Graphs, lines, polynomials, exponentials and logarithms
Chapter 1.2 – Graphs and Lines
General form of a line:
𝑦 = 𝑚𝑥 + 𝑏
Point Gradient Formula
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) (m=rise/run)
Chapter 2.1 – Functions
𝑓(𝑥)
Vertical transformation
Where 𝑓 (𝑥 ) = 𝑓(𝑥 ) − 𝑘  the
function is shifted downwards by “k”
units
Horizontal transformation
Where 𝑓 (𝑥 ) = 𝑓(𝑥 − 𝑘)  the
function is shifted to the right by “k”
units
As such, where 𝑓(𝑥 ) =
, the
2
function rises/falls half as quickly
Reflection
Where f (x) = - f (x), the graph is
reflected about the x axis
Where x and y are inverted (i.e. x = y2 )
the graph is perpendicular to how the
graph will have been
Stretching/squeezing the function
Where 𝑓 (𝑥 ) = 2𝑓(𝑥)  the function
rises/falls double as quickly
Chapter 2.3 – Quadratic Functions
The Quadratic Function
f (x) = ax2 + bx+ c  General form
f (x) = (x± d)(x± e)  Intercept form
f (x) = (x - g)2  vertex form
To find the x-intercepts of a quadratic
function:
Either: factorize to intercept form,
then equate each bracket to zero
Or: use the quadratic formula:
-b± b2 - 4ac
2a
Chapter 2.4 – Polynomials and rational
function
To find the vertex form of a quadratic
function:
1. take “ ( -b )2 ", and add/subtract it
2
to the formula
2. Find numbers that multiplies to
give you “ ( -b )2 ” and adds to give
2
you “b”: these are the roots of
the equation.
3. Analyze the transformations,
see Chapter 2.1 – Functions,
above.
Finding asymptotes
n(x)
Where f (x) =
d(x)
Degree of a polynomial
Is the highest “power” in the
polynomial chain
Vertical Asymptotes:
1. After cancelling common
factors, where d(x) = 0 , there is a
vertical asymptote
Horizontal Asymptotes
1. If the degree of n(x) < the degree
of d(x), y = 0 is the horizontal
asymptote
2. If the degree of n(x) = the degree
of d(x), y = a b is the horizontal
asymptote:
a. a is the leading coefficient
of n(x)
b. b is the leading coefficient
of d(x)
3. If degree of n(x) > degree of d(x) ,
there is no horizontal
asymptote
Chapter 2.5 – Exponential functions
Note: graphs shift as per regular
functions, see
Chapter 2.1 – Functions.
A graph has an exponential shape where f (x) = bx, x ¹1, x > 0 .
Properties of f (x) = bx
1. All graphs go through
for any base b
2. The graph is continuous
3. The x axis is a horizontal
asymptote
4. Where b >1, bx increases as x
increases
5. Where 0 < b <1, bx decreases as x
increases
Exponent Laws
Where a and b are positive, a ¹1, b ¹1 , x & y are real.
1. axay = ax+y
4. (ab)x = axbx
x
2. ay = ax-y
3.
a
(ax ) y = axy
Further:
1. ax = ay iff x = y
5.
a
ax
( )x = x
b
b
2. where x ¹ 0, ax = bx iff a = b
Common bases
10 & e
Note, growth/decay formulae are often in the form y = cekt , c & k are constants, t is
time.
Chapter 2.6 – Logarithmic Functions
y = loga x iff x = ay
that is: a = bc : logb a = c
Log properties
Where b, M and N are positive, b ¹1and p & x are real numbers
1. logb 1 = 0
4. blog x = x, x > 0
5. logb MN = logb M + logb N
2. logb b =1
3. logb bx = x
6. logb M = logb M - logb N
b
N
7. logb M p = plogb M
8. logb M = logb N iff M = N
Changing base of log etc.
Note, calculator has "log" = log10 and "ln" = loge
xln b
x
 e =b
 ln x = logb x
ln b
Financial applications
Chapter 3.1 – simple interest
I = P· R·t , where I is interest, P is principle, r is the annual simple interest rate,
and t is the time in years.
Note: do not forget to add the principle again when working out future value,
since this formula only works out interest
Chapter 3.1 – compound interest
The compound interest formula
A = P(1+ i)n = P(1+
r mt
) : where A is the future value at the end of n periods, P is the
m
principle, r is the annual the annual nominal rate of interest, m is the amount of
compounding periods per year, i is the interest rate per compounding period, n is
the total number of compounding periods.
Note: make sure “i” and “n” are in the same units of time.
Continuous compound interest
A = Pert , r is the annual compounding rate, t is time in years.
Computing growth time
Since A = P(1+ i)n , ln A = nln(P(1+i)).
Annual percentage yield
APY = (1+
r m
) -1, or, if compounded continuously, APY = er -1
m
Chapter 3.4 – Annuities
Strategy
1. Make a timeline of payments
2. If single payment: either simple
or compound interest
3. If multiple:
a. Payments into an account
increasing in value (FV)
b. Payments being made out
of an account decreasing
in value (PV)
c. All amortization is PV.
Future value of an ordinary annuity
(1+ i)n -1
FV = PMT(
) , where FV is the future value, PMT is the periodic payment, i is
i
the rate per period, n is the number of periods/payments.
Present value of an ordinary annuity
1- (1+ i)-n
PV = PMT(
)
i
Derivatives
Chapter 10.4 – the derivative
Slope of a secant between two points
f (a) - f (b)
a- b
Average rate of change (slope of a
secant between x and x+h)
f (a+ h) - f (a)
,h¹ 0
h
The derivative from first principles
lim
h®0
f (x + h) - f (x)
note: it is most probable that the h on the denominator will go
h
Chapter 10.5 – basic differentiation properties
1. Constant
f (x) = c ® f '(x) = 0
2. Just an x
f (x) = x ® f '(x) =1
3. A power of x
f (x) = xn ® f '(x) = nxn-1
4. A constant*a function f (x) = k·u(x) ® f '(x) = k·u'(x)
5. Sum/difference
f (x) = u(x)± v(x) ® f '(x) = u'(x)± v'(x)
Chapter 11.2 – derivatives of logarithmic and exponential functions
1. Base e exponential
f (x) = ex ® f '(x) = ex
2. Base e exponential with constant in power
f (x) = ecx ® f '(x) = cecx
3. Other exponential
4. Natural log
5. Other log
f (x) = bx ® f '(x) = bx lnb
1
f (x) = ln x ® f '(x) =
x
1 1
f (x) = logb x ® f '(x) =
·
ln b x
Chapter 11.3 – product/quotient rule
The product rule
f (x) = F(x)S(x) ® f '(x) = F(x)S'(x)+ S(x)F '(x)
The quotient rule
f (x) =
T(x) B(x)T '(x) - T(x)B'(x)
®
B(x)
[B(x)]2
Chapter 11.4 – the chain rule
m(x) = f (g(x)) ® m'(x) = f '(g(x))·g'(x)
easy to do via substitution
the above formula means: derivative of whole thing times derivative of bracket
The general derivative rules
1. d [ f (x)]n = n[ f (x)]n-1 · f (x)
2.
3.
dx
d
1
ln[ f (x)] =
· f '(x)
dx
f (x)
d f ( x)
e = ef ( x) · f '(x)
dx
Chapter 12.1 – first derivatives and graphs
Make sign charts
Local extrema
Where the first derivative is 0, and the sign of the first derivative changes around
it, it is a local extrema:
1. – 0 +  minimum
2. + 0 -  maximum
3. – 0 – or + 0 +  not a local extrema
Note, where f '(c) = 0 , finding f ''(c) can also identify whether it is a local extrema:
where f '(c) = 0 & f ''(c) = + , it is a local minimum; where f '(c) = 0 & f ''(c) = - , it is a local
maximum. This test is invalid where f ''(c) = 0 .
Chapter 12.2 – second derivatives and graphs
The second derivative describes the concavity of a graph (where f ''(x) > 0 , the
concavity is positive
, and f '(x) (/the slope) is increasing; where f ''(x) < 0 , the
concavity is negative
and f '(x) (/the slope) is decreasing.
Point of inflexion
A point of inflexion is where the concavity of the graph changes (and, as such, the
sign of the derivative, too, will change. This occurs where f ''(x) = 0 (or, if it is a
vertical point of inflexion, undefined) and the sign of the second derivative
changes about that point.
Graph sketching
1. Analyze f (x) , find domain and intercepts
2. Analyze f '(x) , find partition numbers and critical values and construct a sign
chart (to find increasing/decreasing segments and local extrema)
3. Analyze f ''(x) , find partition numbers and construct a sign chart (to find
concave up and down segments and to find inflexion points)
4. Sketch f (x) : locate intercepts, maxima and minima and inflexion points: if
still in doubt, sub points into f '(x)
Chapter 12.6 – optimization
1. Introduce variables, look for relationships among variables, and construct a
mathematical model of the form Maximize/minimize f (x) on the interval I.
2. Find critical values of f (x) .
3. Find absolute maxima/minima: this will occur at a critical value or at an
endpoint of an interval
a. Check that the function is continuous over an interval
b. Evaluate f (x) at the endpoints of the interval
c. Find the critical values of f '(x)
d. The absolute maximum is the largest value found in step “b” or “c”.
Chapter 4.1 – Systems of linear equations in two variables
Simultaneous equations of two lines: isolate a variable and substitute.
Integrals
Chapter 13.1 – antiderivatives and
indefinite integrals
Antiderivative is symbolized by F(x) ,
and may be accompanied by any
constant F(x) + C .
Indefinite integrals of basic functions
1. x to the power of n
òx
n
dx =
n+1
x
+C
n+1
Indefinite integral
ò f (x)dx = F(x) + c is a family of
antiderivatives
3. x as a denominator
ò x dx = ln x + C, x ¹ 0
1
2. e to the power of x
ò e dx = e + C
x
x
Indefinite integrals of a constant multiplied by a function, or, two functions
1. ò k· f (x)dx = k· ò f (x)dx
[ f (x)± g(x)]dx = ò f (x)dx+ ò g(x)dx
2. ò
Chapter 13.2 – integration by substitution
Based on the chain rule: d f [g(x)] = f '[g(x)]· g'(x) (derivative of outside function
dx
multiplied by the derivative of the inside function): thus
ò f '[g(x)]· g'(x)dx = f [g(x)]+ C
General indefinite integral formulae
1.
2.
4.
[ f (x)]n+1
+ C, n ¹ 0
n+1
ò ef (x) · f '(x)dx = ef ( x) + C
ò [ f (x)]n · f '(x)dx =
3.
ò
1
· f '(x)dx = ln f (x) + C
f (x)
Integration by substitution
Sometimes it is hard to recognize the form of the function to be integrated (that
is: to see which of the above formulae apply to it). So, we substitute the messy
part for “u” and integrate with respect to “u”, rather than x.
Where y = f (x) ® dy = f '(x)dx
General indefinite integral formulae for substitution
1.
2.
4.
un+1
+ C, n ¹ -1
n+1
ò eu du = eu + C
ò un du =
ò du = ln u + C
3. u
1
Method of integration by substitution
1. Select a substitution to simplify the integrand: one such that u and du (the
derivative of u) are present
2. Express the integrand in terms of u and du, completely eliminating x and dx
3. Evaluate the new integral
4. Re-substitute from u to x.
Note, if this is incomplete (i.e. du is not present) you may multiply by the constant
factor and divide, outside of the integral, by its fraction.
Chapter 13.4 – the definite integral
b
ò a f (x)dxis the definite integral of f (x) from x=a to x=b. Worked out by subtracting
where x=a from where x=b. Note, these are not absolute values: above x axis is
positive, below is negative: opposite if b<a.
Error Bounds
For right and left rectangles, f(x) is above the x-axis: f (b) - f (a) · b- a
n
Properties of a definite integral
a
1. ò a f (x)dx = 0
2.
3.
ò
ò
b
a
b
a
f (x)dx = - ò f (x)dx
4.
a
b
k· f (x)dx = k· ò f (x)dx , where k
is a constant
b
a
5.
ò
ò
b
a
b
a
[ f (x)± g(x)] =
f (x)dx =
ò
c
a
ò
b
a
f (x)dx± ò g(x)dx
b
a
f (x)dx + ò f (x)dx
b
c
The fundamental theorem of calculus
ò
b
a
f (x)dx = F(b) - F(a)
 You do not need to know C
Average value of a continuous function over a period
b
1
f (x)dx
ò
b- a a
More than 2 dimensions
Chapter 15.1 – functions of several variables
Substitute (x,y,z,…,et.) into the equation given.
Find the shape of the graph by looking at cross sections (e.g. y=0, y=1, x=0, x=1).
Chapter 15.2 – partial derivatives
Derivatives with respect to a certain symbol: watch for signs, all other symbols
count as constants
fx (x, y) derived with respect to x ||| fxy (x, y)  derived first with respect to x, then
y
Chapter 15.3 – maxima and minima
1. Express the function as z= f (x, y)
2. Find fx (x, y) & fy (x, y) , and simultaneously equate them to find critical values
3. Find f xx (a, b), fxy (a, b)& fyy (a, b) (A, B, and C, respectively)
4. Find A, and AC - B2 .
a. IF AC-B*B>0 & A<0, f(a,b,) is local maximum
b. IF AC-B*B>0 & A>0, f(a,b) is local minimum
c. IF AC-B*B<0, f(a,b) is a saddle point
d. IF AC-B*B=0, test fails
Chapter 15.4 – maxima and minima using Lagrange multipliers
1. Write problem in form
a. max/ min ® z= f (x, y)
b. g(x, y) = 0
2. Form the function F(x, y, l ) = f (x, y)+ lg(x, y)
3. Derive with respect to x, y and lambda
4. Simultaneously equate answers
5. If more than 1 answer, find z values and deduce which is max/min
If v = f (x, y, z), derive with respect to that too, and simultaneously equate with
more.