Chapter 3 Definitions

Quadratic Function in Vertex Form

(h,k) is the vertex a determines if vertex is maximum or minimum

Roots

Another name for solutions

Discriminant is Negative Perfect Square

Both components of the complex number will be rational

Quadratic Equations

Equation where the highest degree of any term is 2. There could be up to two solutions to the equation.

Equations with Absolute Value

For a > 0, then |x|=a is the same as x = +a or x = -a

Absolute Value Inequalities with Less Than (<)

For a>0, |x|< a is the same as -a < x < a. Or, when the solutions for x < a and x > -a overlap.

Absolute Value Inequalities with Greater Than (>)

For a>0, |x|> a is the same as x > a or x < -a.

Quadratic Formula

Formula created by Completing the Square

Factor-Zero Property

If A*B=0, then A=0, B=0 or A&B = 0

Quadratic Formula

If ax^2+bx+c=0, then

Square Root Principle

If x^2 = a, then x = +sqrt(a) or -sqrt(a).

Axis of Symmetry

Imaginary vertical line that goes through the x value of the vertex. x= -b/(2a)

Division of Complex Numbers

Multiply the numerator and denominator by the conjugate of the denominator.

Discriminant = 0

One real rational solution One x-intercept

Solution to Absolute Value Inequality Greater Than 0

Since the inequality is looking for only positive solutions, then the solutions will be All Real Numbers except when the expression is equal to 0.

Solution to Absolute Value Inequality Less Than or Equal to 0

Since the output of an absolute value cannot be negative, then the only time the inequality will be true is when the expression inside the inequality is equal to 0. We use {} to represent a single solution

Solution to Absolute Value Inequality Less Than 0

Since the output of an absolute value cannot be negative, then the solution to absolute value inequality less than 0 is No Solution.

Extraneous Solution

Solutions that was generated through the process of solving an equation but does not produces a "True" statment when evaluated with the original equation.

Imaginary Numbers

Square root of a negative numbers

Absolute Value

The distance a number is away from zero. The output of an absolute value cannot be negative.

Vertex

The highest or lowest ordered pair of a quadratic function.

Discriminant is Negative but not Perfect Square

The imaginary component will irrational

Discriminant <0

Two complex solutions (Conjugates) No x-intercept

Discriminant >0

Two real solutions Two x-intercepts

Discriminant is Positive but not Perfect Square

Two real solutions will be irrational (conjugates)

Discriminant is Positive Perfect Square

Two real solutions will be rational

Standard form for Complex Numbers

a + bi, where a is the real component and bi is the imaginary component. fractional complex numbers must be separated into two distinct (and reduced) terms. Standard form of complex numbers are not commutative.

Conjugate of Complex Number, a + bi

a - bi Change the sign on the imaginary component of the complex number.

Discriminant

algebraic expression to identify the number of solutions in a quadratic equation. b^2-4ac

Standard form for Quadratic Equation

ax^2+bx+c=0 where a,b,c are real numbers and a is not zero

Vertex with General Form

if f(x)=ax^2+bx+c, then the vertex is the ordered pair

Complex Numbers

sum of real and imaginary component. Occurs when there are no x-intercepts when graphing a quadratic function.

Highest Point in the Quadratic Function

when a < 0, then the vertex will be the highest ordered pair of the function.

Lowest Point in the Quadratic Function

when a > 0, then the vertex will be the lowest ordered pair of the function.