R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
_
Graphs of Rgeiproeal functions
Recall: L.
V 1
The reciprocal function of f (x) is \
When graphing reciprocal functions, you can use the key characteristics of a linear or quadratic
function to graph the related reciprocal function.
Egg Charactgisties: '
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Function Com osition
If two functions have domains that overlap (i.e. same units), they can be ,
(3 Bl Gilli , or (A U \ d to create a new function on that shared domain.
V
For the polynomial function f(x) and g(x),
(r + gm = f(x) + gcx)
l (fgXx) = f(x)
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
a , 5m g; (01", Are: 29
Exploring Graphs of the Rggiprocal Trigonomgtric Function
call:
V
Each of the primary trigonometric graphs have a corresponding reciprocal function
The graphs of reciprocal functions are related to the graphs of the primary func
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
 JPCriodic Function; El A function whose graph will repeat in some regular way 3%"; 3156033 i
K
Introduction To Qgriodic Functions
CI If one cycle iupon another cycle, a perfect match wor 1
exist (period of the graph will stay the same) ~ , .
ranmple
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
ra
, Estimating Instantaneous Rates of Change
tall:
V
For a function y = f (x), the average rate of change in the interval x1 S x S x2 is
A: = f(x2)f(x1)
Ax x2 x1
The S eC/Qxr is a line that passes through two points on the graph of a relation
The e/ o
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Dividing Polynomials
\Vdecall: Long division can be used to determine the quotient of two numbers
Example 1 Use long division to nd the following quotients
i
107 +74 '4 128 ,A
' K yew! rel
$3 % L A; L
ein; 1
5 R :9 xemef cfw_3
(A it z; _;_ 2. . x
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
. Bellwork  Additional Related Acute Angle Examples
When asked to nd the exact value, use special triangles:
Express as a function of the related acute angle, and then nd the exact value.
(a) will?" A
240180 M? (d) siné35°
5 f 3
" V: v } ITS
0
Aygw
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
W
' Mex OQ relation is the change in the Q g variable (denoted
) divided by the corresponding change in the WA variable (denoted by
3:.)
i a function y = f (x), the average rate of change in the interval x1 3 x 3 x2 is
Average Rate of Change = 5
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Exploring Absolute Value
\* 3 \l ,
The function f(x) = lxl is the 5k: 30 M ii" ' V i J? 1; ofa function
r the absolute value of a function, f (x) will always be a l ' l 7 3 " value.
v
Example 1 Evaluate each of the following
a. 2 b. 3[ c. 572+
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
6x lorin Gra hso the rimar Tri onomztrie Functions
/_
cosx on the
sinx and y =
Complete the table of values below, and use the coordinates to graph y
Vraphs of Sine and Cosine:
axis provided
g
. ,. /
Y. x
O 7 . 7,
w _/ n a 1\ O
. \ ~
O O r _
n y2
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
\
Transformations oiTrigonoqutric functions
TranSfOrmations of the parent function y = sinx are f (x) = a Sin ( k ( x " d) ) + C Where
a<0
la>1
0<la<1
lal
k<0
k>1
0<lkl<1
Ikl Olel
30
\x
d<0
C>0
c<0
31:0
Remember If the independent variable has a coef
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Transformations of Cubic and manic Functions
The polynomials function y = a [k (x  d) ]" + c can be graphed by applying transformations to the
. \
graph of the parent function U3 1/ , where n E N
represents a vertical stretch / compression and possibl
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Solving Problems Involving Rates of Change
V
\ in? : 3 law: .
":r the graph of any function that has a NbXMg mm or i m 1 mm ,
t. tangent line drawn at the point will be a i \Qcag line
f ~ Thus, the instantaneous rate of change is at
both the maximum a
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Page 1 of 5
Skills and Concepts Review
Exponent Laws:
771.
)duct Rule: am X a = am Quotient Law: (1; = a" Power of a Power: lm)" = am"
m m
Power of a Produce: (ab)" = ambm Power of a Quotient: (g) = g;
. o _  . _ 1
Zero Exponents. a 1 Negative Exponent
R S Mclaughlin Collegiate and Vocational Institute
Advanced Functions
MHF 4U1

Fall 2014
Transformations of Functions
The following are all possible transformations that can be applied to a function:
v y=af[k(xd)]+c
Where: I \ (
a _ Q \r 1 I J J \C\ I \ _& I V I
, 4
(Na. v V 9/ 1 A. l
_  .4 ~ r 7, ~
k ,
_  a I \
c _ .
v  a, 