The final Review

The Final Review
Quadratic Functions

​

​​Example:

​2x² + 10x + 8 = 0
2x² + 8x + 2x = 0
2x ( x+8 ) + 2 ( x+8 ) = 0
( 2x+2 ) ( x+8 ) = 0
2x+2=0 x+8
2x=-2 x=-8
x=-2/2
x=-1

Factoring when A NOT= 1

​5x² - 8x + 3 = 0
5x² - 5x -3x +3 = 0
5x ( x-1 ) -3 ( x-1 ) = 0
( 5x-3 ) ( x-1 ) = 0
5x-3 =0 x-1 =0
5x =3 x =1
x=3/5

​16

​15

​8

​10

​16

​-3

​-8

​-5

​A parent function for a quadratic function looks like this:



The qualities that a quadratic function has are:

- The axis of symmetry
- X-intercepts (zeros, roots, solutions)
- Y-intercepts
- Vertex (Minimum/Maximum)
- End behavior
- Increasing/decreasing intervals

Characteristics of a Quadratic Function

Introduction to quadratics

​End behavior
Ex: The parabola is downwards
as x reaches infinity, y reaches -infinity
as x reaches -infinity, y reaches -infinity

Increasing/decreasing intervals
Ex: The vertex is (-2,-2)
Increasing: (- infinity, -2)
Decreasing: (-2, infinity)

Domain and range:
The domain will always be -infinity to infinity
The range depends on the vertex:
Ex: vertex (0,7) and the parabola is downwards
- 7 to -infinity

​​You can use the x method

​ Example:
X² +7x +12

Factoring when A=1

​

​a=1
b=7
c=12

​12
a x c

​b
7
add

​3

​4

y = (x+3) (x+4)

​x- intercepts

1. x+3=0 +=-
x=3 -=+
2. x+4=0
x=4

​x² + 12x + 5
x²+ 12x + 36 -36 +5 12/2=6
( x+6 )² -31
( x+6 )² -31 = 0
( x+6 )² = 31
x+6 = ± √31
x= -6 ± √31

​​when a=1

​​when a NOT= 1

Factoring using completing the square

You are trying to make a perfect trinomial:
(a+b) = a² +2ab+b²

​2x² + 12x + 5
*remember to factor out the a value, and leave the c alone, for example:
2 ( x² + 6x ) +5
2 ( x ⋅ x + 2 ⋅ 3 ⋅ x +9) -9+5
also, only distribute the x2 to the square (9), not the c
2( x+3 )² -18 +5
2 ( x+3 )² -13
2 ( x+3 )² = 13
( x + 3 )² = 13/2
x + 3 = √ (13/2)
x = 3 ± √ (13/2)

​Example:

*The formula is the most important part when using this formula*

Factoring using Quadratic formula

​​Important info.

​Example: The parent function is a (x -h)² +k

ex¹: 2( x-1 )²
Verticle stretch of 2, 1 to the right

ex²: ( x+4 )² -5
4 units to the left, 5 units down

ex: 1/4 (x+5) +3

Transformations

The key: (look at a)

​Now Apply what you have learned

Multiple Choice

What is the parent function for a Quadratic equation?

• y=x²

• y=√x

• y=mx +b

• y = k/x

Multiple Choice

Where is the axis of symmetry for this function?

6x² - 36x + 12

• -3

• x=-42

• 42

• x=3

Multiple Select

How many roots does this Quadratic equation have?

5x² +25

• 5

• 3
• 0

• 1

Multiple Choice

Solve: b² = -35 + 12b RE-ARRANGE AND FACTOR

• b = -5 and -7

• b = 5 and 7

• b = -5 and 7

• b = 5 and -7

Multiple Choice

What is the greatest common factor of 42xy² and 28x³y?

• 84x³y²

• 14y

• 14xy

• 14xy²

Multiple Choice

Factor (hint...take out the greatest common factor first)
2c² + 2c - 84

• (c-6)(2c+14)

• (c+7)(2c-12)

• (c-4)(2c+21)

• 2(c+7)(c-6)

Multiple Choice

What is the first step in solving
(x + 2)(x - 3) = 0

• Plug in zero for x

• Solve for x

• Set each binomial factor equal to zero

• FOIL

Multiple Choice

Factor

5x² - 28x + 32

• (3x - 10)(x + 6)

• (x - 8)(5x - 4)

• (5x - 8)(x - 4)

• (5x - 8)(x + 7)

Multiple Choice

5x²+20x+20

• 5(x+2)(x+2)

• 5(x-2)(x-2)

• 5(x+2)(x-2)

• (5x+2)(x+2)

Multiple Choice

2x² - 13x - 70

• (2x - 7)(x + 10)

• (7x + 5)(x + 2)

• (2x + 7)(x - 10)

• 2(x + 7)(x + 10)

Multiple Select

Complete the square for
x² + 12x + ____

• x² + 12x + 144

• x² + 12x + 36

• x² + 12x - 36

• x² + +12x - 144

Multiple Choice

What do you do to the b value to correctly complete the square?

• square it

• divide it by 2 and take the square root of the result

• divide it by 2 and square the result

• divide it by 2 only

Multiple Choice

What is one of the solutions to this equation?
(2x - 1)² = 49

• 7

• -7

• -4

• -3

Multiple Choice

Solve using square roots.
5b² - 4 = 41

• b = 9, b = -9

• b = 3, b = -3

• b = 8, b = -8

• no solution

Multiple Choice

Solve each equation with the quadratic formula.
2x²−5x+9=−2

• 
  x= (-5 ±√113)/4

• no solution

• x=-2 ±√26

• x=3

Multiple Choice

5x²−80=0

• x=4,−4

• x= 7, -7

• x=1

• no solution

Multiple Choice

−3x² = 9 − 6x² Solve each equation with the quadratic formula.

• x= ±√3

• no solution

• x= (-3 ± 2√11)/7

• x=45, -43

Multiple Choice

Which transformation maps the graph of
f(x) = x² to the graph of g(x) = (x + 4)²?

• a reflection across the line x = -4

• a reflection across the line y = -4

• a translation shifting f(x) 4 units to the left

• a translation shifting f(x) 4 units to the right

Multiple Choice

Identify the vertex of f(x) = -2(x+1)² + 3

• (h,k)

• (3, 1)

• (-1,-3)

• (-1,3)

Multiple Choice

What steps transform the graph y = x² to y = x² + 8

• shifted up 8 units

• shifted down 8 units

• shifted left 8 units

• shifted right 8 units