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.
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
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
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
radical
The expression √s
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.
intercept form
the form y=a(x-p)(x-q), where the x-intercepts of the graph are p and q
radical sign
√
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
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
axis of symmetry
imaginary line that divides parabola into mirror images and passes through the vertex
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
principal square root
the positive square root of a number
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)
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
maximum value
vertex y-coordinate when a<0
minimum value
vertex y-coordinate when a>0
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)