functions Flashcards
when does a system of linear equations not have a solution?
0 0 0 | c; c ≠ 0
when does a system of linear equations have infinite solutions, no unique solutions?
0 0 0 | 0
distance between two points
= √((x2-x1)^2 + (y2-y1)^2)
midpoint of a line segment
(x,y)
x = (x1+x2)/2
y = (y1+y2)/2
when is a line completely vertical? horizontal?
x1 = x2 ⇒ vertical line
y1 = y2 ⇒ horizontal line
when are two lines parallel to each other? perpendicular?
m1 = m2 ⇒ parallel lines
m1m2 = -1 ⇒ perpendicular lines
forms of a linear equation
general form: ax + by + c = 0
slope-intercept form: y = mx + b
slope-point form: y-y1 = m(x-x1)
reciprocal functions
- zeros ⇒ poles
- poles ⇒ zeros
- minimum ⇒ maximum OR maximum ⇒ minimum (attains reciprocal value)
- non-vertical asymptotes attain the reciprocal value
qualities of compostion
- not commutative => the order of operations influences the result
- associative => change of grouping does not influence the result
inverse function
- (fof-1)(x) = x for all xDf and (f-1of)(x) = x for all xDf
- drawn by reflecting across y=x
- calculated by replacing the y and x
self-inverse functions
f(x) = f-1(x)
symmetrical across y=x
when does a function f have an inverse function?
if f is a one-to-one function (every two different originals have two different images, monotonic)
a function is not one-to-one => restrict the domain
transformations of functions
- translation
- reflection
- shrinking/stretching
translation
y = f(x) + k ⇒ up or down for k units
y = f(x + k) ⇒ left or right for k units
translation for vector (a,b) ⇒ y = f(x - a) + b
reflection
reflection through the x-axis ⇒ f(x) = -f1(x)
reflection through the y-axis ⇒ f(x) = f1(-x)
vertical stretching, shrinking
- x-axis is fixed, in the y-direction
- vertical stretching: y = kf(x)
- vertical shrinking: y = f(x)/k
- stretching => dividing y with factor k
- shrinking => multiplying with k
horizontal stretching, shrinking
- vertical stretching: y = f(x/k)
- vertical shrinking: y = f(kx)
- stretching => dividing x with factor k
- shrinking => stretching with 1/k
absolute value of a function
y = |f(x)| ⇒ reflection of what is in the -y into +y
y = f(|x|) ⇒ reflection of what is in the +x on the -x
squared functions
- f(x) → (f(x))2
- values are non-negative
- f(x) = 0, f(x) = 1 doesn’t change
- 0 < f(x) < 1; f(x) > f(x)2
- f(x) > 1; f(x) < f(x)2
even/odd functions
Even ⇒ graph is symmetrical across the y-axis → f(-x) = f(x)
Odd ⇒ graph is symmetrical across the origin → f(-x) = -f(x)
vertex of a parabola
- V (h,k)
- h = (α1 + α2)/2 = -b/2a
- k = f(h) = -D/4a
- x = h => line of symmetry
forms of a quadratic function
general form: f(x) = ax2 + bx + c
vertex form: f(x) = a(x - h)^2 + k
factorized form: f(x) = a(x - α1)(x - α2)
formula of a polynomial
p(x) = anxn + an-1xn-1 + … + a1x + a0
n is the degree of the polynomial (n∈N)
anxn = leading term (an = leading coefficient)
a0 = constant term
cubic, quadratic, linear term of a polynomial
a3x3 … cubic term
a2x2 … quadratic term
a1x … linear term
finding candidates for the roots of a polynomial
- integer coefficients and integer root => ɑ divides the constant coeff. in p
- integer coefficients, rational root a/b => a divides the constant coeff. in p, b divides the leading coeff.
remainder theorem
when dividing polynomial p(x) with linear polynomial (x - ɑ); ɑ∈R, the remainder is a constant p(ɑ)
fundamental theorem of algebra
every non-constant polynomial has a complex root
=> every non-constant polynomial of degree n1 can be factorised in n linear terms
=> a polynomial of degree n1 has n roots in C, some of which may be the same
factor theorem
p(x) has a factor (x - ɑ); ɑ∈R, iff p(ɑ) = 0
graphs of polynomials
smooth, asymptotes, intercepts w axes, turning points
- smooth (no corners)
- no asymptotes
- one y-intercept
- n intercepts with the x-axis
- n-1 turning points
behavior of the graph of a polynomial at the roots
- root of the first degree => crosses the x-axis as a straight line
- even degree => lays down on the x-axis and bounces back (doesn’t change in sign)
- odd degree => lays down on the x-axis and crosses it (changes sign)
drawing a polynomial
- calculate p(0) => first point
- mark roots and their degrees
- draw from the y-intercept into both directions
p(0) = 0 => start drawing with the leading coefficient on the right side of the greatest root
an > 0 => p(x)>0
an < 0 => p(x)<0
Vieto’s formulae
sum of roots = - (an-1/an)
product of roots = (a0/an)(-1)^n
rational function
f(x) = p(x)/q(x); where both p and q are polynomials
where are the zeros, poles, holes of a rational function?
- zeros = roots of the numerator
- poles = roots of the denominator
- hole = root of both the numerator and denominator, no longer a root of the denominator after cancelling out
where is the asymptote and what is the behaviour of the graph around it?
+when does it intersect the asymptote
- asymptote: y=k(x) (quotient of p/q)
- graph always approaches, can intersect
- intersection: where r(x) = 0
behaviour of the graph around roots based on multiplicity
- degree 1 => sign changes, passes the x-axis as a straight line
- odd => sign changes, appears as a straight line on the x-axis
- even => sign remains, appears a straight line on the x-axis
behaviour of the graph around poles based on multiplicity
- odd multiplicity => sign changes
- even multiplicity => sign remains
what is the non-vertical asymptote based on the degree of the num and den
- deg(num) > deg(den) => y≠0
- deg(num) < deg(den) => y=0
- deg(num) = deg(den) => y = ann/and
what is the exponential function?
f(x) = a^x; 0 < a < ∞, a≠1
key points of the exponential function
(0,1) and (1,a)
exponential growth
- a>1
- grows faster than any other polynomial
- greater a, greater growth
exponential decay
- 0 < a < 1
- decays faster than any 1/polynomial
- smaller a, greater decay
exponential model
A(t) = Ao b^t
b > 1 => exponential growth
0 < b < 1 => exponential decay
half-life definition
= time it takes for a certain material to decrease to half its amount
formula for half-life
A(t) = Ao (1/2)^(t/h)
h…half-life
natural constant
e = 2.71828
e = lim(x->∞) (1 + 1/n)^n
e = 1 + 1/1! + 1/2! + 1/3! + … + 1/n!
simple interest
P = P.(1+nr)
n…time
r…interest rate
compound interest
P = P. (1 + r/m)^nm
r … interest rate
m … number of componding
nm … compounding period
continuous compounding
P = P.e^(rn)
n … time
logarithm def
loga(b) = c <=> a^c = b
a > 0, a ≠ 1, b > 0
inverse function of f(x) = a^x
f(x) = loga(x)
corollaries of the inverse function of f(x) = a^x being f(x) = loga(x)
loga(a^x) = x
a^(loga(x)) = x
key points of the logarithmic function
(1, 0), (a, 1)
asymptote of the logarithmic function
x=0
increasing vs decreasing logarithmic function
0<a<1 => decreasing (decreases slowly)
a>1 => increasing (increases slowly)
change of base formula - exponents
a^x = b^((logb(a))(x))
properties of logarithms
loga(xy) = loga(x) + loga(y)
loga(x/y) = loga(x) - loga(y)
loga(x^k) = kloga(x)
loga(x) = 1/logx(a)
change of base formula (logarithms)
loga(x) = logb(x)/logb(a)
when does the sign change in exponential/logarithmic functions
decreasing function is applied
