Chapter 4 Quadratic Functions
imaginary unit i
i=√-1, so i²=-1
square root
if r^2=s, and s is a positive number, s has two square roots- + and -.
zero product property
if the product of two factors is zero, then at least one of the factors must be zero. Ex: If ab = 0, then a = 0 or b = 0.
quadratic equation
An equation that can be written in the (standard) form ax2 + bx + c = 0, where a,b,and c are real numbers and a ≠ 0
intercept form
the form y=a(x-p)(x-q), where the x-intercepts of the graph are p and q
complex number
a number a+bi where and and b are real numbers, a number a+bi where a and b are real numbers and i is the imaginary unit
complex plane
a set of coordinate axes in which the horizontal axis is the real axis and the vertical axis is the imaginary axis; used to graph complex numbers
vertex
lowest or highest point on parabola
best fitting quadratic model
The model given by quadratic regression
Equality between complex numbers
a+bi=c+di if and only if a=c and b=d
monomial
an expression that is either a number, a variable, or the product of a number and one or more variables
conjugates
binomials of the form a√b + c√d and a√b - c√d whose product is always a rational number with no radicals
vertex form
A quadratic function in the form y=a(x-h)^2+k, where (h,k) is the vertex of the parabola and x=h is its axis of symmetry
radical sign
√
radical
The expression √s
zeros of a function
the x-values for which f(x) = 0
complex conjugates
two complex numbers of the form a+bi and a-bi; product is always a real number
Vertical Motion Problem
A function used to model the height of an object in motion. When object is dropped: h=-16t²+h1 When object is launched/thrown: h=-16t²+vt+h1 If object is launched upward, v will be positive If object is launched downward, v will be negative If object is launched parallel to ground, v will equal zero
axis of symmetry
imaginary line that divides parabola into mirror images and passes through the vertex
principal square root
the positive square root of a number
quadratic formula
when a, b, and c are real numbers where a doesn't equal 0, the solution of ax²+bx+c=0 are x=[-b ± √(b² - 4ac)] / (2a)
quadratic function
a function that can be written in the form f(x)=ax^2+bx+c, where a, b & c are real numbers and a is not equal to zero
imaginary number
a number of the form a+bi where b is not equal to zero
pure imaginary number
a number of the form a+bi where b is not equal to zero and a=0
completing the square
a process used to form a perfect square trinomial.
discriminant
b²-4ac; when D>0 there are 2 real solutions and graph will have 2 x-intercepts; when D=0 there is 1 real solution and graph will have 1 x-intercept; when D<0 there are 2 imaginary solutions and graph will have no x-intercepts
absolute value of a complex number
if z=a+bi, then the absolute value of z, denoted |z|, is a nonnegative real number defined as |z|=√a²+b²; this is the distance between z and the origin in the complex plane
quadratic inequality in two variables
it can be written in one of the following forms: y<ax²+bx+c, y>ax²+bx+c (and then when y is equal to or less/greater than) Graph will consist of all solutions (x,y) of the inequality
quadratic inequality in one variable
it can be written in one the following forms: ax²+bx+c<0, ax²+bx+c>0 (and then when y is equal to or less/greater than) Can be solved by tables, graphs, or algebraic methods
radicand
number beneath the radical sign
parabola
the shape of the graph of a quadratic function
roots
the solutions of a quadratic equation
trinomial
the sum of three monomials
binomial
the sum of two monomials (ex. x+4)
maximum value
vertex y-coordinate when a<0
minimum value
vertex y-coordinate when a>0
