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.
