functions

Question 1

when does a system of linear equations not have a solution?

Answer 1

0 0 0 | c; c ≠ 0

Question 2

when does a system of linear equations have infinite solutions, no unique solutions?

Answer 2

0 0 0 | 0

Question 3

distance between two points

Answer 3

= √((x2-x1)^2 + (y2-y1)^2)

Question 4

midpoint of a line segment

Answer 4

(x,y)
x = (x1+x2)/2
y = (y1+y2)/2

Question 5

when is a line completely vertical? horizontal?

Answer 5

x1 = x2 ⇒ vertical line
y1 = y2 ⇒ horizontal line

Question 6

when are two lines parallel to each other? perpendicular?

Answer 6

m1 = m2 ⇒ parallel lines
m1m2 = -1 ⇒ perpendicular lines

Question 7

forms of a linear equation

Answer 7

general form: ax + by + c = 0
slope-intercept form: y = mx + b
slope-point form: y-y1 = m(x-x1)

Question 8

reciprocal functions

Answer 8

• zeros ⇒ poles
• poles ⇒ zeros
• minimum ⇒ maximum OR maximum ⇒ minimum (attains reciprocal value)
• non-vertical asymptotes attain the reciprocal value

Question 9

qualities of compostion

Answer 9

• not commutative => the order of operations influences the result
• associative => change of grouping does not influence the result

Question 10

inverse function

Answer 10

• (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

Question 11

self-inverse functions

Answer 11

f(x) = f-1(x)
symmetrical across y=x

Question 12

when does a function f have an inverse function?

Answer 12

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

Question 13

transformations of functions

Answer 13

1. translation
2. reflection
3. shrinking/stretching

Question 14

translation

Answer 14

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

Question 15

reflection

Answer 15

reflection through the x-axis ⇒ f(x) = -f1(x)
reflection through the y-axis ⇒ f(x) = f1(-x)

Question 16

vertical stretching, shrinking

Answer 16

• 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

Question 17

horizontal stretching, shrinking

Answer 17

• vertical stretching: y = f(x/k)
• vertical shrinking: y = f(kx)
• stretching => dividing x with factor k
• shrinking => stretching with 1/k

Question 18

absolute value of a function

Answer 18

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

Question 19

squared functions

Answer 19

• 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

Question 20

even/odd functions

Answer 20

Even ⇒ graph is symmetrical across the y-axis → f(-x) = f(x)
Odd ⇒ graph is symmetrical across the origin → f(-x) = -f(x)

Question 21

vertex of a parabola

Answer 21

• V (h,k)
• h = (α1 + α2)/2 = -b/2a
• k = f(h) = -D/4a
• x = h => line of symmetry

Question 22

forms of a quadratic function

Answer 22

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)

Question 23

formula of a polynomial

Answer 23

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

Question 24

cubic, quadratic, linear term of a polynomial

Answer 24

a3x3 … cubic term
a2x2 … quadratic term
a1x … linear term

Question 25

finding candidates for the roots of a polynomial

Answer 25

1. integer coefficients and integer root => **ɑ divides the constant coeff.** in p 1. integer coefficients, rational root a/b => **a divides the constant coeff.** in p, **b divides the leading coeff.**

Question 26

remainder theorem

Answer 26

when dividing polynomial p(x) with linear polynomial (x - ɑ); ɑ∈R, the remainder is a constant p(ɑ)

Question 27

fundamental theorem of algebra

Answer 27

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

Question 28

factor theorem

Answer 28

p(x) has a factor (x - ɑ); ɑ∈R, iff p(ɑ) = 0

Question 29

graphs of polynomials | smooth, asymptotes, intercepts w axes, turning points

Answer 29

* smooth (no corners) * no asymptotes * one y-intercept * n intercepts with the x-axis * n-1 turning points

Question 30

behavior of the graph of a polynomial at the roots

Answer 30

* 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)

Question 31

drawing a polynomial

Answer 31

1. calculate p(0) => first point 2. mark roots and their degrees 3. 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

Question 32

Vieto's formulae

Answer 32

sum of roots = - (an-1/an) product of roots = (a0/an)(-1)^n

Question 33

rational function

Answer 33

f(x) = p(x)/q(x); where both p and q are polynomials

Question 34

where are the zeros, poles, holes of a rational function?

Answer 34

* 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

Question 35

where is the asymptote and what is the behaviour of the graph around it? | +when does it intersect the asymptote

Answer 35

* asymptote: y=k(x) (quotient of p/q) * graph always approaches, can intersect * intersection: where r(x) = 0

Question 36

behaviour of the graph around **roots** based on multiplicity

Answer 36

* **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

Question 37

behaviour of the graph around **poles** based on multiplicity

Answer 37

* **odd** multiplicity => sign **changes** * **even** multiplicity => sign **remains**

Question 38

what is the non-vertical asymptote based on the degree of the num and den

Answer 38

* deg(num) > deg(den) => y≠0 * deg(num) < deg(den) => y=0 * deg(num) = deg(den) => y = ann/and

Question 39

what is the exponential function?

Answer 39

f(x) = a^x; 0 < a < ∞, a≠1

Question 40

key points of the exponential function

Answer 40

(0,1) and (1,a)

Question 41

exponential growth

Answer 41

* a>1 * grows faster than any other polynomial * greater a, greater growth

Question 42

exponential decay

Answer 42

* 0 < a < 1 * decays faster than any 1/polynomial * smaller a, greater decay

Question 43

exponential model

Answer 43

**A(t) = Ao b^t** b > 1 => exponential growth 0 < b < 1 => exponential decay

Question 44

half-life definition

Answer 44

= time it takes for a certain material to decrease to half its amount

Question 45

formula for half-life

Answer 45

A(t) = Ao (1/2)^(t/h) h...half-life

Question 46

natural constant

Answer 46

e = 2.71828 e = lim(x->∞) (1 + 1/n)^n e = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n!

Question 47

simple interest

Answer 47

P = P.(1+nr) n...time r...interest rate

Question 48

compound interest

Answer 48

P = P. (1 + r/m)^nm r ... interest rate m ... number of componding nm ... compounding period

Question 49

continuous compounding

Answer 49

P = P.e^(rn) n ... time

Question 50

logarithm def

Answer 50

loga(b) = c <=> a^c = b a > 0, a ≠ 1, b > 0

Question 51

inverse function of f(x) = a^x

Answer 51

f(x) = loga(x)

Question 52

corollaries of the inverse function of f(x) = a^x being f(x) = loga(x)

Answer 52

loga(a^x) = x a^(loga(x)) = x

Question 53

key points of the logarithmic function

Answer 53

(1, 0), (a, 1)

Question 54

asymptote of the logarithmic function

Answer 54

x=0

Question 55

increasing vs decreasing logarithmic function

Answer 55

0 decreasing (decreases slowly) a>1 => increasing (increases slowly)

Question 56

change of base formula - exponents

Answer 56

a^x = b^((logb(a))(x))

Question 57

properties of logarithms

Answer 57

loga(xy) = loga(x) + loga(y) loga(x/y) = loga(x) - loga(y) loga(x^k) = kloga(x) loga(x) = 1/logx(a)

Question 58

change of base formula (logarithms)

Answer 58

loga(x) = logb(x)/logb(a)

Question 59

when does the sign change in exponential/logarithmic functions

Answer 59

decreasing function is applied