squares Practice: 7) 9x2 − 16y2
1) 16m2 − 9 8) 4u2 − 9v2

2) − 4 + 25v2

3) 2m − 16x2

4) 25 − m2

 
17 

Sum/Difference of Cubes Practice: 3) a3 + 8b3
1) 125 − 27x3 4) a3 − 64b3
2) a3 + 216
5) 3v2 − 8v − 60
Trinomials Practice: 6) 5r2 + 52r + 63
1) v2 − 17v + 70 7) 2k2 + 21k + 10
8) 3v2 + 17v + 20
2) n2 − 11n + 24

3) a2 + 12a + 27

4) k2 − 5k − 36

18 

1) m2 − 8m Factoring Review 
2) m3 − 49m
3) 6v2 − 24v + 18 7) n3 − 12n2 + 35n
4) 6x2 − 120x + 600 8) 5r4 − 10r3 − 315r2
5) m2 − m − 12 9) − 5x2 + 80
6) − k2 + 4k + 21 10) v4 − 7v3 + 6v2
11) 5n2 + 5n2 − 150n
12) 6n3 − 60n2 + 96n

19 

Lesson 2.5 - Quadratic Formula + Imaginary/Complex Numbers 

Converting from standard form to factored form reveals _​ _____________​ aka ​______________________________​ ​. 
But if the quadratic is not factorable, ​_____________________________________________________________​ ​. 

Quadratic Formula  Imaginary Number 
Complex Number 

Solve each equation using the quadratic formula. 3) 8r2 + 3r = − 2

1) 2n2 − 9n = 3

2) 7r2 = − 4 + 3r 4) 3m2 − 6 = 5m

20 

Simplify. 3) √ − 75
4) 5 √ − 484
1) √ − 36
2) − 5i √ − 180  

5) (7 − 7i) + (3 + 4i) 10) (− 7 + 2i)(− 1 − 2i)
6) (2 − 4i) − (− 6 + 8i)
7) (7 − i) + (1 − 5i) 11) (2 − 2i)(− 2 − 3i)
8) (− 4 − 7i) − (6 + i)
9) (7 − 4i)2 12) 3i(2 − 6i)

   
21 

Graphing Quadratic Functions Review 

Graph each quadratic function (include a minimum of 5 points).
Then identify the vertex, axis of symmetry, and intercepts.
1) g(x) = − 3x2 + 6x + 5

2) h(x) = 2x2 + 4x − 6

22 

Lesson 2.6 - Identifying the Nature of Roots from the Discriminant 

Quadratic Formula  Discriminant 

Fundamental Theorem of Algebra 

23 

Find the discriminant of each quadratic equation. Then state the nature (number and type) of roots.

1) − 4x2 + 4x − 1 = 0 5) − 6p2 = − 2p − 1

2) − 10v2 + 7v − 10 = 0 6) 2n2 − 4n = − 2

3) − 4n2 − 9n + 10 = 0 7) 9x2 = − 7 − 4x

4) − 6n2 + 7n + 5 = 0 8) 4n2 = 5 + 8n

 
24 

Lesson 2.7 - Finding Roots from the Vertex Form 

Find the roots for each quadratic equation.

1) y = 3(x − 7)2 − 1 2) y = 3(x − 7)2 − 1 3) y = (x − 3)2 4) y = − 4(x − 2)2 + 4

Graph each quadratic function (include a minimum of 5 points). Then identify the key features.
1) y = − 2(x + 2)2 + 4

Vertex:
Axis of symmetry:
Concaves:
Extrema:
Intercepts:

2) y= 1 (x − 5)2 − 2
5

Vertex:

Axis of symmetry:

Concaves:

Extrema:

Intercepts:

25 

Lesson 2.8 - Writing Quadratic Equations from a Scenario 

What do we need to write a quadratic function in... 

Standard Form?  Factored Form?  Vertex Form? 

Key Words:  Key Words:  Key Words: 

Based on the given information, determine the most efficient form you could use to write a quadratic function.

1) Minimum (6, -75) _________________________________________
y-intercept (0, 15)

2) Points (2, 0), (8, 0), (4, 6)

_________________________________________

3) Points (100, 75), (450, 75), (150, 95)

_________________________________________

4) Points (3, 3), (4, 3), (5, 3)

5) x-intercepts (7.9, 0) & (-7.9, 0) _________________________________________
Point (-4, -4) _________________________________________
_________________________________________
6) Roots (3, 0) & (12, 0) _________________________________________
Point (10, 2) _________________________________________

7) Max hits a baseball off a tee that is 3 feet high.
The ball reaches a maximum height of 20 feet
when it is 15 feet from the tee.

8) A grasshopper was standing on the 35-yard line.
He jumped and landed on the 38-yard line.
At the 36-yard line, he was 8 inches in the air.

26 

Using the given information, write a quadratic equation in the most efficient form.
1) Roots: (3, 0) & (12, 0)

Point (10, 2)

2) Minimum (6, -75)
y-intercept (0, 15)

3) Points (3, 3), (4, -1), (8, 3)

4) Max hits a baseball off a tee that is 3 feet high.
The ball reaches a maximum height of 20 feet when it is 15 feet from the tee.

5) A grasshopper was standing on the 35-yard line of a football field.
He jumped and landed on the 38-yard line. At the 36-yard line, he was 8 inches in the air.

 
27 

Write a quadratic equation for each scenario.
1) Sasha is training her dog, Bingo, to run across an arched ramp, which is in the shape of a parabola. To

help Bingo get across the ramp, Sasha places a treat on the ground where the arched ramp begins and
one at the top of the ramp. The treat at the top of the ramp is a horizontal distance of 2 feet from the first
treat, and Bingo is 6 feet above the group when he reaches the top of the ramp. Write a function to
represent Bingo’s height above the ground as he walks across the ramp in terms of his distance from the
beginning of the ramp.

2) Hector’s dog, Ginger, competes in a waterfowl jump. She jumps from the edge of the water, catches a toy
duck at a horizontal distance of 10 feet from the edge of the water and a height of 2 feet above the water,
and lands in the water at a horizontal distance of 15 feet from the edge of the water. Write a function to
represent the height of Ginger’s jump in terms of her horizontal distance.

 
28 

3) Gary is a fire jumper attempting to run and jump through a right of fire. He runs for 10 feet and begins his
jump 4 feet from the fire and lands 3 feet from the fire ring on the other side. One foot before Gary lands on
the ground, he is 3.5 feet in the air. Write a function to represent Gary’s height above the ground in terms of
his distance from the beginning of his jump.

4) Michelle is a human cannonball. She would like to reach a maximum height of 30 feet during her next
launch. Based on her previous launches, she estimates this will occur when she is 40 feet from the cannon.
When she is shot from the cannon, she is 10 feet above the ground. Write a function to represent

Michelle’s height above the ground in terms of her distance from the cannon. 

 
29