15 functions - 16 quadratics Flashcards
define
a function is like a
a machine that takes an input, transforms it, and spits out an output
example
in f(x) = x^2 + 1
every input (x) is squared and added to one to get the output f(x)
define
because a fraction can’t be divided by 0, when the denominator is zero, a function is
undefined
define
Domain is
the set of all possible input values (x) to a function (values that don’t lead to an invalid operation or an undefined output)
define
Range
the set of all possible output values (y) from a function
define
what are vertical asymptotes?
a vertical line that guides the graph of the function but is not part of it
define
what are horizontal asymptotes?
a horizontal line that is not part of a graph of a function but guides it for x-values
technique: functions
to find the domain, start with
all real numbers and exclude the values of x for which the function is invalid or undefined
technique: functions
to find the range,
graph the function on your calculator and figure out the possible values of y, taking note of any horizontal asymptotes
technique: functions
anytime f(x) is used in a graphing question,
think of it as the y
define
what is a point?
an input and an output, an x and a y
confusing concept
zeros, roots, and x intercepts of a function are all
different terms for x that makes f(x) = 0
define
a constant is a
function
no matter the input, the same output always results
confusing concept
solutions to f(x) = k refers to
the intersection points of f(x) and the horizontal line y = k
confusing concept
consider constants as ___________ lines
horizontal lines
e.g. f(x) > 5 means the entire graph of f is above the horizontal line y=5
define
f(x) = ax^2 + bx + c
a quadratic in the form of a function
define
the roots/x intercepts/solutions refer to the
values of x that make f(x) = 0
key formula
sum of the roots
-b/a
in quadratic ax^2 + bx + c
key formula
product of the roots
c/a
define
vertex is
the midpoint of the parabola
confusing concept
the x coordinate of the vertex is always
the midpoint of the two roots
confusing concept
vertex form is one way of
representing a quadratic function
key formula
vertex form equation
y = a(x-h)^2 + k
technique: functions (vertex form)
to get a quadratic function into vertex form,
must complete the square
example
complete the square of
y = x^2 - 4x - 21
A. b = b/2
-4 = -4/2 = -2
=> y = (x-2)^2 - 21
B. (-2)^2 = 4
=> y = (x-2)^2 - 21 - 4
= y = (x-2)^2 - 25
C. vertex = (2, -25)
technique
vertex form allows
to find the vertex without knowing the roots of a quadratic
formula
discriminant is equal to
b^2 - 4ac
confusing concept
the sign of the discriminant indicates
how many solutions there are for a parabola
technique
if D > 0,
there are two real roots (two solutions)
technique
if D = 0,
there is one real root
technique
if D < 0,
there are no real roots
key formula
quadratic formula
x = (-b±√(b²-4ac))/(2a)
explanation
when b²-4ac > 0
the “±” in the quadratic formal takes effect and results in two different roots
explanation
when b²-4ac = 0, what is its effect in the quadratic formula
since the “b²-4ac” part in the quadratic formal equal to zero, we’re essentially adding and subtracting 0, both of which gives the same root; hence, one real root/solution/x intercept
explanation
when b²-4ac < 0
we’re taking the square root of a negative number, which is undefined and gives us no real roots
technique
when asked for the maximum or the minimum of a quadratic,
find the vertex
define
what is an asymptote?
An asymptote is a line that is never crossed by the function
technique
where do you find the vertical asymptote?
vertical asymptotes are found where the bottom of the fraction is zero (because you are never allowed to divide by zero)
confusing concept
even if a quadratic doesn’t have any real roots,
it can have imaginary roots
