#### Algebra II - Unit 4 Exam ( 4.1 - 4.4 )

In a quadratic equation, the greatest exponent is ___________, and the value of a in ax� is never ______________.

two, zero

What are the types of terms in a quadratic equation?

bx = linear term
c = constant term

A quadratic equation forms a ___________________________. To graph the quadratic function, you must use __________________________________________________.

parabola, order pairs that satisfy the equation

What is the equation for the axis of symmetry and the x coordinate of the vertex?

-b
----
2a

If a graph opens upward then . . .

it has a minimum value where the y - coordinate is a > 0

If graphs open downwards then . . .

it has a maximum value where the y - coordinate is a < 0

Name the types of solutions of a quadratic equation?

- one real solution ; discriminant is zero
- two unique real solutions ; discriminant is positive
- no real solutions ( two unique imaginary/complex solutions ) ; discriminant is negative
REMEMBER
Do not just say no solution for an answer, there could be

To have a solution to a quadratic function, the ______________________________________________________________. No real solution ____________________________________________.

- parabola must touch or pass through the x - axis
- is when the parabola does not touch the x - axis

The roots of a problems refers to . . .

the solutions of the x - intercepts

When solving a quadratic equation, make sure to set the equation equal to . . .

zero.

When factoring, if the coefficient of the first term is not one, then you must ________________________________________________. However, the target sum is always _______________________________-.

- multiply the coefficient of the first term to the coefficient of the last term to find the target product
- the middle term

What is the equation for standard form?

Ax + By = C, which is basically ax� + bx + c

The imaginary unit ?? is the principal square root of ____________________, which is ________________________.

- ( -1 )
- ?? = ?-1

Numbers of the form 6??, -2??, and ???3 are called ________________________________________, which are square roots of negative real numbers. For any positive real number b, _____________________________________________________________.

- pure imaginary numbers
- ( ?-b� = ?b� x ?-1 or b?? )

What is the pattern for imaginary numbers?

R 1 : ?? = ?-1
R 2 : ??� = -1
R 3 : ??� = -??
NO R : ??? = 1
R = REMAINDER

Two complex numbers in the form ____________________ and _____________________ are called __________________________________. The product of this is _____________________ a real number.

- a + b??, a - b??
- complex conjugates
- always

Foil Trick:
If the two parentheses being multiplied have the exact SAME expression with the SAME signs then . . .

1. multiply the first term by the first term
2. multiply the two terms in the parentheses and then double them
3. last terms multiplied by the last term
--------------------------------------------------------------
exs:
( 6x + 5 )�
1. (6x)6x = 36x�
2. (6

Foil Trick:
If the parentheses being multiplied contain the SAME expression, but there is a DIFFERENT sign between the terms in the parentheses then . . .

Note: there will be no middle term
1. first term times first term
2. last term times last term
--------------------------------------------------------------
ex:
( 5x + 2 )( 5x - 2 )
1. (5x)5x = 25x�
2. (2)(-2) = -4
= 25x� - 4

If the leading coefficient is positive, then ____________________________________________________.

- the graph opens upwards and has a minimum

If the leading coefficient is negative, then ____________________________________________________.

- the graph opens downwards and has a maximum

You can find the roots of a quadratic function by . . .

looking at the y values of the chart and seeing when they turn positive to negative or negative to positive

The bigger the leading coefficient . . .

the narrower the parabola is.

The smaller the leading coefficient . . .

the wider the parabola is.

If there is an exponent other than one on the ??, then it is . . .

not fully simplified
--------------------------------------------------------------
ex:
?-15 x ?-25
= -5?-15 ( ??� = -1 )

When completing the square, the leading coefficient always has to be . . .

one, even if the coefficient does not evenly divide the others terms, simply convert them to fractions using the leading coefficient.

When factoring the leading coefficient ( which is not one ), do not forget to . . .

multiply the last term of the trinomial by the leading coefficient when moving and adding the coefficient to the other side of the equation.

How do you know an inequality will be a "and" or "or" statement.

Method 1:
- less thAND
- greater OR
Method 2:
Use zones and choose values that are in the zone and plug them into the original equation to test validity ( if zone 1 is bad, zone 3 is bad, and zone 2 is good )

What are the steps for completing the square in a quadratic equation?

1. Find one half of b, the coefficient of x
2. Square the result in Step 1
3. Add the result of Step 2 to x� + bx
Equation : x� + bx + ( b / 2 )� = ( x + ( b / 2 )�)

What is the quadratic equation, and what variable can not equal zero?

a ? 0

In a quadratic equation, you must set the equation equal to . . .

zero

When entering a quadratic equation into a calculator, then you must . . .

put parentheses around the numerator

Quadratic equation can be . . .

simplified farther after finding the plus or minus equation
( - 6 � 2?23 / 4 ) ? 2 = ( -3 � ?23 / 4 )

The discriminant is . . .

the equation under the square root.

What are the types of solutions you can get from an inequality discriminant?

b� - 4ac > 0; b� - 4ac is a perfect square : 2 real, rational roots
b� - 4ac > 0; b� - 4ac is not a perfect square : 2 real, irrational roots
b� - 4ac = 0 : 1 real rational root
b� - 4ac < 0 : 2 complex roots

What is the equation for vertex form?

y = a(x - h)� + k ( easiest for graphing )
vertex : ( h,k )
x = h : axis of symmetry
min or max : ( +a ) or ( -a )
no y-int is shown

When putting a trinomial into vertex form, you must . . .

move the third term ( and potentially multiply it depending if you factored anything out of the trinomial ) and add it to the outside of the parentheses.
ex:
y = -2x� + 8x - 3
= ( -2x� + 8x + ______ ) - 3
= -2(x� - 4x + 4 ) - 3 - (-2(4))
y = -2(x� - 4x +

How to you express the type of line and the type of dot using braces and parentheses?

( ) = o = - - -
[ ] = � = -----

How do you write set notation?

{ x | x? < x < x? }

How do you write int notation?

( x?, x? )